رياضي عمومي 2(Calculus II )

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Author: مهدي نجفي خواه

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9 O " $ O = (, , ) . . . . . . . . . . . . . . . . >&j L Ǳ [

y = , z = . . . . . . . . . & x . &# . −x x = , z = . . . . . . . . . & y . &# . −y x = , y = . . . . . . . . . & z . &# . −z z = . . . . . . . . . . . . . xOy T &# T−xy y = . . . . . . . . . . . . . xOz T &# T−xz x = . . . . . . . . . . . . . yOz T &# T−yz x ≥ , y ≥ , z ≥ . . . . . . . . Z 6 N# x ≤ , y ≥ , z ≥ . . . . . . . . 4$ Z 6 N# x ≤ , y ≤ , z ≥ . . . . . . . . 4 Z 6 N# x ≥ , y ≤ , z ≥ . . . . . . 4.& Q Z 6 N# x ≥ , y ≥ , z ≤ . . . . . . . Z3 Z 6 N# x ≤ , y ≥ , z ≤ . . . . . . . Z6O Z 6 N# x ≤ , y ≤ , z ≤ . . . . . . . Z Z 6 N#

\ &. $0 A, .8 −−−−−−−−−−−−→ x −x = ()−( ) #E " "+ ( , , ) (, , ) *(" x − x = () − ( ) 0 0 → , + *( \ . $0 −AB M x= & −→ CD \ &. $0 + #&+ &0 B . AB &0 "+ −→ −→ → O&0 OC !VO 0 q, . $ , + R# *Z & −AB 5F .$ M −−−−−−−−−−−−−−→ ( , − , ) (, , )

D

C

B

A

. $0 N#c= h E 1) 2 3 4 . → 0 U# *O&0 B 0 A E 5&V v U 0 −AB M . (M' 5 R# 0 *O&0 !C% ^- *$ $ ^- O&0 &#&0&) E , + E >.&[, }

C = (xB − xA , yB − yA , zB − zA )

)*"9 ! ? 5F 0 $ $ 5&6 C $&+ &0 . . Mn , +

R $&+ &0 . R .$ $ EF &. $0 + , + *Zg S C 9Z$ 5&6 −−−−→ R = (x, y, z) x, y, z ∈ R

0 &0 . $0 0 9$M RQ 5 $ EF . $0 Z" 0 $ EF 5F 0 M M x= ]< =S h .$ . C & &0 !T&' . $0 M $S . 1 . @ . + 5F & ( *O&0 "+ −→ OC −−−−−−−−−−−−−−→ ( , , − ) (, , ) \ . $0 A& '() * −−−−−→ *( (, , ) $ EF . $0 #&+ −(−−, −−, −−−−)−(−−,−−,−→) \ &. $0−−−−A, .8 −−−−−−−−→ U# M oL6 . $ EF . $0 N# (, , ) (, , ) *−(−−−−,−−,−→) . $0 → 6= $ EF . $0 #&+ −AA &F O&0 LB$ A S A .8 *$ 0 O = −(−,−−,−→) → → #&+ −BA &F O&0 C = −(a,−−b,−→c) #&+ −AB S A .8 −−−−−−−−−→ *( −C = (−a, −b. − c) \ &. $0 E N# 0 h $ EF . $0 0 M oL6 . #E O

−−−−−−−−−−−−−−→ ) ( , , − ) (, , )

)

−−−−−−−−−−−−−−→ ( , − , ) (, , )

−−−−−−−−−−−−−−→ ( , − , ) (, , )

)

−−−−−−−−−−−−→ ( , , ) (, , )

)

M 0&0 . @ . z y x $ , AH −−−−−−−−−−−−−−−−−−−−−→ −−−−−−−−−−−−−→

(x − y, y − z, x − z) ( , , ) ∼ (, , − ) (z, y, x)

\ &. $0 R0 N[V# & M $ 5&6 A8 1 T x \ . $0 N# &0 "+ *$. $ $ )

\ . $0 AGB 9H* !VO "+ &. $0 A% −→ CD AB M x= \ . $0 $ −→ M " DEF . $0 $ R# Zg S . T .$ *O&0 −→ −→ CD AB U# O&0 `?-_ E ABCD U-.& Q Z"# >. T R# .$ *O&0 & . Z @ Z ( ) Z → −→ ∼ CD *A% H* !VOC −AB . $0 $ M (O $ .& 5 . @ R# # j E !T&' &y.&3 @ M " "+ . T .$ .$ "+ Y0 . *O&0 V# &j L . 0 & F E 0 U# *( E. Z Y0 . N# \ &. $0 R0 −→ −→ −→ EF CD AB → −→ *−AB ∼ AB AGB −→ −→ −→ CD ∼ AB &F AB ∼ CD S A% *−→ → −→ −→ −→ −→ −→ ∼ EF &F CD ∼ EF AB ∼ CD S A: *−AB −→ CD

→ −AB M R# 0 =&M 4E_ pO $ M ( 5F O&0 "+ xB − xA

=

xD − xC

yB − yA

=

yD − yC

zB − zA

=

zD − zC

−(−−, −−−−,−−)−(−−,−−,−→) \ &. $0 A& −−−'() % −−−−−−−−−→ #E " "+ ( , , ) (, , ) () − ()

= () − ( ) ,

() − (− ) = () − () , () − ( )

= () − ()

d0 . E O D = . $ 0 $&+ a0 .$ l&Q *+ = . d0 $. Ǳ&O 5$ 0 . $0 5

E _&0 $&U0 0 . _&0 d'&[ 4&+ / $ X R# 0 *$ O 4& T& .&M VF0 $ $ Z+U 5

. $0 &q= R U0−n 1 &q= E R S RL *( E& R 1 U0−n n

M x= 52 2 '6 −−→ >. T R# .$ *α ∈ R . $ . 1 R .$ v = −− (x, y, z) #E >. T 0 X 0 . u .$ α %qT&' v u ` +

9ZM G#U

−−−−→ u = (a, b, c)

−−−−−−−−−−−−−→ u + v = (a + x, b + y, c + z) −−−−−−−→ αu = (αa, αb, αc)

n

0 %

$ v = −(−−−−,−−,−−,−→) u = −(−−, −−, −−−−,−→) M x= A u + v −v u [& ("0 Y *O&0 R .$ . $0 *u − v M 0&0 . @ .−−d−−−c−−b−−a→$ , A; −−−−−−−−−→ −−−−−−−−−→

&F a, b ∈ R u, v, w ∈ R S

( , − , , ) = a(− , , , ) + b( , − , , ) −−−−−−−−−→ +c( , , − , ) + d(, , , )

T #E Y0 . .$ M c b a $ , M $ 5&6 A (=&# 5 + M −−−−−−−−−→ −−−−−−−−−−−−−→ (, , , , ) = a( , , , , ) −−−−−−−−−→ −−−−−−−−−→ +b(, , , , ) + c(, , , , )

!"# $ $ O S M B&' .$ ( U0 &q= Zg S Q ( U Q 0 &q= N# U0 &z& ( U0 $ T *( u3 R# 0 &3 uL0 R# E m *. $0 u , · · · , u ∈ R M x= !VO 0 ( .&[, &. $0 R# E !"# G N# E . h

* " $, a, · · · , a ∈ R 5F .$ M au + · · · + a u @ H ;< . &. $0 R# Y >&[M + , +

*Z$ 5&6 span{u , · · · , u } $&+ &0 & & F y '() v = −(−−, −−−→) ∈ R ZM x= A& −−−−→ −−−→ u + v−=−−(→, ) >. T R# .$ * u = (, ) −−−−−→ u − v = (, ) u + v = ( , − ) *O&0 . $0 $ R# Y >&[M E #& + −−−−−→ v = (, , ) u = −(−−, −−, −−−→) M . T .$ A, .8 Z#. $ w = −(−,−−,−→) n

k

k

k

k

k

span u, v, w = =

9e+) 5$ 0 "0 A& u + (v + w) = (u + v) + w 9e+) #n< MO A, u+0=0+u=u 9U+) a $ ) A

u + (− ) u = 0 9U+) #1 $ ) A u+v=v+u 9e+) #n
M 0&0 . @ . z y x A& 0 *−(−−, −−, −−−→) = x−( −,−−,−→) + y(−−,−−,−→) + z(−−,−−,−→) v = −(−−−−,−−,−→) w = −(−,−−,−→) ∈ R M x= A; −−−−−→ −v u u+v #$&\ >. T R# .$ u = ( , , ) [& . u + v + w u + v − w u − v + w u *M pO .$ $&T y x $ , M $ 5&6 A u + v ∈ R

au + bv + cw a, b, c ∈ R (a + b + c)i + (a + b + c)j +(−a + b + c)k a, b, c ∈ R

−−−−−−−→ −−−−−→ −−−−−−−→ ( , , − ) = x(, , ) + y ( , − , )

*. $ ) M XY R# R S &0 ]3 R# E → → U# *ZM $& C = −AB $&+ E ( C #&+ N# −AB *C = B − A C . $0 ( C . $0 Ǳ& C Y\ 0 *( C Y\ 0 Ǳ [ 5$ + !j E !T&' . $0 E >.&[, S *Z# O+ !#&1 1= C C R0 $. XB&b .$ !B$ R+

!\ " w v u &. $0 R# 0&0 a = b = c = ]3 * Y −−−−−−−→ −−−−−−−−→ v = ( , , , ) u = (, , , ) &. $0 A, .8 −−−−−−−−−→ $ O x= S #E Y "0 R .$ w = ( , − , , ) "#&0 &F au + bv + cw = O

&F β = a + b + c α = a + b + c ZM x= S #.F (0 #E !VO 0 c β α X"' 0 . b a 5

.$ *b = −c + α − β a = c − α + β

⎧ ⎪ ⎪ ⎨

>.&[, span{u, v, w} , + β = b α = a x= &0 ]3 E (

⎪ ⎪ ⎩

⎧ a + b + c = ⎨ a + b = b − c = c = b ⇒ ⇒ a + b + c = ⎩ a + b = a + b + c =

c = b a = −b

−a + b + c = α −

=

R#0&0 c = a = − &F b = $ O x= S * " Y "0 . $0 R# U# *−u + v + w =

=

O $ $ &. $0 , + E N# 4 M 0 % Q " Y !\ "

= =

−−−−−→ −−−→ ) u = ( , − ), v = (, ),

)

−−−−−→ −−−−−→ u = (, −), v = (−, )

)

−−−−−−−→ −−−−−→ −−−−−→ u = ( , , − ), v = (, , ), w = (, , ),

)

−−−−−→ −−−−−−−−−→ u = ( , , ), v = (−, , ),

⎧ ⎫

−−−−−−−−−−→ ⎨−− ⎬ α, β, α − β | α, β ∈ R ⎩ ⎭ −−−−−−−−−−−→ (a, b, a − b) | a, b ∈ R −−−−−→ −−−−−−−→ a( , , ) + b(, , −) | a, b ∈ R −−−−−→ −−−−−−−→ span ( , , ), (, , −)

R# .$ v = −(−−−−,−→) u = −(−−, −−−→) M x= A .8 >. T span{u, v} =

−−−−−−−−→ w = (−, , ),

)

=

−−−−−−−→ −−−−−−−→ u = ( , , , ), v = (, , , ),

b

−−−−−−−−−→ −−−−−−−−−→ w = (, , , −), n = (, , −, )

=

α+β

{au + bv | a, b ∈ R} −−−−−−−−−−→ (a − b, b − a) | a, b ∈ R = R

&F β

=

−−→ 1 v = −−→ (c, d) u = (a, b) &. $0 M $ 5&6 A8 d = (B&' .$ *ad = bc M Y !\ " 1 & *( a/b = c/d U 0 pO R# b =

[M >. T 0 . R E LB$ . $0 M $ 5&6 AJ v = −(−−, −−−−,−→) u = −(−−−−,−−,−→) &. $0 E Y *(O 5 w = −(−−, −−, −−−→) &. $0 . T .$ & M M H . &q= [M >. T 0 . R E q, .$ *R = span u, · · · , u U# (O 5 0 & F E Y )I N# !V6 u , · · · , u ∈ R &. $0 Zg S . T . &q= &f .$ O&0 Y !\ " _z M $ R 0 &. $0 E Y !\ " , + M (B&' .$ *M B * O & V 0 . #&3 M B . V . $0 &q= M u , · · · , uk ∈ Rn

n

k

k

n

n

b − a

α

= a−b

S #E

*a = α + β

&F u, · · · , u ∈ R S (T& U# *( . $0 &q= N# span{u, · · · , u } R *$. $ . *;* q1 E −→ .$ . u , · · · , u ∈ R &. $0 O $ #0 b T XM N# M Zg S !"# )*E . T " T 5& c+ + M a, · · · , a $ , U# *O&0 *$ O T a u + · · · + a u M O&0 O $ $ ) S 0 * O S !"# >9E O&[ Y "0 #&. $0 S v = −(−,−−,−→) u = −(−−, −−−−,−→) A& '()−−$−−−−−→ ZM x= S #E Y !\ " R .$ w = ( , − , ) &F au + bv + cw = O n

k

n

β

k

n

k

n

k

k

k

−−−−−−−−−−−−−−−−−−−−−−−→ −−−−−→ (a + b + c, −a − c, a + b + c) = (, , )

.$

⎧ ⎧ ⎧ ⎨ a + b + c = ⎨ a + b = ⎨ − a = −a − c = ⇒ c = a c=a ⇒ ⎩ a + b + c = ⎩ a + b = ⎩ b = −a

9$ O ⎧ ⎧ a + a + a + a = a ⎪ ⎪ ⎪ ⎪ ⎨ ⎨ a + a + a = a ⇒ ⎪ a + a = ⎪ a ⎪ ⎪ ⎩ ⎩ a = a

= −a − a − a = −a − a = −a =

⇒ a = a = a = a =

x= E M B . &q= ?U0 −−−−−−→ (a, b, c, d) = a u + a u + a u + a u

M $$S ⎧ ⎧ a + a + a + a = a a ⎪ ⎪ ⎪ ⎪ ⎨ ⎨ a + a + a = b a ⇒ a + a = c a ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ a = d a

=a−b =b−c =c−d =d

&F B = {u, u , u, u} S R# 0&0

−−−−−−→ −−−−−−−−−−−−−−−→ (a, b, c, d) = (a − b, b − c, c − d, d) B

M $ 5&6 #E $. E N# .$ 0 #&3 0 . v . $0 ]< $ #&3 N# !V6 e &. $0 9M 5&0 & F

&z\1$ R &q= E #&3 &. $0 $ U * *Z & R ;< J . w 6 $, R# *( n 0 0 "0 R .$ &. $0 E $ U k &F n < k S R# 0&0 * " Y #&3 B = {u, · · · , u } ⊆ R S $0 j $ , &F O&0 LB$ . $0 v ∈ R $ 0 *v = a u + · · · + a u M $ O (=&# . @ a , · · · , a ZM $ $ . 1 Z & B #&3 0 v >&j L . $ , R# *[v] = −(a−−,−·−·−· −, a−→) *j = −(−,−→) i = −( −,−→) ZM x= A& '() *$ O & R 0 . & #&3 B = {i, j} >. T R# .$ −−→ (a, b) = ai + bj ?U0 * −−−−−→ −−−−−→ j = (, , ) i = ( , , ) ZM x= A, .8 0 . & #&3 B = {i, j, k} >. T R# .$ *k = −(−,−−,−→) *−(a,−−b,−→c) = ai + bj + ck ?U0 *$ O & R .8 v = −( −,−−,−→) u = −( −,−−,−→) &. $0 A

− − − _z #E *$ !V6 R 0 #&3 N# w = (, −,−→) M $ O au + bv + cw = O x= E 9 Y !\ "

n

n

n

n

n n

⇒ a=b=c=

x= E M B . &q= &f .$

−−−−−−−→ −−−−−−−→ ) e = (, , − ), e = (, − , ), −−−−−−−→ −−−−−→ e = (− , , ), v = ( , , ).

(α, β, γ) = au + bv + cw ⎧ ⎨ a+b=α a+c=β ⎩ b+c=γ

−−−−−−−→ −−−−−→ ) e = ( , , − ), e = (, , ), −−−−−→ −−−−−→ e = (, , ), v = (x, , y). −−−−−→ −−−−−→ e = ( , , ), e = (, , ), −−−−−→ −−−−→ e = (, , ), v = (x, y, z).

−−−−−−−→ −−−−−−−→ ) e = ( , , , ), e = (, , , ),

a=

−−−−−−−−−→ −−−−−−−−−−→ e = (, , , −), e = (, , − , −),

−−−−−−−→ −−−−−−−→ ) e = ( , , , ), e = (, , , ), −−−−−−−→ −−−−−−−→ −−−−−−→ e = (, , , ), e = (, , , ), v = (x, y, z, t).

R0 .$ Y !\ " &. $0 $ U aM ' M . T .$ span u , · · · , u &q= U0 . u *** u &. $0 k

k

n

⎧ ⎧ ⎨ b = −a ⎨ a+b= a+c= ⇒ c = −a ⎩ ⎩ b+c= − a =

−−−→ −−−→ −−−→ ) e = ( , ), e = (, ), v = (, ). −−−→ −−−→ −−−→ ) e = ( , ), e = ( , ), v = (x, y).

−−−−−−−→ v = (, , , ).

n

B

i

)

n

n

α+β−γ

−−−−−→ (α, β, γ) = B

, b=

⎧ ⎨ a+b=α b−c=α−β ⇒ ⎩ b+c=γ ⎧ ⎨ a=α−b b = α − β + γ ⇒ ⎩ c = −α + β + γ α−β+γ

.$

−α + β + γ , c= B = u, v, w

&F

Z#. $

S U#

−−−−−−−−−−−−−−−−−−−−−−−−−−→ (α + β − γ, α − β + γ, −α + β + γ)

&. $0 A .8

−−−−−−−→ −−−−−−−→ u = ( , , , ) u = ( , , , ) −−−−−−−→ −−−−−−−→ R u = ( , , , ) u = ( , , , )

0 #&3 x= E Y !\ " _z #E $ !V6 a u + a u + a u + a u = 0

)

A = A.

&#F.$ + M ( T ]#& O = [o ] &# .$ M E >.&[, A = [a ] ]#& U+) #1 ?U0 *T &#F.$ + 5$M E M ( (− ) A = [−a ] *( F ($ 0 A ]#&

Mat (n × m) Ǳ&q, n = S A& '() z R = Mat ( × ) ?+, * & "+ E . *R = Mat ( × ) E . Mat (n × m) Ǳ&q, n = m S A, .8 h & !J ij

ij

ij

[ ],

−

⎣

,

⎡

−

−

⎦

Z#. $ Mat ( × ) .$ A .8

−

=

− X 0 C B A M x= A; R# .$ * " − − [& . A − C A + B + C B − C A >. T *M &#F.$ + M O&0 n×m ]#& E M x= A *( N# 0 0 M 5F 4 (i, j) o L c0 T 5F 0 #&3 N# !V6 & E , + M $ 5&6 ( .\Q Mat (n × m) U0 *Mat (n × m) 0 #&3 N# O $ $ & "#& $ 5&6 A LB$ ]#& ]< $ !V6 Mat ( × ) 9M 5&0 & F X"' 0 . xz yt ij

ij

A = A =

− −

A = A =

&$0.&M . $ ( . $0 4 U[@ Z+U ]#&

5&0 0 *O&0 (UT c -&#. & 6L0 #& .$ = *( $ , E X # .F ]#& $& XM # $S n × m ]#& E . h !Y " !VO 0 $, nm E ⎡

a ⎢ a ⎢ A=⎢ ⎣ an

***

a a

***

an

⎤ am am ⎥ ⎥ ⎥ ⎦ · · · anm ··· ···

***

0 >. T R# .$ * O X 5 m Y n .$ M ( A 4 (i, j) #F.$ . a A = [a ] Z"# T? !VO ]#& E 4 j 5 4 i Y .$ e1 #F.$ U#C Z &

0 & F .$ M (O $ Z #& "#& $ .$ [B *AA q1 >. T 0 &a 5 U0C . $0 &# e0& +, & a E n × m & "#& + , + *AM U) H*7* ]M *Z$ 5&6 Mat (n × m) $&+ &0 . A = [a ], B = [b ] ∈ E 0 S A + B := [a + b ] ZM G#U a ∈ R c Mat (n × m) e+) !+, + 0 Mat (n × m) >. T R# .$ *aA := [aa ] U R# 0 *( . $0 &q= N# _&0 .$ K6 %- 9Z#. $ a, b ∈ R A, B, C ∈ Mat (n × m) E 0 M ij

ij

ij

ij

ij

ij

ij

ij

) A + B ∈ Mat (n × m) ,

) A + (B + C) = (A + B) + C, ) A + O = O + A = A, ) A + B = B + A,

−

%& ' (

0

& "#& &#F A Y !\ " − −

−−−−−→ −−−→ −−−−−→ u = ( , − ), u = (, ), u = ( , − ). −−−−−−−→ −−−−−→ −−−−−−→ ) u = ( , , − ), u = (, , ), u = ( , , ). −−−−−−−−−→ −−−−−−−→ ) u = ( , , , − ), u = (, , , ), −−−−−−−−−−→ −−−−−−−→ u = (− , , , − ), u = (, , , ), −−−−−−−−−−→ u = ( , − , , − ).

)

⎤

− Z#. $ Mat ( × ) .$ A .8 − − − − = − − +

&. $0 E $ y O B &q= U0 ZM G#U 9M oL6 . #E

−

) A + (−

) A = O,

) a (bA) = (ab) A,

)

aA ∈ Mat (n × m) ,

) a (A + B) = aA + aB,

]#& A& '() ( #n< VU ]#& A ZM x= S #E ( #n< VU

α β U 0 AA = I x= &F A = γ η + γ β + η .$ *( α −α = −β

A =

−

−

−

⎧ α + γ = ⎪ ⎪ ⎨ −α = β + η = ⎪ ⎪ ⎩ −β =

⇒

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

γ = α= η −

]#& M R# >&[f 0 *A = / −/ ]3 Z$ 4& . #E >&[& #&0 ( A / VU . Mn

−

−

AA

=

−

/

=

/

− /

− /

−

=

= A− A

#E (" #n< VU A = − − ]#& A, .8 &F AB = I B = αγ βη $ O x= S

α + γ β + η −α − γ

−β − η

=

2

=

ai bj + ai bj + · · · + aim bmj ⎡ bj b ⎢ ⎢ j ai ai · · · aim ⎢ ⎣ bmj

***

−

−

−

−

⎡ ⎣

⎤ ⎦

⎡

⎡

−

−

B=

& 1 5F / VU ?U0

]#& M $ 5&6 A; M ( #n< VU

1 *( ad − bc −cd −ba

Y 4 i Y # U E !T&' ]#& P M x= A 9$ 5&6 >. T R# .$ *O&0 I &+ ]#& 4 j *P P = P A%C *P = I AGB C P A:C *$ O !T&' I ]#& 4 j 4 i & # U E E (".&[, P A &F O&0 × ]#& A S A$C *A ]#& 4 j 4 i &Y # U E !T&' ]#&

E (".&[, AP &F O&0 × ]#& A S AC *A ]#& 4 j 4 i & # U E !T&' ]#&

*( #n< VU P ]#& AC ij

jk

ik

⎥ ⎥ ⎥ ⎦

⎦ =

=

= ⎣

−+ +− − − − − −

=

+ +

= ⎤⎡

⎦⎣ ⎤ ⎦

⎤ ⎦

*CAB = BCA = I &F ABC = I S M $ 5&6 A

ij

⎣

M 0&0 B&' .$ . A BA %qT&' A

⎤

⎡

=

= ⎣

A=

⎤

E 4 j 5 .$ A ]#& E 4 i Y %qT&' E M ( *( O !T&' B ]#&

9Z#. $ & "#& %- G#U 0 ) &0 '() %

0

a b A = c d ad = bc

ij

=

ij

*( o1& M −α − γ = α + γ = "#&0 ]3

cij

⎧ α= ⎪ ⎪ ⎨ β=− ⇒ γ = / ⎪ ⎪ ⎩ η = /

=

β=−

]#& A = [a ] S $ n×l ]#& . AB K;'= &F O&0 m×l ]#&

0 0 5F 4 (i, j) #F.$ M ZM G#U B = [bij ] n×m

5F Y1 M O&0 n × n ]#& I S & S c An × n &+ ]#& C T 5F Ǳ&q, #& N# ?U0 *I A = A AI = A &F O&0 n × m ]#& A >. T R# .$ O&0 $, a $ 0 ]#& C B A S n

n

m

) A (B + C) = AB + AC

)

a (AB) = (aA) B = A (aB)

) )

A (BC) = (AB) C A=A

ij

ij

ij

ij

. T .$ . A U0 ]#&

* $$S (=&# 5&Q #$ B U0 ]#& M # S #n< VU

&0 & A / VU . B >. T R# .$ *AB = BA = I M *Z$ 5&6 A $&+ n

−

?z[1 (n − ) × (n − ) & "#& 5& $ M x= A *O&0 n × n ]#& N# A = [a ] O&0 O G#U Y mn' E !T&' ]#& 5& $ 0 0 A S ZM G#U &F O&0 A ]#& 4 j 5 ij

j

|A| = a A − a A + · · · + (− )n+ an An

$& Y E *H* G#U .$ 78 $ &# #$ Y N+M 0 5 . G#U R# *( O $ ) Y [) E q1 *$ + g . ]#& &5 E N# Z#. $ e1 .$ *$ 5&6 . G#U R# E. Z M $. $ O&0 n × n ]#& N# A = [a ] S $ 4 j 5 4 i Y mn' E !T&' ]#& 5& $ A &F O&0 A ]#&

ij

A = − $. .$ BA AB [& &0 AH $ R# M (B&' .$ M $ 5&6 B = *(" #n

A8 Q = P = − M x= >. T R# .$ *E = R = − M $ 5&6 a) E = E

b) P = −E

c) Q = E

d) R = E

e) EP = P E = P

f ) EQ = QE = Q

g) P Q + QP = O

h) RP + P R = O

i) QR + RQ = Oj) P + Q + R + E =

ij

i+

|A| =

(− )

i+

ai Ai + (− )

ai Ai + · · ·

i+n

· · · + (− ) j+

=

(− )

i+

aj Aj + (− )

j+n

· · · + (− )

ain Ain

#& "#& X 0 D C B A S M $ 5&6 AJ &F O&0 q k × n p × m × k

× A O C O AC O * O B O D = O BD ⎛

aj Aj + · · ·

A =

anj Anj

&F A = ⎝

*n ≥ M A

n

Z#. $ 5& $ G#U 0 ) &0 −

−

−

=

'() $

( ) ( ) − () (− ) =

= () −

+ (− )

= () ( + ) − () ( + (− −

−

= ()

−

− ) ) (− − ) =

− ()

+ (− )

=

−

⎞ ⎠

=O

S M $ 5&6 A7 ⎛

⎝

⎞ ⎠

) *'& +

+=

− ()

− () ( − ) − () ( + ) + (− ) ( − ) = −

G#U *$$S G#U U0 &]#& 0 & 5& $ G#U 0 M X R# 0 *( 6SE&0 !VO 0 5& $ z [1 #&0 n × n ]#& 5& $ (n − ) × ]#& 5& $ ? *O&0 O G#U (n − ) |A| $&+ &0 . A U0 ]#& 5& $ $ 9ZM G#U #E >. T 0 $ $ 5&6 a = a 9 × U0 ]#& 5& $ A a c

b = ad − bc d

a d g

b c e e f = a h h i

9 × U0 ]#& 5& $ A; 9 × U0 ]#& 5& $ A f d f d e −b +c i g i g h

Z#. $ 8*H* q1 N+M 0 A&

−

() − = − + (− ) (− )

=

0 F $. $ $ ) T q, $ #E ]#& E 4 Y .$ 5 Q Z$ y"0 4 Y X"'

'() & $

−

= −

−

() − = = ( )(

−

) () ( ) =

5 0 0 k3 Z# $c= 4 5 0 . 5 AC .$ M 4$ 5 0 0 ( A;C .$ Z# $M ZM 4.& Q 5 E . 4 Y E . 4.& Q Y AC .$ Z# $M ZM 4.& Q 5 E . *Z# $M ZM a a ()

=

()

=

=

Z#. $ 8*H* q1 N+M 0 A .8

() b c = a b−a c−a a b − a c − a b c (b − a) (c − a) a a b + a c + a (b − a) (c − a) a a b + a c − b

(b − a) (c − a) (c − b)

E A;C .$ Z# $M ZM #$ 5 $ E . 5 AC .$ M Z# =S . M&= . c− a $, 4 5 E b − a $, 4$ 5 *Z# $M ZM 4 5 E . 4$ 5 AC .$ 0 . #E & "#& E N# 5& $ 0 * $ M [& 5& $ i N+M ⎡

) ⎣

−

−

⎤ ⎦

⎡

)

⎣ −

−

⎤ ⎦

= (− )+ () A +

+

(− ) A + (− ) = + =

. #E & "#& E N# 5& $

Z#. $ 8*H* q1 N+M 0 A, .8 () − = − − (=) − − − − −

−

+ (− ) = A = −

Z# $c= Y 0 0 4 Y 0 0 0 AC .$ M *Z# $c= 4$ Y 0 . 4 Y 0 0 $ ]< −

) ⎡

)

⎣ ⎡

)

⎢ ⎢ ⎣

−

⎤

⎦ ⎤ ⎥⎥ ⎦

)

⎡

)

⎣ ⎡

)

⎢ ⎢ ⎣

() A

0 $ $

−

M [&

⎤ ⎦ ⎤ ⎥ ⎥ ⎦

G#U . E 5& $ Z\ " [& $. XB&b .$ 5$M $& 0 _z +U !B$ R+ 0 *( 1 ]0 .&M *M $& 5& $ i E & F [&

% $

( ?, ZM x , . Y $ &) & $ .$ S A *O x , 5& $ ZM - $, .$ . ]#& N# E Y S A; *$ O %- $, R# .$ ]#& 5F 5& $ ]#& 5F E #$ Y 0 . Y N# E 0q S A *M+ v ]#& 5F 5& $ Zg c=&0 *( ^T c & 0 _&0 i A $ O T T Y1 Rg&3 &# T Y1 _&0 S AH 0 0 T Y1 T&, %qT&' &0 ]#& 5& $ *( #$ Y N# E 0q 0 0 ]#& N# E Y S A8 *( T ]#& 5& $ &F O&0 5F #$ 5 N# E 0q 0 0 ]#& N# E S AJ *( T ]#& 5& $ &F O&0 5F O&0 T "#& E 5 N# &# Y N# S A7 *( T ]#& 5F 5& $

)

⎤

⎡

A−

⎡

.$

(− ) −(− ) () ⎣ −( ) ( ) −(−) ⎦ ( ) −() (−) ⎡ ⎤ / / − / ⎣ − / / / ⎦ / −/ − /

=

=

/ VU 5 Q *(I + A)

M (0&f A, .8 ( $= 0 j $ ) >. T .$

−

= A(A + I)−

(I + A− )(A(A + I)− )

= (I + A− )A(A + I)− = (A + A− A)(A + I)− = (A + I)(A + I)− = I

*( O >&[f ZV' .$ . #E & "#& E N# / VU 0 $ 90&0 ⎤

) ⎡

)

×

−

−

−

⎣ −

⎡

)

⎣ ⎡

⎤

−

−

β β β

γ γ γ

⎢ α ⎢ ⎣ α α A = [aij ] ⎦

−

)

⎤ θ ⎥ ⎥ θ ⎦ θ

−

a a

a a

a a a a

a a

a a

a a a a

a a

*(I + AB)

−

*A

−

a a

a a a a

⎤ a a ⎥ ⎥ ⎥ ⎥ a ⎥ ⎥ a ⎥ ⎥ ⎥ ⎥ a ⎦ a

M (0&f AJ M (0&f A7

A = A(I + BA)−

+ B − = A− (A + B)B −

x= 9 8 :; ) < 2 =3) $ $M E = CA DB >. T 0 5 0 . S LB$ ]#& M A S >. T R# .$ * " U0 &]#& D A M Z#. $ O&0 #n< VU

A C

A C

B D

B D

= |A||D − CA− B|

Z#. $ O&0 #n< VU D S = |D||A − BD− C|

⎢ ⎢ ⎣

−

⎤

⎡

⎥ ⎥ ⎦

)

⎢ ⎢ ⎣

−

−

−

⎤ ⎥ ⎥ ⎦

M (0&f 8*H* N+M 0 & AH

a b c d −b a d −c + b + c + d = a −c −d a b −d c −b a

a+b a a

a a+b a

a a a+b

M (0&f H*H* N+M 0 & A8

= (a + b)b

$ 1 & 1 A U0 ]#&

0 0 A ?U0 *O&0 T GB&L |A| M ( #n< VU

*(AB) = B A ( /|A| ]#& N# A = [a ] M x= $ ]#& 5& $ A ( T GB&L Δ 5& $ &0 U0

R# .$ *( A ]#& 4 j 5 4 i Y mn' E !T&' &0 ( 0 0 A >. T −

−

−

−

ij

ij

⎦

]#& N# S M $ 5&6 AH &0 ( 0 0 A &F O&0 #n< VU

⎡ a ⎢ a ⎢ ⎢ ⎢ ⎢ a ⎢ |A| ⎢ ⎢ a ⎢ ⎢ ⎣ a a

−

⎡

½+½

(− ) A½½ ⎢ (− )½+¾ A½¾ ⎢

*** ½

⎢ Δ⎣

(− )

+n

⎡

A = ⎣

−

···

*** ¾

···

(− ) An½ n+¾ (− ) An¾ ⎥ ⎥

···

(− )

(− )

A½n

⎤ ⎦

⎤

¾+½

(− ) A¾½ ¾+¾ (− ) A¾¾ +n

A ¾n

M x=

n+½

***

n+n

⎥ ⎦

Ann

> '() $

>. T R# .$

Δ = |A| = − − − − − = = − + = A = A =

−

A = − A = A =

?U0 ( #n< VU A ]#& nB

= − , = ,

= ,

= , = −.

A = − A =

A = − A =

=−

= ,

= , = −,

,

= = = ⎡

−

− −

−

−

⎡

⎤ −

⎢ =⎢ ⎣

⎥ ⎥ ⎦

⎡ ⎢ =⎢ ⎣ − ⎡ ⎢ =⎢ ⎣

!− − − − − − / / = / − /

−

−

# ×

−

⎡

+

⎢ − ⎢ ⎣

⎢ +⎢ ⎣

−

−

⎤

⎤

⎥ ⎥ ⎦×

−

⎡

⎤−

⎥ − ⎥ ⎦

−

− "

−

− ⎤ ⎡ ⎤ ⎥ ⎥⎥ + ⎢⎢ ⎥× ⎦ ⎣ − − − ⎦ − − × / − / − ⎡ ⎤ ⎢ ⎥ ⎥ =⎢ ⎣ − ⎦ ⎤ ⎡ − − ⎢ / − / − / ⎥ ⎥ +⎢ ⎣ / −/ − / ⎦ −/ / / − ⎡ ⎤ − − ⎢ / − / − / ⎥ ⎥ =⎢ ⎣ / −/ / ⎦ −/ / / −

− −

−

− − − (− ) − [ ]

− / +/ − = − − − [ ] +/ − /

/ = − − − [ ] −/

= − − + = =

−

×

⎥ ⎢ ⎥ ⎥+⎢ ⎥ ⎦ ⎣ − − ⎦ − −

=

" −

⎤

'() $

− = × − − × − − / −/ = − / / A − / / = − / / / / = − / / −/ −/ = −/ −/ = − = −

A C

B D

A C

B D

A &]#& S −

=

+

A−

−

−A B I

$ $

&F O&0 #n< VU

(D − (A− B)− )[−CA− I]

Z#. $ O&0 #n< VU D S 0&6 >. T

A C

B D +

− =

I −D− C

D − (A − BC − C)− [I − BD− ]

, -. ' / 0 &$0.&M .$ . + u\ Y >_$&U & & $ T 0 . & F t1$ UB&Y nB *M Ǳ&# >&-&#. *ZM

'() % $

−

( )−

−( )− ()

= + − − $ − ( )( )− × (− ) − ( )( )− ()

%

×

M $$S B$&U E *#0 h .$ . 4$ B$&U Z$ . 1 x &0 U0 B$&U $ .$ S *y − z 0 c 4 B$&U $ O !#[ y − z = − !VO 0 #E & $ &0 O $ $ & $ ]3 *$$S !#[ B$&U 5&+ 9( $&U

x =

+

x=

+ y − z,

y − z = −

. 5F S M y = (z − ) / $ O 4$ B$&U E 9(O $ Z Z$ . 1 B$&U .$ x=

+

z − − − z =

z+

E ( >.&[, & $ % ) , + .$

S=

−

? @A) 9BC < D ) E38 %

Y >_$&U & $ 5 & "#& %- 4 N+M 0 "#& !VO 0 . A;* C AX = B A* C 5F .$ M (O ⎢ A=⎣

a

···

an

· · · anm

***

m

am

***

⎤

⎡

⎥ ⎢ ⎦, X = ⎣

x

***

xm

⎤

⎡

⎥ ⎢ ⎦, B = ⎣

b

***

m m

m

m

m

bn

L+ . B MN C L+ . X GO P . A E !T&' ]#& *Z & /+ ! Q RS &# M'J *C /+ !<O P . A ]#& 0 B 5 &B *Z$ 5&6 AB $&+ &0 & A RQ D. R# "= F+G HIJ K+ % &v #& X"' 0 B$&U E . v N# 0 9( 5 M *Z$ . 1 U0 >_$&U .$ ]< $.F (0 +M B$&U N# c +M v N# . $ &+1&0 >_$&U

S *$ O $& & $ =. =. .&M R# $ &0 *O&0

% ) $M !' . F & $ && 1& 0 . R# .$ . F (0 % ) ]< Z$ . 1 [1 B$&U .$ . *F B Z$ . 1 5F E ![1 B$&U

"#& 4= E D. R# S.&V0 5$M $& 0 9ZM $& #E Z #. B & $ 9E .&[, ! 9 "+ .FT

m

⎧ a x + a x + · · · + am xm = b ⎪ ⎪ ⎨ a x + a x + · · · + am xm = b .................................. ⎪ ⎪ ⎩ an x + an x + · · · + anm xm = bn

A;* C

K% N# E . h *Z & !H C m )*HJ n /+ . X #&−m N# A;* C >_$&U & $ 0 !=U# B$&U n E N# % ) 5& c+ . Y0 M ( (α , · · · , α ) T j & 0 ) + , + *O&0 & $ .$ $ )

*Z & 5F K% . A;* C >_$&U & $ A& '() % m

x − y = −,

⎤ ⎥ ⎦

%

>.&[, a x + · · · + a x = b A* C m !"# )*HJ

U (0&f $ , b a *** a M . B$&U R# T j % ) N# E . h *Z & x= 1 M ( $ , E (α, · · · , α ) X #&−m N# *C &" &F x = α · · · x = α ZM

. B$&U T j & 0 ) , + *$ 0 . 10 A Y B$&U Q E XM # $S *Z & )HJ L K% Z & Y >_$&U & $ N# . w 6 >_ &0 & $

z + z − , , z z ∈ R

T j % ) (#& 0 >_$&U & $ R# M $ O ) *$. $

⎡

*v x * * * x x M x=

x + y =

*x = y − $$S B$&U E *#0 h .$ . &" 0 Z0 . 1 . >.&[, R# x &0 4$ B$&U .$ S ]3 *x = y = R# 0&0 *Z. y − = (".&[, 5F % ) ( & $ T j % ) & (, ) *S = (, ) E >_$&U & $ A, .8 x − y + z = , x − y + z =

x + y + z = ,

M $ O B$&U E *#0 h .$ . $ . 1 . >.&[, R# z &0 U0 B$&U $ .$ S *x B$&U $ O !#[ −x + y = − !VO 0 4$ B$&U

o1& N# R# *$ O !#[ −x + y = !VO 0 4 *S = U# *( T j % ) 1&= & $ ]3 ( >_$&U & $ A &a

z=

+ y −

x − y + z = ,

x − y + z =

x + y + z = ,

>_$&U & $ A, .8 x − y + z = ,

*Z &0 Y $ # U AGB *T GB&L $, .$ Y N# 5$M %- A% *#$ Y 0 Y N# E 0q 5$M =&- A:

x + y − z = ,

x + y + z =

E (".&[, 5F 0 h =&- ]#& *#0 h .$ . ⎡ ⎣

⎡ ⎣

−

−

−

⎦

*ZM !+, #E D. 0 &'

⎤ ⎦

⎤

− ⎡

()

−→ ⎣ ⎡ ()

−→ ⎣

−

−

−

− −

⎤ ⎦ ⎤ ⎦

ZM 4$ Y E . Y 0 0 $ AC .$ M R# ^- .$ *Z# $ + ZM 4 Y E . Y 0 0 .& Q Z# $ + 0 h & $ *Z# $ + ZM 4 Y E . 4$ Y A;C E (".&[, F ]#&

x − y + z = ,

nB A* C & $ 0 h =&- ]#& AB ZM x= E ' & $ U 0 E& 5 M *O&0 A;* C & $ aa _&0 !VO 0 AB & $ & ZM $& _&0 !+, * O T + 5F T Y1 #E T&, U# *$$S !#[ . & $ !T&' =&- ]#& . E z $ F ' .$ *ZM !' . !T&' $& & $ $ + E&E&0 >_$&U & $ A& '() $ % x − y + z = , −x + y + z = .

5 >_ 5 X# - ]#& *#0 h .$ . E .&[, X 0 5F =&- ]#& >&

U

⎡

y − z =

y − z +

y

=

S=

=

(−

z+

)

E ( >.&[, & $

+, % ) ]3

( − z) , ( z + ) , z z ∈ R

>_$&U & $ A .8

−x + y + z = ,

−

−

⎣ −

⎡ ⎣

−

−

⎤

⎡

⎡

⎤ ⎦

ZM !+, #E D. 0 &' *

⎦

ZM !+, #E D. 0 &' ⎤

− ⎦ − −

−

⎣

− ⎡

()

→⎣ ⎡ →⎣

⎤

− ) ⎣ − ⎦ (→ − ⎤ ⎡ ⎡ − () ) ⎣ ⎦ (→ →⎣

⎡

()

E ( & $ 5F 0 h =&- ]#& *#0 h .$ . ⎡

−

( z + )

x + y − z = , x + y − z =

⎤

⎤ x ⎦, X = ⎣ y ⎦, A=⎣ − z ⎡ ⎤ ⎡ − − B = ⎣ ⎦ , AB = ⎣ −

.$ x =

x + y − z = ,

⎦

⎤

⎡

⎤ ⎦

Z# $M ZM 4$ Y E . Y 0 0 $ AC .$ M R# ^- &) A;C .$ *Z# $M =&- 4 Y 0 . Y 0 0 .$ . 4$ Y AC .$ *Z# $M x , . 4 Y 4$ Y 4 Y E . 4$ Y 0 0 k3 AC .$ *Z# $ + %- − *Z# $ + %- / .$ . 4 Y AHC .$ *Z# $ + ZM E (".&[, F ]#& 0 h & $ x − y + z = ,

⎤

⎤

() − − ⎦ → ⎣ − ⎦ −

⎤ ⎤ ⎡ − − ) ⎣ − ⎦ (→ − − ⎦ − − ⎤ ⎡ − () − − − ⎦ → ⎣ − −

y−

z = − ,

z=

*x = y − z + = y = z − = z = R# 0&0 *S = (, , ) E ( >.&[, & $ % ) U#

#E ( 0 0 $ [. = =

|A|

.$ *Z# $ + x , . 4$ Y AC .$ M R# ^- 0 0 Z# $M =&- 4$ Y 0 . Y 0 0 $ A;C 4 Y E . 4$ Y 0 0 $ AC .$ *4 Y 0 . Y *Z# $ + ZM E (".&[, F ]#& 0 h & $

− − − = − = −

−x + y + z = ,

m − rank(A) = − = 5 Q *$. $ % ) & $ ]3 *$. $ T j % ) N# & & $ R# 0&0 >_$&U & $ A, .8 x − y + z = , x + y − z =

A=⎣

−

− −

x + y − z = ,

⎡ ⎦ , AB = ⎣ ⎤

−

− −

#E (" A [. |A| =

= =

− − − − − − − − = − − −

5F 5 $ Y $ E !T&' ]#& 5& $ & &0 ( 0 0

− =+

= =

( 0 0 AB [. M B&' .$ *( $ 0 0 A [. ]3 5 AB ]#& Y E !T&' ]#& 5& $ #E ( T GB&L 5F F − − −

= = =

− − − − − − − − − − − =−

=

*( % ) 1&= h $. & $ 7*8*q1 t0&Y ]3

=

*S = $. % ) & $ ]3 *( 1& F B$&U M D. 0 . #E & & $ E N# 0 % % M !' /&S

⎤ ⎦.

x − y = x + y =

x − y + z = x − y + z = ⎧ ⎧ ⎨ x + y + z = ⎨ x − y + z = ) ⎩ x − y − z = ) ⎩ x + y − z = x + y + z = − x − y + z = ⎧ ⎨ y −z + t = − x + y − z + t =

x+ y − z + t = ) ) x − y − t = ⎩ x−y +z −t = )

&0 0 0 5F =&- ]#& X# - ]#& *#0 h .$ . ⎡

y + z = ,

)

5F % ) $ ) 4, &# $ ) .$ 5 & $ !' 50 &#F [. 4 R " $ S .$ u3 R# &3 $M d0 & $ *( ( m × n ]#& N# A M x= &&% ' A E k × k P B N# E . h * ≤ k ≤ min m, n A ]#& E Y k 5 k $. 0 ! E M ( "#&

*$ O !T&' O&0 T GB&L A E k × k ]#& #E N# 5& $ S T A E (k + ) × (k + ) & "#& #E M 5& $ $ O O ( k 0 0 A )*$ M $ O S &F O&0 *rank (A) = k & $ M R# 0 =&M 4E_ pO * % 5F X# - ]#& [. M ( 5F O&0 O $ % ) A;* C &v $ U *O&0 0 0 AB D =&- ]#& [. &0 A *( m − rank (A) 0 0 % ) .$ $ ) !\ "

>_$&U & $ A& '() % x − y + z = ,

x + y + z = ,

x + y − z =

]#& X# - ]#& >. T R# .$ *#0 h .$ . &0 0 0 X 0 >_$&U & $ R# =&- ⎡

A=⎣

−

−

⎡ ⎦ , AB = ⎣ ⎤

−

−

⎤

⎦.

y − z + t = −,

*S

=

>_$&U & $ A .8

x + y − z + t =

(, −, − , )

>. T R# .$ *#0 h .$ . M D. /& 0 M Q

− − − − = = −

− − − |A | = = , − − − − − = − , |A | = − − −

− − = − , |A | = − − − = , |A | = − − − |A| =

)

|A | = |A|

x + y − z =

(O 5 ]3 ( 4 U z ZM x= *#0 h .$ . x + y = + z

.$

|A| =

(z +

− − z |A | = + z x = x = |A | =

=

|A | = |A|

) , (z − ) , z

− = z + ,

z +

− z + z

z ∈ R

#E =

= z − ,

)

x + y = −x − y =

x − y + z = x + y − z = ⎧ ⎨ x − y + z + t = y − z + t = ⎩ x + z + t = −

A;* C & $ .$ S )L K+ % 5& $ (m = n) O&0 0 0 >_ >_$&U $ U &F O&0 T GB&L c 5F X# - ]#&

# U E !T&' ]#& E ( >.&[, A M *x = |A |/|A| *B >&

U 5 &0 A ]#& 4 i 5 >_$&U & $ A& '() % i

>_$&U & $ AO&[ U0 & $ S C A .8

x − y = x + y =

x − y = ) ) −x + y = − ⎧ ⎨ x − y + z = ) ⎩ x + y + z = − ) x − y + z =

t = x =

' − x − y, ) x, y ∈ R

$ ) .$ & $ !' 50 $. .$ 0 % M d0 5F % ) $ ) 4, &#

|A | z = x = = − |A|

S=

x+y+z =

& S = (x, y,

|A | y = x = = − |A|

x − y = − z,

y+z+t= ,

X# - ]#& [. >. T R# .$ *#0 h .$ . & $ ]3 *( ; 0 0 >_$&U & $ R# =&- ]#&

5F % ) .$ !\ " &v $ U ( % ) . $ e1 .$ *O&0 m − rank(A) = − =

|A | x = x = = |A|

x − y + z = ,

x + y + z + t = ,

x − y + z = , x − y + z =

#E *S = {(

i

i

x + y − z = −

,

>. T R# .$ *#0 h .$ .

, , )} − − = = − − − = , x = x = |A | = |A | = − |A| −

|A | − = , y = x = = |A | = − |A| ⎤ −

|A | |A | = − ⎦ = , z = x = = |A| − |A| =

>_$&U & $ A, .8

x + y − z + t = ,

x − y − t = ,

1 & 1 λ ∈ R $, 0C )5 & M ( A ∈ Mat (n × n) U0 ]#& # . \ N# P )*UV )*HJ . B$&U R# *p(λ) = |λI − A| = A )*UV )'F% . p(λ) 4 n [ +) Q A *Z &

|A | y = x = = |A|

>_$&U & $ AO&0 T & $ 5& $ S C A .8

n

R# .$ *A =

M x= A& '() & E (".&[, A jL6 B$&U >. T

= |λI − A| = λ−−

− λ−

x y x + y = x x + y = y

Av = λ v ⇔ ⇔

=

x y

x −x

⇔ λ =

x y

R# .$ *A

= =

x x

x ∈ R = span

⎡ = ⎣

−

−

M x= A, .8 E ( >.&[, A jL6 B$&U >. T ⎦

− − λ− − λ−

(λ − ) (λ − ) (λ − )

|A | = |A|

− z

y = x =

|A | = |A|

z −

(0 &3 (T $ $ . 1 4 B$&U .$ . y x #&0 &' $ + t\ . F

S=

R# 0&0

( − z) , (z − ) , z z ∈ R

*$ 0 % ) O+ . 10 F &" l&Q S D. 0 . #E & & $ E N# 0 % M !' M )

)

⎤

λ− |λI − A| = −

x = x =

)

0 h # &. $0 , + U# *y = x R# 0&0 E (".&[,

x = y x + y = x x + y = y

x + y = − z

R# 0&0 *( AC &a 0 [O ?z &M M

Z#. $ c λ # . \ $. .$ ⇔

x − y = − z,

x ∈ R = span −

Av = v

T & $ R# X# - ]#& 5& $ *#0 h .$ . $ E !T&' & $ O nS .&M 0 . 4 B$&U &z 1 *( 9ZM !' . B$&U

x − y + z = − z − z − + z =

# &. $0 , + U# *y = −x nB x+y = ]3 &0 ( 0 0 λ = 0 h

x + y + z = ,

x − y + z =

= (λ − ) −

U# *λ = ± λ − = ± (λ − ) = ]3 λ # . \ $. .$ ?U0 * " A # #$&\ λ = Z#. $

x − y + z = ,

λ =

z −

⎧ ⎨ ⎩ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

x − y = − x + y =

)

x − y + z = x + y − z = ) x − y + z =

x−y+z+t= x + y − z − t = x − y − z + t = ) x + y + z − t =

x + y = − x + y =

⎧ ⎨ x+y−z = x − y − z = ⎩ x + y − z = ⎧ −x + y + z + t =

⎪ ⎪ ⎨ x−y+z+t= ⎪ x+y−z+t= ⎪ ⎩ x+y+z−t=

/1 #' 2 n × n ]#& A M x= & $, v ∈ Mat (n × ) . $0 S *( . λ &F Av = λv M $S (=&# . @ λ \\' h *W N# . v A P *W 9 N# T GB&L v &. $0 Xb *Z & A 0 λ 0 B$&U .$ U[@ . Y0 T . $0 #E ( ) $.

*$. $ T λ E 0 Av = λv

]#& M R# 0 =&M 4E_ pO % & n . $ A M ( 5F O&0 O Y1 A ∈ Mat (n × n) #$&\ . $ A ]#& S ?U0 O&0 Y !\ " # . $0 . . . v v $ 0 A. V &0 _z&+ ' C λ . . . λ λ # !T&' ]#& C O&0 & F 0 h # &. $0 v R# .$ O&0 v . . . v v & 5$ $ . 1 Z .&M E ( Y1 C AC >. T n

.$ *λ = λ = λ = E .&[, A # #$&\ U# Z#. $ λ = # . \ $.

Av = v ⇔

n

n

⇔

−

⎡ ⎢ ⎢ C − AC = ⎢ ⎣

λ

··· ···

· · · λn

*** ***

⎥ ⎥ ⎥ ⎦

***

A

R# .$ ;*J* B A&X .8 )* A&

C=

v

v

'() & &

−

− −

−

=

(B&' R# .$ A;*J* E A;C &a $ C A, .8 C=

v

v

v

⎡ =⎣

⎤ ⎦

−

.$ ⎡ ⎣

⎡ =

⎣

⎤− ⎡

⎦

⎣

−

⎤

−

−

⎤⎡ ⎦⎣

⎤ ⎦

−

⎦

⎤ − − − ⎦ A = ⎣ − − −(λ + ) = ⎡

R# .$ * M x= A .8 .$ ( jL6 B$&U . $ A >. T x= *$. $ . V &0 λ = − # . \ N# & >_$&U & $ 0 AX = −X

x − y + z = x − y + z =

5F Y !\ " % ) & M $ O

*(" O Y1 A ]#& R# 0&0 *(

v = ( , , − )t

=

⎤ x y ⎦ z

0 h # &. $0 , + U# E .&[,

⎧⎡ ⎫ ⎧⎡ ⎤ ⎨ x ⎬ ⎨ ⎣ ⎦ x ∈ R = span ⎣ ⎩ ⎭ ⎩ x

.$

]#& E λ

(B&'

=

⎡

⎤

x ⎣ − ⎦ ⎣ y ⎦ = ⎣ −

z ⎧ ⎨ −x + y + z = x − z = ⎩ −x + y + z =

⎧ ⎨ x=y+z z=x ⇒ ⎩ y + z = x ⎧ ⎨ y= x=z z=x ⇒ ⇒ y= ⎩ z=x

⎤

λ

⎤⎡

⎡

λ

=

⎧⎡ ⎨ span ⎣ ⎩

⎤⎫ ⎬ ⎦ ⎭

0 h # &. $0 , + 0&6 >. T 0 &0 0 0 X 0 A ]#& E λ =

⎤⎫ ⎬ ⎦ ⎭

⎧⎡ ⎨ span ⎣ ⎩

⎡

−

⎤⎫ ⎬ ⎦ ⎭

⎤

⎦ M x= A .8 >. T R# .$ *A = ⎣ − (O λ − λ + λ + = E ( >.&[, A jL6 B$&U

:$c y L &6#. λ = − \\' 6#. . $ M !V6 &, + *( λ = − i λ = + i &0 0 0 X 0 & F 0 h # &. $0 E ⎧⎡ ⎨ span ⎣ − ⎩

⎧⎡ ⎧⎡ ⎤⎫ ⎤⎫ ⎤⎫ ⎬ ⎬ ⎬ ⎨ ⎨ ⎦ , span ⎣ i ⎦ , span ⎣ −i ⎦ ⎭ ⎭ ⎭ ⎩ ⎩ i

]#& N# # #$&\ + %qT&' & *( ]#& 5F 5& $ 0 0 5F 5& $ f 0 0 ]"#& N# # #$&\ + ` +

*]#& 5F T Y1 T&, ` + 0 0 U# ( ]#&

h Y !\ " # &. $0 $ U $ & .$ # . \ 5F . V .&+O 0 0 aM ' # . \ 0 *( jL6 B$&U

Y !\ " >& # #$&\ 0 h # &. $0 * "

. #E & "#& E N# . \

n

− − ⎡ − ) ⎣ − −

)

)

⎤n

)

⎦

0 &

9M [&

− −

n

n

λ

λ

) #' ' ' 3 & ' 4 *A = [a ] ∈ Mat (m × n) M x= * M $ O G#U A = [a ] >. T 0 A P a T&, 5&V # U E ( Mat (m × n) E q, *(A ) = A M ( RO. >. T R# .$ *$$S !T&' a U# *AA = I M Zg S &U . T .$ . A ]#&

*O&0 D$& 0 0 5F / VU

M $ O h'? '() * ij

t

E N# # &. $0 # #$&\ 4&+ 0 * & Y1 !VO 5&V >. T .$ $.F (0 . #E &]#&

*#.F (0 F

ji

ij

t t

⎤

−

⎦

⎡

⎤ − − − ) ⎣ − − − ⎦

) ⎤ ⎡ −

) ⎣ − − ⎦ − − ⎤ ⎡ ⎥ ⎢ ⎥ ) ⎢⎣ ⎦

_&0 ]#& Y1 4= # &. $0⎡ # #$&\ ⎤A5 a b d *M [& . ⎣ c e ⎦ aa

ji

t

A−

= = ⎡

A−

⎣

=

⎡ ⎣

=

/ −/

/ / / −/ / / / / / / / /

/ − / −/ / − / −/

−

n

.$ A =

= At

=

A

n

=I

ij

ji

x x− −

'() &

>. T R#

−

x+

−

x −

=

x x−

=

x − x +

p (x) =

in nj

ij

M x= A&

n+

ni jn

j

n ≥ E 0 R# 0&0 A = I .$ =A *A &F A = − S A, .8

in

i

i j

−

nj

j

i

=

ij

j

p (x) =

⎤− / −/ ⎦ / ⎤ / −/ ⎦ = At = A /

f

( n × n ]#& N# A M x= & *p (A) = O >. T R# .$ *P (x) = |xI − A|

pO &F A = [a ] ∈ Mat (n × n) S * M ( 5F O&0 &U A M R# 0 =&M 4E_ *a + a + · · · + a = ≤ j ≤ n E 0 AGB *a + a + · · · + a = ≤ i ≤ n E 0 A% *a a + · · · + a a = ≤ i = j ≤ n E 0 A: *a a + · · · + a a = ≤ i = j ≤ n E 0 A$ .$ . A = [a ] ∈ Mat (n × n) ]#& * Ǳ E 0 U# O&0 0 0 D$& &0 M Zg S L9 . T *a = a j i $ * n . $ n × n 5.&\ ]#& Y !\ " # . $0 n A. V %&" ' &0C \\' # . \

0 h # &. $0 ?U0 *( O Y1 R# 0&0 ( * &U >& # #$&\

i

) ⎡ ) ⎣ −

B =

−

− x−

−

*A = A − I &# A − A + I = O .$ x= &0 A = S A .8 − .$ *B AB = Z#. $ −

n

A

= = =

B n B n−

n

n + n −

−

n

B B −

n − n +

R# 0&0 C

⎡ C

−

AC = ⎣ ⎡

B$&U >. T R# .$ *⎣

−

= Ct

√

− √ − − −

⎤

>. T R# .$

n

⎦

i

M x= A .8 E ( >.&[, A jL6

E ( >.&[, λ = 0 h # &. $0 &q= 0 h # &. $0 &q= span ( , , ) , ( , , − ) R# 0&0 *span ( , , ) E ( >.&[, λ = t

t

t

⎤

C=⎣

⎡

⎦ , C − AC = ⎣

−

*C

−

−

⎤ ⎦

(B&' R# .$ M $ O ) − A . \ *n ∈ N A = M x= A .8 − *#.F (0 . B$&U *ZM Y1 . A 0 . h R# 0 *!' λ = − 5F # #$&\ λ − λ − = 5F jL6

λ 0 h Y !\ " # &. $0 * " λ = &0 *v = ( , − ) v = ( , ) E .&[, X 0 λ ]#& 0 v/v v /v &. $0 5$ $ . 1√ Z .&M) ( √ R# 0&0 *Z. C = √// −√// −

= C t

n

t

t

C t AC =

−

.$ *A = C n

A

= = = =

−

n − t C C n − t C C (− )n + n (− )n − n C Ct (− )n − n (− )n + n − n n (− ) + −

Ct

&#

& "#& E N# M $ 5&6 0 * * 9( E. Z Y1 ]#& N# &0 #E 5.&\

) ⎡

−

−

)

−

−

) ⎣ −

−

⎡

⎦

(λ + ) (λ − ) =

⎡

n

n

⎤

5.&\ ]#& N# A ∈ Mat (n × n) S MC % * v V# # &. $0 λ . . . λ λ # &. \ &0 & v 5$ $ . 1 Z .&M E !T&' ]#& $ 0 v . . . v &F Z &0 C = [vv · · · v ] .

⎤

⎡

⎦

) ⎣

−

⎤ ⎦

⎢ ⎢ C − AC = ⎢ ⎣

⎤

λ

··· ···

· · · λn

λ

*** ***

⎥ ⎥ ⎥ ⎦

***

&. $0 E C .$ O&0 c#&+ A # #$&\ + S R# 0&0 *$ 0 &U C ]#& &F ZM $& V# − ]#& A& '() & * . A = − 5F jL6 B$&U ( 5.&\ ]#& R# *#0 h .$ λ = λ = 5F # #$&\ ]3 *( λ − λ = 0 0 X 0 λ = λ = 0 h * # &. $0 " x= Z ]3 *span − span &0 ZM ( √ ) / √ v = , / C=

.$ C

( √ ) / √ v = , − / ( √ ) √ / / √ v = √ / − /

v

−

= Ct

c#&+ A # #$&\ 5 Q >. T R# .$

−

C − AC = C t AC = ⎡

⎤

*#0 h .$ . A = ⎣ − ⎦ ]#& A, .8 5F jL6 B$&U ( 5.&\ ]#& R# λ λ − =

λ = −√ λ = &0 0 0 A # #$&\ ]3 *O&0

X 0 λ λ λ 0 h # &. $0 *λ = √ ⎧⎡ ⎨ span ⎣ ⎩

⎡ √ v = ⎣ ⎡ v = ⎣

⎧⎡ ⎧⎡ ⎤⎫ ⎤⎫ ⎤⎫ ⎬ ⎬ ⎬ ⎨ √ ⎨ √ ⎦ , span ⎣ ⎦ , span ⎣ − ⎦ ⎭ ⎭ ⎭ ⎩ ⎩ − − ⎤

⎡

$M x= 5 ]3 *O&0

⎤

/ √ / ⎦ , v = ⎣ / ⎦ , √ − / / ⎡ √ ⎤ / √ / √/ ⎢ ⎦ − / , C = ⎣ √ / − / / − /

⎤ / √ ⎥ − / ⎦ − /

F ((x, y, z) + (α, β, γ)) = F (x + α, y + β, z + γ)

t

= x + α − y − β + z + γ = (x − y + z) + (α − β + γ) = F (x, y, z) + F (α, β, γ)

R 0 R E Y & O& G F S aF >. T R# .$ a ∈ R O&0 R 0 R E Y O& H Y !#[ / VU ?U0 * Y (O& H◦F F +G *(A ) = A ( #n< VU Y !#[ N# #n< VU

M ( . $0 e ∈ R M x= E ∈ ?U0 T 5F >&j L #& N# 5F 4 i o L

T u&j L #& N# 5F 4 i o L M ( . $0 R *O&0 E 0 ( Y O& F : R → R M x= ZO&0 O $ ≤ j ≤ n m

n

l

−

&F O&0 LB$ A ∈ Mat (n × n) S M $ 5&6 AH *( 5.&\ ]#& N# M = A + A ! Q ( 5.&\ 5.&\ ]#& $ %qT&' &#F A8 Q ( 5.&\ &U ]#& $ %qT&' &#F AJ − [& n E 0 . A − l&Q A7 *M *A = O &F AA = O S M M (0&f A

m

n

t

−

, 56& 7

n

i

i

m

n

m

F (ei ) = aj E + aj E + · · · + amj Em

>$ F P . A = [a ] ]#& >. T R# .$ S R# 0&0 *Z$ 5&6 M (F) $&+ &0 Z & F !"# v ∈ R E 0 &F O O !VO 0 &. $0 ij

n

∀v ∈ Rn

" 5$ 0 Y (T& =&' M . $0 &&q= R0 e0 U# * O & Y . T .$ . F : R → R (O& a ∈ R u, v ∈ R E 0 M Zg S Y F (au) = aF (u) F (u + v) = F (u) + F (v) −−→ *f (x, y, z) Z"# f (−− (x, y, z)) &0 _z +U

(O& A& '() n

m

n

−−−−−−−−−−−−→ F (x, y) = (x, x − y, y − x)

: F (v) = M (F ) v

E Y & O& R0 N[V# & $ *$. $ $ ) m × n & "#& , + R 0 R 0 R E Y & O& G F S % E #n< VU Y O& J R 0 R E Y O& H >. T R# .$ a ∈ R O&0 R 0 R

#E *( Y R 0 R E

Rn

m

m

F (a (x, y)) =

m

n

= =

aF (x, y)

=

n

l

F (ax, ay) −−−−−−−−−−−−−−−−−→ (ax, ax − ay, ay − ax) −−−−−−−−−−−−→ (x, x − y, y − x)

n

) M (aF ) = aM (F )

)

F ((x, y) + (s, t)) = F (x + s, y + t) −−−−−−−−−−−−−−−−−−−−−−−−−−−−→ = (x + s, x + s − y − t, y + t − x − s) −−−−−−−−−−−−→ −−−−−−−−−−−→ = (x, x − y, y − x) + (s, s − t, t − s)

M (F + G) = M (F ) + M (G)

)

M (H ◦ F ) = M (H) M (F ) ) M J − = M (J)−

a = M x=

−−−−−−−−−−−−−−−−→ F (x, y) = (x − y, x + y, x + y) −−−−−−−−−−−−−−−→ G (x, y) = (x, x + y, x + y)

−−−−−−−−−−−−−−−−−−→ H (x, y, z) = (x − y + z, x + y − z) −−−−−−−−−−→ J (x, y) = (x − y, x + y)

=

'() &

F (x, y) + F (s, t)

Y R 0 R E F (x, y, z) = x − y + z (O& A, .8 #E *( F (a (x, y, z)) = F (ax, ay, az) = ax − ay + az = a (x − y + z) = aF (x, y)

Z#. $ 8** q1 E A t\ 0 A

9ZM t\ (B&' R# .$ . 8** q1 Z

A &. $0 !VO 0C 5 Q A&

(H ◦ F ) (x, y) = H (F (x, y)) = H (x − y, x + y, x + y) = (x − y) − (x + y) + (x + y) i + (x − y) + (x + y) − (x + y) j

F

⎡

=⎣

M (H ◦ F )

=

⎡

−

−

⎡

⎣

⎤

−

⎦

= M (H) M (F )

x= &0 8** q1 E A (+"1 t\ 0 A Z#. $ J (x, y) = (u, v) −

u−v =x u + v = y ⇒

⎡

M (G) = ⎣ ⎛⎡ H ⎝⎣

J

u = x + y v = −x + y

M (J)

=

=

/ −/

−

−

=

/ /

F (x, y) = (x − y, x + y, x + y)

)

F (x, y) = x − y

)

F (x, y) = (x + y, x − y)

)

F (x, y, z) = (x + z, y + x, z + x)

⎡

=⎣

⎤

−

⎦

⎤⎞

⎦⎠ =

⎦

⎤

0&6 >. T 0

⎦,

⎛⎡

⎛⎡ , H ⎝⎣

−

,J

− − = ,

⎦⎠ = ⎤⎞ ⎦⎠ =

⎤

⎦,

,

−

=

M (J) =

=⎣

⎤⎞

H ⎝⎣

−

⎡

G

⎦,

⎤

−

,

−

, .

Z#. $ 8** q1 E A (+"1 t\ 0 A, (aF ) (x, y) = = =

aF (x, y) −−−−−−−−−−−−−−−−→ (x − y, x + y, x + y) −−−−−−−−−−−−−−−−−−−−−→ (x − y, x + y, x + y) ⎡

= ⎣

M (aF )

⎡ =

⎣

−

−

⎤ ⎦ ⎤ ⎦ = aM (F )

Z#. $ 8** q1 E A; t\ 0 A

(F + G) (x, y) = F (x, y) + G (x, y) −−−−−−−−−−−−−−−−→ −−−−−−−−−−−−−−−→ = (x − y, x + y, x + y) + (x, x + y, x + y) −−−−−−−−−−−−−−−−−−−−→ = (x − y, x + y, x + y)

a = M M t\ . T .$ . 8** q1 A4 F (x, y) = (x − y, x + y) H (x, y) = (x + y, x − y) G (x, y) = (x − y, x − y) J (x, y) = (x + y, x + y)

(A

⎤

= M J −

)

= B − A−

=⎣

−

) F (x, y, z) = (x − y + z, x + y − z)

−

⎡

#E & O& E N# M $ 5&6 0 * 9M oL6 . N# u#&+ ]#& ]< * Y

*(AB)

M (H) =

−

−−−−−−−−−−−−−→ x + y −x + y − ⇒ J (x, y) = ,

−

G

−−−→ −−−→ J J − (x, y) = (x, y) ⇒ J (u, v) = (x, y) −−−−−−−−−−→ −−−→ ⇒ (u − v, u + v) = (x, y)

F

M (F ) = ⎣

−

⎦

=

E (".&[, F u#&+ ]#& R# 0&0

−−−−−−−−−−−−−→ = (x + y, x − y)

⎤

− −

)

=A

M M (0&f A5

⎡ M (F + G)

= ⎣ ⎡ = ⎣

−

−

⎤ ⎦ ⎤

⎡

⎦+⎣

= M (F ) + M (G)

⎤ ⎦

!<

]#& $& !<

det ( A ) −−−−→ . . . . . . . . . . . . . . . . . . . . A ]#& 5& $ >. T 0 . I** R#+ E ; (+"1 '() 9Z$ 4& !< N+M 0 #E transpose(A) −−−−→

with(linalg) : A := matrix(2, 2, [[1, −2], [2, 3]]); B := matrix(2, 2, [[0, 2], [3, 1]]); C := multiply(inverse(A), B); multiply(C, A);

M x=

58' 9 / :

............... A

? @A) 9BC EJ $

>_$&U E Sy & $

eq_i:a_{i1}*x_1+a_{i2}*x_2+...+a_{im}*x_m=b_i

.$ #E !VO 0 . & $ R# *

≤i≤n

M O&0 O !V6 9ZM $. !<

;H7 T 0 !< . c= 4 E $& >& \ &6 0 *$ O U) doc.pdf !#&= &# Y [) $. .$ > . $ : 9 . $ &0 M $. $ $ ) linalg 4&0 . c= 4 .$ !<[ .$ *$ + .&M 0 $& F h=&' .$ . 5F 5 with(linalg) N# E $& 0 ; ) + 2 =U i jL0 + &0 . 5F ( 0 ]#& N# &# . $0 &#F.$ &0 v 4&0 B −n . $0 G#U 0 *ZM . $ E v_n * * * v_2 v_1 X 0 !<

−−−−−−−−−−→ v := vector([v1 , v2 , ..., vn ]) −−−−→ v := (v , v , · · · , vn )

#F.$ &0 A 4&0 n × m ]#& G#U 0 *ZM $& . $ E A 4 (i, j) ij

A := matrx(n, m, [[A 11, A 12, ..., A 2m]

with(linalg) :

, [A 21, A 22, ..., A 1m],

eq 1 := a 11 ∗ x 1 + ... + a 1m ∗ x m = b 1; .. .

... , [A n1, A n2, ..., A nm, ]])

eq n := a n1 ∗ x 1 + ... + a nm ∗ x m = b n; Sy := {eq 1, eq 2, ..., eq n}; Va := {x 1, x 2, ..., x m}; solve(Sy,Va)

. $ E ( =&M & $ R# !' 0 5 M *$ O $&

5$.F (0 0 9N+ 2 + 1) % . $ E ( =&M A ]#& jL6 +)Q eigenvectors(A) . $ N+M 0 *$ O $& charpoly(A) !VO 0 [ *$.F (0 . A # &. $0 #$&\ 5

[a,b,(v_1,...,v_k)] 4= 0 & .&[, E E jL6 +)Q .$ 5F . V .&+O b # . \ a M $ 0 # &. $0 E Y !\ " , + (v_1,...,v_k) *O&0 a 0 h # /.$F .$ 78 & http://webpages.iust.ac.ir/m_nadjafikhah/r2.htm

*( O $.F MnB = d'&[ 0 p 0 e0& & B&a

]#& G#U 0

⎡

A ⎢ A ⎢ A := ⎢ ⎣ An

***

A A

***

An

⎤ Am Am ⎥ ⎥ ⎥ ⎦ · · · Anm ··· ···

***

c &. $0 G#U 0 ' M $ O T *ZM $& $&# 0 #$ > . $ [B *$ O $& ]#& . $ E 0 !< help N+M 0 M $. $ $ ) G#U .$ (B *#E $<0 & F UB&Y

]#& B A M x= OD ) 2 '6 >. T R# .$ O&0 LB$ $, c !<

A + B −−−−→ . . . . . . . . . . . . . . . . . . . B A & "#& e+) !<

A − B −−−−→ . . . . . . . . . . . . . . . . . . . . . B ]#& A !<

c ∗ A −−−−→ . . . . . . . . . . . . . . . . . . . . . . . . . . A ]#& .$ c !<

B .$ A & "#& %qT&' multiply ( A , B ) −−−−→ !<

A ∧ n −−−−→ . . . . . . . . . . . . . . . . . . . . . . n 5 0 A ]#&

!<

inverse ( A ) −−−−→ . . . . . . . . . . . . . . . . . . . . . . . . A / VU

) u •v = v •u

)

u • (v + w) = u • v + u • w

)

a (u • v) = (au) • v

)

u •u ≥

G#U 5 &VO i UB&Y Z, >. T 0 . Z &0 da $ R0 # E e- $ S V# ?z a *$ + ZV' N# 1&[Y !0&1 da $ 5F O&0 0 0 *$. $ $ ) !#&" !' 0 $U & O. *( ( tY 0 [ M , - !T +) E !" #$ %&"' 0 [ M !" #$ &# !#&" !' .$ Y [) E $& !j= R# E m *O&0

* & . Z, R# *(

;**; q1 E AC S# !B$ 0 >. T 0 . u ∈ R x . $0 AB &# Y C .S u =

√ u •u

U# *ZM G#U + *−−−−→* *(a, b, c)* := a + b + c

9&F a ∈ R u, v ∈ R S ) u ≥

)

u = ⇔ u = 0

)

|u • v| ≤ uv

;< M . @ 5&+ #E ( & .&"0 uL0 R# ` -

4&+ &z[#\ M ( (.$ &,$ R# #$ *(& 5 $ %- R O $ &0 & . ! v = −a,−−b,−→c!, w = −−x,−y,−−→ z ∈ R S !VO 0 $ $ 5&6 v w $&+ &0 . w .$ v $ %qT&'

)

au = |a|u

)

u + v ≤ u + v

O M &" & aa &" & X 0 . AH A &" & * &

Z#. $ O M &" & $ ) !B$ 0 $ v ≤ &. $0 R0 # E .$ *− ≤ uu v >. T 0 . u, v ∈ R

•

v • w = ax + by + cz

*ZM G#U 9$ + G#U 5 . R .$ $ %- 0&6 >. T 0 −−−−−−−→ −−−−−−−−→ R .$ . $0 $ w = (b , · · · , b ) v = (a , · · · , a ) S ZM G#U O&0

•

∠ (u, v) := arccos

u•v u v

n

n

n

n

v • w := a b + · · · + an bn

*$ + G#U 5 &F O&0 α 0 0 v u R0 # E S R# 0&0

RL R .$ $ %- i $. .$ & ]3 R# E R 0 #$ 50 XB&Y R# & (S Z * " ^T 9&F a ∈ R u, v, w ∈ R S n

u • v = uv cos α

E N# / g. D C B A M x= A .8 &0 `?- @ 4$ 5 ` + M M (0&f *,?-_ 9AGB *; !VOC *( 0 0 Y1 $ @ 4$ 5 ` +

AB + BC + CD + DA = AC + BD −→ >. T R# .$ *v = −→ DA u = AB ZM x= *!' −→ −→ −→ −→ DB = DC + CB = u + v CB = v DC = u −→ → −→ −→ h $. ZV' R# 0&0 *−CA = CB − AB = v − u

u &. $0 V# 0 =&M 4E_ pO % *u v = M ( 5F O&0 J v •

>. T R# .$ *A, B ∈ R M x= & → . $0 E >. T 0 . B A )*'=< $&+ &0 $ + G#U −AB #$ 5&0 0 *Z$ 5&6 d (A, B) ! d (x, y, z) , (a, b, c) :=

u + v = u + v + v − u

9$ O $& E G#U E ( =&M >&[f 0

) d (A, B) ≥

)

(u + v) • (u + v) + (u − v) • (u − v)

=

u • u + u • v + v • v + u • u − u • v + v • v

=

u

•

d (A, B) = d (B, A)

)

d (A, B) + d (B, C) ≥ d (A, C)

*

* & !8'8 E . A &" & }

#E $. E N# .$ 0 9M [& . ∠ (u, v)

$ %- AGB 9*; !VO #$ 0 . $0 N# # j A%

−−−−−→! −−−−−−−→! ) u = , , − , v = , , u=

u + v • v = u + v

u • v v u

)

d (A, B) = ⇔ A = B

)

u + v + v − u = =

(x − a) + (y − b) + (z − c)

9&F A, B, C ∈ R S

u + − v + − u + v = u + v + v − u

#$ 5&0 0 &# *(

,

−−−−−→! −−−−−→! , , , v = , ,

&. $0 R0 # E A&

−−−−−→! −−−−−→! ) u = , , , v = , , −−−−−−−→! −−−−−−−−→! ) u = , , − , v = − , ,

B = (, , ) A = ( , , − ) M x= A $. R# .$ . 7**; q1 E AC (+"1 *C = (, , ) *M t\ GB&L . $0 u ∈ R M x= *ZM G#U u/u >. T 0 . u V# . $0 *( T ( N# . $0 R# E M ( 5F . nS Z R# !B$ u *u = u u 9?U0 x= B 2 2 2 < =P 0 0 u 0 v # j >. T R# .$ *u = 0 u, v ∈ R M u •v u proju v = u •u

*$ O ) % *; !VO 0 *(

u=

−−−−−−−√ −−→ , −,

'()

−−−−−−−√ −−→ v = −, ,

&0 ( 0 0

∠ (u, v) = arccos ⎛

u •v uv

⎞ √ √ ( )(− )+(− )( )+( )( ) ⎠ , = arccos ⎝ , √ √ +(−) +( ) (−) + +( )

− = arccos = arccos ( ) ≈ ×

5F / g. M aa A, .8 9 #E *( `?-_ &" " C = (, , )

B = ( , , ) A = ( , , )

, √ ( − ) + ( − ) + ( − ) = , √ d (A, C) = ( − ) + ( − ) + ( − ) = , √ d (B, C) = ( − ) + ( − ) + ( − ) = d (A, B) =

*( u 0 v # j E >.&[, l 5F .$ M h = v − l U# U#

0 0 u 0 v . $0 # j @ ZM$Q proju v =

* * |u • v| | + − | =√ l = *proju v* = = √ u ++ ,

/

0 0 O @ .$

#E *O&0

h

+

=

++

−

=

−

=

90&0 v 0 . u # j $. .$

√

0

= =

S M ( RO. *( ∠ (u, v) # E α M ( R+ 0 *$ O @ nB cos α < &F α ≤ π ≤ S =@ E *( O & | cos α| _&0 [& .$ !B$ S ( 0 0 u V# . $0 &0 # j V# . $0 &F α < π/ .$ *( 0 0 −u &0 &F π/ < α ≤ π M π/ <

u proju v = sgn (cos α) proju v u

−−−−−−−→! −−−−−−−→! ) u = , − , , v = , , −

)

u=

9Z#. $ ` + .$ ]3

−−−−−→! −−−−−→! , , , v = , ,

/ g. &0 da ('&" A *M [& . C = (, , − ) $ +, $ 0 $ V# . $0 w v u S M $ 5&6 A 9 x LB$ . $0 E 0 &F O&0 Z 0

proju v

B = (− , , ) A = ( , , )

a) proju x = (u • x) u b) projv x = (v • x) v c) projw x = (w • x) w d) x = (u • x) u + (v • x) v + (w • x) w

& x . R0 # E M x= A$& & "MC AH # E β . u . $0 & y . R0 # E α . u . $0 ;*; !VOC *Z &0 γ . u . $0 & z . R0 &0 u V# . $0 u/u >. T R# .$ M $ 5&6 *A% −−−−−−−−−−−−−→ α, cos β, cos γ) i jL0 *( 0 0 (cos cos α + cos β + cos γ =

∠ (u, v) = π/ v = u = M . T .$ A8 9M [& . #E > .&[, E N# . \

a) u • v

b) u • u

c) u + v

d) u − v

e) (u − v) • (u + v) f ) u + v

u + v + w = 0 V# w v u &. $0 M x= AJ *#.F (0 . u v + u w + v w ` + . \

•

•

•

v.| cos α| |u • v| v u.v |u • v| u

proju v = proju v proju v u |u • v| sgn (cos α) = u u uv| cos α| cos α u = u | cos α| u uv cos α = u u u •v = u u •u

. $0 0 v = −−−, −−, −−−→! . $0 # j A& '() *#.F (0 . u = −−−, −−−−,−→! Z#. $ G#U 0 ) &0 *!' proju v

u •v

u u •u ( ) ( ) + (− ) () + () (− ) −−−−−−−→ = ( , − , ) ( ) + (− ) + () −−−−−−−−−−→

= − , ,− =

B = ( , , ) A = ( , , − ) M x= A, .8 . BC e- 0 A / . E $. `&. @ *C = (, − , ) *M [&

ZM x= 0 . h R# 0 *!' −−−−−→ −→ u = CB = B − C = ( , , ) −−−−−−−→ −→ v = CA = A − C = ( , , −)

. H & A T&= & H . BC e- 0 A # j S CAH # cB Zg&1 da .$ &F AGB ;* ; !VOC Z &0 h *$M [& . CH e- CA e- 5 h [& 0

&U #&3 B = &v, v , · · · , v ' S E&0 >. T R# .$ O&0 R E V . $0 &q= #E 0 v ∈ V k

n

v=

v • v v • v

v +

v • v v • v

v + · · · +

ZM x= v =

vk • v vk • vk

vk

√

∠ (u, v) = π/ v = u = M . T .$ A7 *#.F (0 . u − v u + v R0 # E . $0 w v u LB$ . $0 E 0 M $ 5&6 A *( $ +, u 0 v (u w) − w (u v) •

}

'()

−−−−−→! −−−−−−−−→! −−−−−−−→! , v = − , , , v = , −, , ,

*( R 0 #&3 B >. T R# .$ *B = v , v, v >. T R# .$ v = −a,−−b,−→c! ∈ R S ?U0

v • v v • v v • v v = v + v + v v • v v • v v • v

a+b+c −a + c a − b + c = v + v + v

5 Q #&3 B M x= 78 QGK+ $ O&0 R E V . $0 &q= #E 0 v, v, · · · , v ZM G#U n

k

w

=

v

w

=

v −

w

=

wk

•

*** =

v • w w w • w v • w v • w v − w − w w • w w • w vk • w v k • w vk − w − w − ··· w • w w • w vk • wk− ··· − w wk− • wk− k− B = w , w , · · · , wk

>. T R# .$ *( R E V . $0 &q= #E 0 ! v = −−,−−,−−,−→ M x= A& '() % −−−−−−−−−−−→! ! ?U0 v = − , , −, v = −−,−−,−−−→ , 4 S D. N+M 0 >. T R# .$ *B = v, v , v #E 0 V# &U #&3 N# c &U

#&3 (+O *#E&"0 R E v, v, v &q= Z#. $ 8*;*; q1 0 ) &0 . h R# 0 *!' V# &U #&3

n

#$ . $0 0 . $0 N# # j AGB 9;*; !VO $& & "M A%

=>?@AB C ' ' D !#&" 0 .$ ( (' . _z +U &U &#&3 &0 5$M .&M & ( 4E_ 4$ ).$ &#. &# & E&"B& h R# 0 &3 uL0 R# E m *ZM &U . O $ $ #&3 E d0 $ .$ S $&# 50 *( >&)& ' *$ + h mT 5 uL0 R# UB&Y

v , v , · · · , vk

M x= . T .$ *O&0 R E V . $0 &q= #E 0 #&3 i E&0 M ( V J )*I [ B Zg S

# S )? J . T .$ . #&3 R# *v v = j = i ***v ** = i E 0 M v = −−−, −−, −−−→! M x= A& '() &U #&3 N# B = &v , v' >. T R# .$ *v& = −−,−'−,−→! *( V = span v , v 0 M x= A, .8 B =

n

i

•

j

i

−−√ −−−−√ −−−− −−→ # " √ v =

, ,

−−√ −−−−−−−− # " √−−→

, , −

v =

0 &U #&3 N# B = &v, v' >. T R# .$ *( span&−(−−, −−, −−−→), −( −,−−,−→)' &# .$ + 0 O&0 R .$ . $0 e M x= A .8 >. T R# .$ *( N# 0 0 M 4 i #F.$ c0 T 5F *( R 0 V# &U #&3 N# B = &e , e , · · · , e ' V =

w

= =

w

= =

v −−−−−−−→! , , , v • w v − w w • w −−−−−−−→! + + + −−−−−−−→! , , , − , , , ++ +

n

n

i

n

w

v • w v • w v • w w − w − w w • w w • w w • w −−−−−−−→!t + + + −−−−−−−−−−→!t , , , − − , , , +++ −

−−−−−−−−−−−→ t +++ − , , , − − / + + + / −−−−−−−−−−−−−−→ t

+++ − − , ,− ,− / + + / + / −−−−−−−→!t , , , B = v , v , v , v v −

= =

=

.$ *( &U #&3 u

w = ||w ||

=

u

w = ||w ||

=

u

w = ||w ||

=

u

X R# 0 ZM V# . #&3 R# $

−

, , ,

=

= =

l&Q ( &U #&3 N#

−−−√ −−−−−−−− −−→ # "t √−−−−−√ −

, , , −

u

,

− #−−−−−−−−−−−−−−−−−−−−→ "t √

−

√

√

u

√

, , − , −

,

>. T R# .$ *(

w

−−−−−−−→! −−−−−−−→! , , , , v = , , , ,

w

M !#[ Y1 !VO 0 . #E & "#.& E N# )

⎡

)

⎣

−

= =

−−−−−−−−−−→!t − , , , −

=

−−−−−−−−−−−−→

t − , , , −

w

=

= =

−−−−−−−→! −−−−−−−→! , , , , v = , , ,

) ⎤ ⎦

−

⎡

)

⎣

,

−−−−√ −−−−−−√ −−→ " #

, , ,

,

−

−−−−−−−−−−−−→ ,− ,− ,

= u , u , u

v −−−−−−−−−−→!t − , , , v • w v − w w • w −−−−−−−−−−→!t − , , , −

= =

−−−−−→! −−−−−→! −−−−−→! , , , v = , , , v = , ,

−−−−−−−→! −−−−−−−→! v = , , , , v = , , ,

, , ,

t

−−−→! −−−→! , , v = ,

v =

=

w = ||w ||

−−√ −−−−−− −−−−→ # " √

>. T R# .$

t

−−−−−−−→! −−−−−−−→! ) v = , , , , v = , , , ,

v =

=

w = ||w ||

.$ $ O x=

−−−−−−,−−,−−,−→! 0 0 X 0 v v v M x= A, .8 (+O 4 S D. *O&0 −−−−−−,−−,−−,−→! −−−−−−,−−,−−,−→! 9ZM ) B = v , v, v, v #&3 0 .

−−−−−→! −−−−−→! −−−−−→! ) v = , , , v = , , , v = , ,

)

=

w = ||w ||

*( V# &U #&3 N# B

−−−−−→! −−−→! ) v = , − , v = , v =

−−−−−−−→! −−−−−−−→! −−−−−−−→! , , , − , , , = , , , v • w v • w v − w − w w • w w • w −−−−−−−−−−−→! − + − + −−−−−−−→! − , , −, − , , , ++ + + + + −−−−−−−→! , , , − + ++ −−−−−−−→! −−−−−−−→! −−−−−−−−−−−→! − , , −, + , , , − , , , −−−−−−−−−−→! ,− ,− , B = w , w , w

u

(+O 4 S D. N+M 0 $. .$ 0 & 90&0 V# &U #&3 N# ]< &U #&3 N#

)

=

,

V# &U #&3 N#

) v =

w

−−−− # "t √−−−−−−−− √−−→

−−−−−−−−−−→ t

w = , , , . ||w || B = u , u , u , u

=

=

=

−

+ + + −−−−−−−−−−→!t − , , , +++ −−−−−−−−−−→!t − , , ,

v • w v • w v − w − w w • w w • w −−−−−−−−−−→!t + + + −−−−−−−−−−→!t − , , , − , , , − +++ −−−−−−−−−−−−→

t / + + + − , , , − − / + + + / t t −−−−−−−−−−→! −−−−−−−−−−→! − , , , − , , , −

− t

−−−−−−−−−−−→ − − , , , −

⎤ ⎦

=

t

−−−−−−−−−−−−−−→

t − , ,− ,−

= = = (u × v) × w = = = =

−−−−−−−−−→ u × (− , −, −) i j k − − − −−−−−−−→ (, , −) i j k × w − −−−−−−−→ (, , − ) × w i j k − − −−−−−−−−→ (, −, −)

E ;< $ M x=

−−−−→ −−−−→ v = (x, y, z), u = (a, b, c) ∈ R

#E >. T 0 . v .$ u . $0 A!%# &#C ! K;'= 9ZM G#U u×v = = =

*(" #n< MO ).& %- R# 0&0 9Z#. $ R $. & #&3 &. $0 $. .$ A .8

w =

v

= = =

v

k c z

(bz − cy) i + (cx − az) j + (ay − bx) k −−−−−−−−−−−−−−−−−−−→ (bz − cy, cx − az, ay − bx)

u, v, w ∈ R E 0

u × (v + w) = u × v + u × w

j × k = −k × j = i

)

a (u × v) = (au) × v

k × i = −i × k = j

)

u⊥u × v , v⊥u × v

)

u × v = uv sin (∠ (u, v))

)

u • (v × w) = (u × v) • w

M x= A .8 γ) 9>. T R# .$ −(α,−−β,−−→

i j k u • x y z α β γ −−−−−−−− −−−−−−→ − −−−−−−−−−−−

y z x z x y u• , − , α γ α β β γ y z x z x y a −b + c β γ α γ α β a b c x y z α β γ

.$ u ).& %qT&' $. .$ −−−−−−−→ −−−−−→ ) u = ( , − , ), v = (, , )

)

u = i − j, v = i + j + k

)

−−−−−→ −−−−−→ u = (, , ), v = ( , , )

)

−−−−−−−→ u = i − j + k, v = ( , − , )

λ ∈ R S & S uv S & S u × v = 0 AJ *u = λv M $$S (=&# *$ !V6 $ . kM u × v v u &. $0 A7 &F O&0 $ +, v u x &. $0 0 w . $0 S A *wu × v ZM x= A& '() −−−−−−−→ −−−−−−−→ u = ( , , − ) , v = ( , , − )

0

9M [& .

v = −(−−−−,−−,−→) u = −(−−, −−−−,−→) M x= AH −−−−−→ ;**; q1 ( & N# $. *a = w = ( , , ) *M t\ $. R# .$ .

) u × v = −v × u

)

u • (v × w) =

a∈R

i j a b x y

i × j = −j × i = k

−−−−→ −−−−→ = (x, y, z) u = (a, b, c)

u×v

i =

9>. T R# .$

j k − − −−−−−−−−−−−−−−−−−−→ = (− + , − + , − ) −−−−−−−−→ = (− , , − )

v = −(−−, −−−−,−→) u = −( −,−−,−→) M . T .$ A, .8 9Z#. $ w = −(−−, −−−−,−→) u × (v × w)

i j = u × − −

k

}

U * (u − v) × (u + v) = u × v M M (0&f A8 ("Q &" R# √ . \ u v = v = u = M . T .$ AJ *M [& . u × v u × v = v = u = M . T .$ A7 *M [& . u v . \

*u × v + (u v) = uv M (0&f A Q .$ &" *u × v ≤ uv . + M $ 5&6 AI ( . 10 . T M (0&f u + v + w = 0 M x= A *u × v = v × w = w × u * u × (v × w = (u w)v − (u v)w M M (0&f A; M M (0&f A * (u + v) × (v + w) (u + w) = (u × v) w M M (0&f A *u × (v × w) + v × (w × u) + w × (u × v) = 0 &F u ⊥ v S M $ 5&6 AH & ' *u × u × (u × [u × v]) = uv . $0 x 4 U &. $0 v u M x= A8 B$&U & $ O &+, v u 0 #&0 Y# O Q O&0 . 10 >. T .$ O&0 % ) . $ u×x = v . $0 *M oL6 . B&" % Q y# O R# •

%- E [U AGB 9*; !VO N+M 0 ('&" [& A% ).& ).& %0 0 X 0 C B A M x= '() % da ('&" * " (− , , ) (, , ) ( , −, ) *M [& . ABC −→ −→ ( RO. *Z#S h .$ . AC AB &. $0 *!' . $0 $ R# E !T&' `?-_ E M A% *; !VOC ( =&M 0&0 *( ABC da ('&" 0 0 $ . $ → −→ 9ZM [& . −AB × AC Gj M −→ AB = −→ AC =

−−−−−→ B − A = ( , , ) −−−−−−−→ C − A = (−, , )

= =

=

*−→ −→* *AB × AC * j k * * i * * * − * √ −−−−−−−−−−−−→ (− , − , ) =

.$ . ABC da ('&" $. .$

0 &

9M [&

) A = ( , , − ) , B = (, − , ) , C = (, , − )

)

A = (, , ) , B = (, , ) , C = (− , , )

F B ;< ( & F .$ . $0 $ U# #& $ ).& $ %- &+, *( #& !+, N# &S %- M B&' .$ *. $ u\ *$. $ .&"0 &$0.&M #) % u, v, w ∈ R . $0 E 0 G#U u (v × w) >. T 0 . R# ! )+ K;'= E A8C (+"1 0&0 *Z$ 5&6 [u, v, w] $&+ &0 $M •

•

•

•

•

R# 0&0 Area(ABC)

•

•

u S R ST 9: U5 . $ x= Ǳ [ .$ . & F $ 0 O&0 LB$ . $0 $ v N# u $ .$ v c v $ .$ u 5$ $ (M' &0 ZM ('&" *AGB * ;!VOC $ O !T&' `?-_ E

E $. `&. .$ %- u . $0 @ 0 0 `?-_ E R# v . $0 @ 0 0 h $. `&. & *O&0 u e- 0 v / . .$ *( v u R0 # E / .$ %A

= u.h = uv sin (∠ (u, v))

E U# *(" u × v . $0 E c) cQ R# M `?-_ E ('&" &0 ( 0 0 u× v ).& %qT&' 0 u × v M $ O ) *v u &. $0 y O & *( $ +, $. $ . 1 5F .$ `?-_ E R# M T

−→ −→ 9ZM [& . −→ AD AC AB &. $0 &S %−→ AB = −→ AC = −→ AD =

−−−−−−−→ B − A = (, −, ) −−−−−→ C − A = (, , ) −−−−−−−→ D − A = (, −, )

= =

−→ −→ −→ [AB, AC, AD] * * * − * * * * *= ** − **

−→ −→ −→ AB, AC, AD =

= = =

yu yv yw

zu zv zw

a ∈ R O&0 . $0 w v u u S

9&F

) [u + u , v, w] = [u, v, w] + [u , v, w]

}

&S %- [U AGB 9*; !VO &S %- N+M 0 XUV Z' [& A% B = (, , ) A = ( , , − ) p&\ &#F A, .8 T Z D = (, , ) C = (, , − ) M ZM ) V R# 0 u3 R# 0 &3 0 *!' M . $ . 1 T N# .$ 1 & 1 . Mn Y\ .& Q %qT&' ( =&M ]3 *O&0 T & F E !T&' 4 Z' −→ −→ AD AC AB &S T %- R# S *ZM [& . −→ & *]VUB&0 T N# 0 p&\ O&0

Z#. $ ;**; q1 Z#. $ **; &a E AC (+"1

= (u × v) • w

xu [u, v, w] = xv xw

.$ V

0&0 ?U0 *[u, v, w]

$ % B − A, C − A, D − A −−−−−→ −−−−−−−−→ −−−−−−−→ (, , ), (− , − , ), (, − , ) − − − − + + =

*U1 T N# .$ . Mn p&\ ]3 *M >&[f . ;**; q1 A& 0 $ B = ( , , ) A = ( , , ) / g. &0 Z' A; *0&0 . D = (, , ) C = (− , , ) E .&[, 5F / g. M 0&0 . Z' A C = (, , ) B = ( , , ) A = ( , , ) *D = ( , , )

)

[au, v, w] = a[u, v, w]

)

[u, w, v] = [v, u, w] = −[u, v, w]

)

[u, v, w] = [w, u, v] = [v, w, u]

v u &. $0 M [u, v, w] = 1 & 1 A *O&0 Y "0 w v u . $0 &0 G ST U5 . *AGB *; !VOC $$S oL6 K Y"B E N# w `?-_ E ('&" &0 ( 0 0 K Y"B E R# Z' / . E $. `&. .$ %- v u &. $0 y O & `?-_ `&. R# 0 u × v 5 Q *`?-_ E R# 0 w 0 w . $0 # j @ &0 ( 0 0 h $. `&. ( $ +, U# u × v h = = u

proju×v w | (u × v) • w| u × v

&. $0 y O & `?-_ E ('&" =@ E 9&0 ( 0 0 K Y"B E Z' ( u × v 0 0 v V

= = =

u × v h (u × v) • w [u, v, w]

5F / g. M 0&0 . Z' A& C = (, , ) B = ( , , ) A = (

'()

E .&[, *D = ( , − , ) −→ −→ −→ oL6 ' Y"B E AD AC AB . $0 &0 *!' 4 Z' 4 K Y"B E i 0&0 *$$S

K Y"B E Z' Z6O N# 0 0 D C B A / g. &0 Z6O N# M ( =&M ]3 *$ O ) % *; !VO 0 *( , , − )

p&\ E M . y B$&U A& '() $ 0&0 . T .$ . $.nS X = (x , y) X = (x , y) *X = X M −−−→ *Z#0 Z $& . XX . $0 &S V . X *!' E .nS y B$&U .$ *$ O ) % H*; !VO 0 9&0 ( 0 0 X X p&\ : X = ( − t) X + tX ; t ∈ R x = ( − t) x + tx : ; t∈R y = ( − t) y + ty y − y x − x = : x − x y − y

B = (− , , ) A = ( , − , ) Y\ .& Q &#F A U1 T N# .$ D = (, , ) C = ( , , ) B = (, , −) A = ( , , ) Y\ k3 &#F AH .$ E = (, , ) D = ( , , ) C = ( , , ) . $ . 1 T N# *[u + v, v + w, w + u] = [u, v, w] M M (0&f A8 M M (0&f AJ *(u × v) × (w × x) = [u, v, x]w − [u, v, w]x *[u × v, v × w, w × u] = [u, v, w] M M (0&f A7

∗

}

+ y G#U AGB 9H*; !VO Y\ $ E .nS y A% y $ (U- A, .8

: x + y =

:

x=t+

, y = t −

;t∈R

*M oL6 Z 0 ([" . .$ E b y $ & 0$ d'&[ 0 h *!' . p Y R# 5$ 0 E ( =&M ]3 *U@&\ &z cB R 9Z"# $. & !VO 0 . 0 *ZM t\

: : :

x + y = x − = −y y− x− = −

?U0 *v = −(−−, −−−→) X = (, ) Z#. $ R# 0&0 ([" y $ E pO *v = −( −,−→) X = ( , − ) U 0 v v & ( $ 5F $& &. $0 E Z 0 * " E y $ ]3 *( yb M O&0 = − 9?U0 *O&0 e@&\ R# 0&0 ∩

⎧ ⎨ x + y = x=t+ : ⎩ y = t − x= / : y = /

*O&0 e@&\ X

=(

⎧ ⎨ t + + t − = x=t+ : ⎩ y = t −

/ , / )

.$ . Mn y $ U#

5F 0 # O&0 e1 Ǳ? .$ .{ S G#U 0&0 .$ . " .{ $ O $. 5F 0 0- 5& S& $ 6 $. N# &0 y R# 0&0 * & y . " R# *$S 3 . $0 ( )C $& . $0 N# A.{ B 5&V C &S V *$$S oL6 A.{ 0 $. X = (x , y ) 3 )? &0 ( . y $ 0 p&\ + , + E ( >.&[, v = −−→ (a, b) 9Z"# (B&' R# .$ *t ∈ R M −X→ + tv !VO : X = X + tv ; t ∈ R

\ )*HJ . B$&U R# *$ O ) GB H*; !VO 0 * & -# I E .&[, -# I MNJ : x = x + at , y = y + bt ; t ∈ R

-# ! )*HJ O&0 T GB&L b a M . T .$ E ( >.&[, :

y−y x−x = a b

$& X &S V &0 y M x= $ R# .$ *O&0 v $& X &S V &0 y ( v O&0 v E v M V# 1 & 1 >. T *O&0 O $ tU 0 X $& . $0 M ( R# ;*H*; q1 E i& N# x , 5 5&+ E T b 0q &0 . x y N# *(=S x= y 0 Y\ 5 . &S V ?U0 *$M

Z#. $ % (B&' 0&6 O&0 T GB&L a b = S A: *h = |ax|a|+ c| *( . 10 h = |ax+ +by + c| = (B&' .$ ]3 a +b

}

0 $

p&\ E .nS y B$&U A *0&0 . p&\ E .nS y E . X = ( , ) Y\ T&= A; *0&0 . X = (, ) X = ( , − ) . . −y a .$ . . −x y S M $ 5&6 A x/a + y/b = 5F &M B$&U &F M eY1 b .$ *$ 0 : : a x + b y + c = S M $ 5&6 A B$&U &F O&0 e@&\ y $ ax + by + c = !VO 0 . $ 5F 1? ! E .nS = y 5 : ax + by + c + λ (a x + by + c) = . p Y , + *( \\' $, N# λ M (O * & y O oL6 y $ i = , , M X = (x , y ) Y\ M $ 5&6 AH 9M U1 y N# 0 1 & 1 X = (, )

X = (

i

x x x

y & Y\ T&= 98*; !VO

, )

i

y y y

B$&U 0 y & X

Y\ T&= A .8 *0&0 . ax + by + c = 0C O&0 y 0 X Y\ # j H ZM x= *!' −−−→ −−−→ d = X H X X `?- da R# .$ *A$ O ) 8*; !VO 9 [& !0&1 X X

=

d = =

, (x − x ) + (y − y ) −−−→ * −→* |X X • v| *projv −− X X * = v |a (x − x ) + b (y − y ) | + a + b

*−−−→* *X X * =

(a, b) &S V X = (x , y ) &# .$ M *( $& v = −−→ 9$ . ( RV+ (B&' (O 5 O&0 T GB&L b a S AGB

i

=

: ax + by + c = p Y M $ 5&6 A8 5.&\ . −y 0 ([" : −ax + by + c = * " : ax + by + c = p Y M $ 5&6 AJ * " 5.&\ . −x 0 ([" : ax − by + c = : ax + by + c = p Y M $ 5&6 A7 * " 5.&\ Ǳ [ 0 ([" : −ax − by + c = y y O & da ('&" M $ 5&6 A &0 0 0 . −y . −x l : ax + by + c = *( c/|ab| y $ 1? ! E M "# 0 . Y B$&U AI Y\ c : x + y = : x − y + = *$.nS X = (, )

= (x , y )

:

y + c/b x− = −b a

−−−−→ a) . v (, −c/b) . X R# 0&0 x= 5 (−b, 9 .$ *$M h

= = =

, X X − d c (−bx + a (y + c/b)) (x ) + y + − b b + a |ax + by + c| + a + b

Z"# 0 Z O&0 T GB&L b a = S A% X = (, −c/b) R# 0&0 l : x = t , y = −−c/b 9 .$ *v = ( −,−→) h = = =

, X X − d c |x | (x ) + y + − b + |by + c| c y + = b |b|

Y\ E M "# 0 . T B$&U A .8 *$.nS C = (, − , ) B = ( , , ) A = ( , , − ) −→ −→ AC AB &. $0 U1 P T 0 C B A 5 Q *!' 7*; !VO 0C * T E R# 0&0 . $ . 1 T 0 X = M ZM x= ![1 &a & Z ]3 *A$ O ) → −→ × AC A 9=@ E *n = −AB n = =

−→ −→ AB × AC = (B − A) × (C − A) −−−−−→ −−−−−−−−→ −−−−−−−−→ (, , ) × (− , −, ) = ( , −, )

E ( >.&[, T B$&U R# 0&0 (x −

) − (y − ) + (z + ) =

*y = x − &#

} x Y\ E .nS T 97*; !VO

0 $ +, T 0 Y S & 0$ E q1 0&0 R# *( $ +, T .$ $ ) p Y e+) 0 y 5F O&0 9Z#S T G#U &[ . X = (x , y , z ) 3 )? &0 P T % & ( @&\ + , + n = −(a,−−b,−→c) . −→ 9U# ( $ +, n 0 −− X X M X = (x, y, z) P : n • (X − X ) =

!VO 0 * & P )*(6= )*HJ . . Mn B$&U

E ( >.&[, B$&U R# &j L

P : a (x − x ) + b (y − y ) + c (z − z ) =

GB J*; !VO 0 *$ O & )(6= ! )*HJ M *$ O ) Y\ E M "# 0−− .− T B$&U A& '() % −−→ *( $ +, n = (, , ) . $0 0 OnS X = ( , , − ) Z#. $ *8*; 0 ) &0 *!' (x −

) + (y − ) + (z + ) =

*x + y = &#

T $ (U- A .8 *M oL6 Z 0 ([" . P : x + y = z + pO *E &# " e@&\ &# &q= .$ T $ *!' B&" R# .$ & *( & F−−& E T−−$−−−E −−−−−→ . T .$ n n *n = (, , − ) n = ( , −−,−→) #$ 5&0 0 *O&0 0 0 5&6&j L ([" M #E

P

: x −

y

+

z

=

=

−

=

−

*U@&\ . Mn T $ R# 0&0 *(" RQ M 0 %

c x = y x + y + z = T 1? ! A *0&0 . x + y + z + = A = (

p&\ E M "# 0 . T B$&U A; *$.nS C = ( , − , ) B = (, , )

, , )

Y\ E M "# 0 . T B$&U A *$.nS P : x − y = z T > E 0

X = ( , , − )

}

G#U G L & . T 9J*; !VO &q= .$ T Y\−−−−E −→M "# 0 . T B$&U A, .8 v = (, , ) x . $0 $ &0 $.nS X = ( , − , ) *$ O ) % J*; !VO 0 *( E w = −( −,−−,−→) v E b . $0 $ 0 n = v × w ).& %qT&' *!' 0 *O&0 $ +, c P $. 0 nB ( $ +, P T E w 9R# 0&0 *#F .&+O 0 T & !B$ R+ i n = v × w =

j

k

⎤

−−−−−−−−→ ⎦ = (− , , − )

P : − (x − ) + (y + ) − (z − ) = : z+x=y+

9>. T 0 5F .&3 >_$&U ( : x = x + at , y = y + bt , z = z + ct ; t ∈ R

E .&[, 5F &M >_$&U O&0

:

y−y z−z x−x = = a b c

>&j L 0 p&\ E .nS y B$&U A& '() & *"# 0 . X = (x , y, z) X = (x , y, z) −−−→ −−−−−−−−−−−−−−−−−−−→ X X = (x − x , y − y , z − z ) &S V . X *!' y R# B$&U ]3 *Z#0 Z $& . $0 . :

x − x y − y z − z = = x − x y − y z − z

*O&0

0 T $ $. 0 E !T&' y &M >_$&U A, .8 . P : x + y + z = P : x − y + z = >_$&U

*#.F (0 . T $ 5F B$&U y E &S V R =&# 0 *!' % ) (#& 0 & $ R# 5 Q [B *ZM !' & $ .$ & $ $M . nS . \ . &v E V# Z ]3 *$. $ z = ZM x= ]3 *ZM [& 5F X"' 0 . #$ 9A$ O ) GB *; !VO 0C R# 0&0

x − y = x + y = ⇒

x y

= ⇒X = =

, ,

. $0 0 v 5F $& . $0 nB ( e1 P T 0 y 5 Q $ +, n 0 v !B$ R+ 0 *( $ +, P T n & $ + x= 5 .$ *( v = n × n = ( , −, ) × (, , ) = (−

x − / y − / z− = = −

%qT&' @ [U AGB 9*; !VO T y R0 # E A% .&M$

x x x x

y y y y

z z = z z x − x = x − x x − x

i

y − y y − y y − y

}

i

z − z z − z z − z

i

i

=

X 0 . z y x . P T S M $ 5&6 AH B$&U &F M eY1 c b a T GB&L . \ .$ *$ 0 x/a + y/b + z/c = >. T 0 P T & (x , y, z) Y\ T&= M $ 5&6 A8 &0 ( 0 0 ax + by + cz + d = h=

x + y

=

|ax + by + cz + d| + a + b + c

z + >_$&U 0 T $ R0 # E *M oL6 . y + z = x +

AJ

T $ T&= A7 *M [& . #V# E a x + b y + c z + d = T $ S M $ 5&6 A T &F O&0 e@&\ ax + b y + cz + d = $.nS $ 5F 1? ! E 1 & 1 P = P 9(O 5 0 λ E 0 M x + y = z + x + y − z =

∗

P : a x + b y + c z + d +λ (a x + b y + c z + d ) =

& # E y E (.$ M "# 0 . T B$&U AI 9$.nS O $ $ T $ O

P : x + y + z = , P :

, , )

9E ( >.&[, y B$&U R# 0&0 :

Y\ E .nS T B$&U M $ 5&6 A 9&0 ( 0 0 i = , , M X = (x , y , z )

x + y + z +

=

2 U# *$$S G#U T .$ y 0 [O ?z &M &q= .$ y *$ O G#U 5F &0 E . $0 N# Y\ N# &0 Y\ .$ 3 )? &0 ( . y & −−−−→ + , + v = (a, b, c) X = (x , y , z ) *( v E −X−−→X . $0 &0 M ( X = (x, y, z) @&\ X = tv M $. $ $ ) t ∈ R U# )*HJ R# 0&0 *−X−−→ >. T 0 -# I\ −→ : X = X + tv ; t ∈ R

h $. T&= &F Z &0 H . 0 X # j S *!' −−→ # cB Zg&1 X X H da 5 Q *( 0 0 −HX e- @ &0 −−−→ −−→ y $& X X R0 α # E / .$ 0 0 −HX *( 9( v h

h

−−−→ ! −−−→ = X X sin ∠ X X , v −−−→ −−−→ X X × v −−−→ X X × v = X X −−−→ = v X X v

9Z#. $ M (B&' .$ R# 0&0 >. T 0 &M >_$&U i& B&" R# .$ 5 Q l : x = t, y = t, z = t +

T&= R# 0&0 *v = (,

, )

X

v *−−−−−−−→ −−−−−→* * * *(, , −) × (, , )* √ + + . *−−−−−−−−→!* * * √ * , − , * =

= =

x + y + z + =

l :

]3 ( 9&0 ( 0 0 h $.

= (, , )

* * ! * * * X − X × v*

d (X , l) =

y # j B$&U A4 .8 x − y + z − =

x + y + z + + λ (x − y + z − ) =

q, 5F $. 0 ! E ( >.&[, P 0 # j 5 M *( ( =&M nB *A% I*; !VO 0C *( $ +, P 0 M P E t\ U 0 R# *ZM t\ . P P 5$ 0 $ +, pO −−−−−−−−−−−−−−−−−−→! P 0 & n = + λ, − λ, + λ . $0 5$ 0 $ +, & ( P 0 & n = −−,−−,−→! . $0 λ

λ

λ

: :

x = y x+y+z =

x−

=

y−

:

=

z− −

x = y x + z =

]3 *Z"#

?U0 * O&0 E v v M #E . T .$ y $ R# * " E ]3 *( yb M = = − U# & $ N# .$ . & F >_$&U & F 5$ 0 e@&\ t\ 0 .$ >_$&U E . t X"' 0 z y x #$&\ *ZM !' 9Z$ . 1 >_$&U

*v

−−−−−−−→ = ( , , −)

t − = t + t − + t +

X

= (, , − ) −−−−−→ v = (, , ) X = (− , , −)

+t−=

⇒ ⇒

*"# 0 . P : x + y + z + = T 0 >&T $ B$&U ( T $ $. 0 ! 5 Q *!' 9>. j0 E OnS Pλ :

*M oL6 Z 0 ([" E 9 " M (U- . $ &q= .$ y $ *!' y $ $& &S V 60 d0 E ![1 *=& e@&\

9>. j0 . 9Z0&# . O $ $

−−−→ X X × v d (X , l) = v

*

x = y x+y +z = >_$&U 0 y (U- A .8 . z = t − y = t + x = t − >_$&U 0 c

t= t =

t=

$ O >& t #$&\ S C U@&\ y $ ]3 Y\ E ( >.&[, & F $. 0 ! A$ 0 =& y *X = (, , ) : x = y + , x = z − y R0 # E A .8 *M [& . P : x + y + z + = T R0 # E #F 0 % *; !VO E M . @ 5&+ *!' # E R# *A U#C P 0 # j R0 # E &0 ( 0 0 P ! & ( T & y $& R0 β # E !+V α ).& %qT&' 0 0 5F $& ]3 ( T $ $. 0 9O&0 T $ 5F &

−−−−−−−→ −−−−−−−→! −−−−−→ v = n × n = (, −, ) × , , − = (, , )

9R# 0&0

λ

n • nλ = ( + λ) + ( − λ) + ( + λ) = − λ +

(0 P B$&U .$ λ . \ 5$ $ . 1 &0 *λ λ

9R# 0&0 *P : x − y +

= P ∩ P

: :

=

#&0 ]3 Z#.F

z − = x + y + z + = x − y + z − = − y + z = − x − y =

∠ (l, P ) = α = = = =

>_$&U 0 y & X

π

π

−β

− ∠ (v, n)

v •n π − arccos vn

π − arccos

= (, , − )

Y\ T&= A .8

*0&0 . : x = y, z = y +

}

Z E y $ T&= AGB 9*; !VO y $ w 6 $ +, A% $ 6 % *; !VO .$ M . @ 5&+ >. T R# .$ &0 R# 0&0 ( ! &O . p Y P T 5 Q ( p Y P *n = v × v .$ ( E & F $ .$ ( E & F $ &0 R# 0&0 ( ! &O . y $ 0 y M R# E *n = v × v ` + .$ ]3 *v = v × v M $$S ( $ +, n

=

=

n

= =

i j k v × v × v = × v i j k −−−−−−−−−−−−→ − = (− , −, ) i j k v × v × v = − −

−

) ∈ ⊂ P

(x − ) − (y − ) + (z −

T B$&U (, , −

) ∈ ⊂ P

5 Q

)=

: :

:

x = t −

:

x=y−

, y = t + , z = =

−t

(z − )

$& X &S V . $ y ZM x= *!' −→ X X . $0 X Y\ 0 X Y\ E *O&0 ZM !j . −− O&0 e1 T N# 0 y $ R# S *AGB *; !VOC &. $0 M " j+ 1 y $ R# ]3 *]VU0 −→ U# *O&0 Y "0 −− X X v v vi

i

i

−−−→ X X , v , v = =

5 Q 0&6 >. T 0 E ( >.&[, P

E ( >.&[, O&0 P P eY\ M .$

X = (− , , ) Z#. $ i jL0 (B&' R# .$ & .$ *v = (−−,−−,−→) X = (, , ) v = −(−−, −−, −−−→) 9

−(x − ) − (y − ) + (z − ) =

y & Y\ T&= AGB 9I*; !VO T 0 y # j A% Q U1 T N# .$ y $ &#F A5 .8

−−−→ X X , v , v =

−−−−−−−−−−−−→ (−, −, )

E ( >.&[, P T B$&U (, ,

}

− (x − ) − (y − ) + (z − ) = −(x − ) − (y − ) + (z − ) =

x + y − z − = x + y − z − =

0 &

> E 0 (, , −) Y\ E M "# 0 . Y B$&U A *$.nS x = y = z − y ( , , −) (, − , ) p&\ E .nS y &#F A; M eY1 . x + y = z + T

$−−−−−−−→ −−−−−−−→ −−−−−→% ( , − , ), (, , − ), ( , , ) ⎤ − − ⎦ = =

* " j+ . Mn y $ ]3

90&0 . O $ $ p Y w 6 $ +, B$&U A0 .8

⎧ ⎨ x = s + : y = s + ⎩ z = s −

;

⎧ ⎨ x = t + : y = t + ⎩ z = t +

( Y E >.&[, y $ w 6 $ +, G#U 0&0 *!' *; !VOC O&0 c $ +, & F 0 $M eY1 . $ M *( y 5F &3 ( O !' B&" ZM x= *A% . $0 X 0 v = −(−,−−,−→) v = −(−,−−,−→) v 9ZM x= $. 0 ! X 0 X X *O&0 y $& y E M O&0 T P *O&0 y &0 y X Y\ y E M O&0 T P *$.nS X Y\ *O&0 P P T & . $0 X 0 n n *$.nS

Y\ E M @&\ + 5&V 9 * cM . (0&f Y\ * & # $ . T&= N# 0 i jL0 ) GB ;*; !VO 0 *Z & # $ `&UO . (0&f $, x= R . `&UO X = (x , y ) . cM >&j L S *$ O . 1 C # $ 0 1 & 1 X = (x, y) Y\ &F ZM 0 &# R = −X−−→X U# $ O R 0 0 X & X T&= M $. $ 9&j L !VO

C : (x − x ) + (y − y ) = R

}

!j= y &0 x + = y−− = z y M M (0&f A x = y + z + x + y − z = T $ w 6

*( E

x− y + z = T $ w O E !T&' y B$&U A *0&0 . x + y = z y $ OnS (−, −, ) Y\ E M 0&0 . Y AH x+ y+ z− : = = 9M eY1 . #E =&

− − * : x − = y + = z−− >_$&U 0 y 0 ([" X = (, −, ) Y\ #1 A8 *0&0 . x + = y − = z 0&0 y $ R0 T&= R# & M [& 0 B = AJ 9#[0 .&V0 . O $ $ y $ $. .$ ]< : x = t − , y = − t , z = −t −

R# VQ M A% # $ G#U AGB 9;*; !VO x Y\ $ E .nS # $

`&UO X = ( , − ) cM 0 # $ A& '() * &# ( (x − ) + (y + ) = B$&U 0 R = x + y − x + y =

# $ N# mU

: x = t − , y = −t + , z = −t +

N N 0 . x −

a x − x x − x

B$&U A, .8 B$&U 0 5F 5$M ! &M e0 &0 #E *(

x + y − x + y + =

R X

A = (

= = , )

d (A, B) =

(A + B) =

*−→* *AB * =

*−−−−−→* *(, −)* =

−−−→ −−−→ (, ) = (, )

Y\ E M "# 0 . # $ B$&U A .8 *$.nS C = (, ) B = (− , )

b y − y y − y

c z − z z − z

=

A E 3**' 4

(x − ) + (y + ) =

Y\ .$ cM &0 R = `&UO 0 # $ B$&U M Z.

*( X = (, −) y .&3 5F Y1 M "# 0 . # $ B$&U A .8 `&UO (B&' R# .$ M ( RO. *( AB = (, ) (, ) Y\ &z\1$ c 5F cM ( B & A T&= Gj 0 0 # $ R .$ *$ O ) % ;*; !VO 0 *( AB y .&3 y X *(x − ) + (y − ) = E ( >.&[, # $ B$&U #E

y− z− = −

y # j A7 *0&0 &j L &T (x , y , z ) Y\ E OnS T B$&U M $ 5&6 A 0 v = (a, b, c) $& (x , y , z ) &S V &0 y 9( #E >. T =

E M . @ 5&+ @L e@&\ &# 4$ ).$ &

$. & pL $. 0 E !T&' #&j #F 0 5&6+ 0 * " T−xy 0 LB$ T N# x + y = z \1$ 5&0 !SV "9 &# Y )% !( * !VO 0 B$&U & 0 ) , +

C : ax + by + cxy + dx + ey + f =

c = l&Q *a + b + c = 5F .$ M ( *Z & + . Z#E $3 $. & & + 0\[@ 0 $ .$ . & S R# M >. T c = pO R O $0 &0 ]< *$ + Z .0

C

` + M @&\ + 5&V 2 $ * . ( 4 U $, 0 0 oL6 Y\ $ & & F !T = $, !; . O oL6 p&\ * & !; (B 0 *AGB *; !VOC & 5F Y . 4 U

B = A = (−c, ) & &M M $M x= 5 d0 .$ h $. q0 C S ]3 *( a (0&f $, (c, ) 9Z#. $ G#U 0&0 &F O&0 e1 C 0 X = (x, y) O&0

# $ B$&U ZM x= *!' M $ O X 0 A, B, C ∈ S x= E *(

S : x + y + ax + by + c =

⎧ + + a + b + c = ⎨ a + b + c = − + − a + b + c = ⇒ −a + b + c = − ⎩ + + a + b + c = ⎩ a + b + c = − ⎧ ⎨

c =

d (X, A) + d (X, B) = a , , (x + c) + y + (x − c) + y = a

b

= −

:

y + = (x − ) + (y − ) = 0 *

. −y &0 ±a 0 0 . −x &0 C $. 0 ! M $ O ) y x 0 h Y1 Z X 0 . b a $, *( ±b 0 0 q0 M (B&' .$ *( (, ) q0 R# cM * &

#E B$&U 0 X = (x , y ) .$ cM &0 b a &Y1 Z &0 9AR#+C ( C :

#$&\ & $ !' E ]3 M

: x + y −

S

x y + = a b

=

9E (".&[, h $. # $ R# 0&0 *$ O !T&'

Z#. $ b = c − a x= &0 5$M $& E ]3 M C :

a

(x − x ) (y − y ) + = a b

}

y (U- A *M oL6 . Z 0 ([" 0 X = (, − ) E M 0&0 . @&\ 5&V A; e1 5F . 0 X = (, ) &#F * " R = T&= ( e1 5F 5.$ X &#F ( X = (, ) .$ 5F cM M "# 0 . # $ B$&U A *( /&+ x + = y y 0 ( 9E (".&[, 5F B$&U M 0&0 . # $ `&UO cM A C : x + y =

# $ l

: x+y =

GB C x + y + x − y = %C q0 G#U AGB 9*; !VO !Y " N# .$ p& q0 A% + C & &M &F c = a − b b < a ZM x= S S M B&' .$ *( a c 5F 4 (x ± c, y ) E .&[, + .&[, C & &M &F c = b − a ZM x= a < b *$ 0 b 5F 4 (x , y ± c) E B$&U 0 N# A& '() % * C :

x + y = x + y +

*M oL6 . 5F ` *( O $ $ B$&U 0 5$M ! &M e0 &0 *!' C :

&Y1 Z X

(x − )

+

(y − )

=

= ( , ) .$ cM &0 q0 N# M Z.

*( b = a =

x + y − y =

# $ A = (

, −)

C : x + y − y =

Y\ (U- AH

*M oL6 . Z 0 ([" C : x + y − x + y = # $ (U- A8 oL6 . Z 0 ([" C : x + y = x + y + *M . R `&UO X = (x , y ) cM 0 # $ M $ 5&6 AJ 9$M .&3 5 !#{ >. j0 *C : x = x + R cos t , y = y + R sin t ; ≤ t < π x y $ 0 /&+ # $ + cM 5&V A7 *0&0 . l : x = y l : x + y = # $ 0 /&+ &# $ + cM 5&V A *0&0 . x = y x + y =

x + y = q0 X

Y\ (U- AJ *M oL6 Z 0 ([" .

x +y + x+

= ( ,− )

q0 x+y = y (U- A7 *M oL6 Z 0 ([" .

= y

C : x + y = M x= A F (x, y) = x + y − , x + y +

!

& &M &0 q0 B$&U A, .8 *"# 0 . 4 0 & &M ]3 ( V# Y\ $ x o L 5 Q *!' + c = b − a a < b (B&' M " (x , y ± c) !VO Z#. $ .$ *( u &M $ y q0 cM 5 Q *( c = − = ?U0 *X = (A + B) = ( , ) b = R# 0&0 *b = √x= t0&Y & *c+= &# >.&[, q0 B$&U ]3 *a = .$ = − a E (

x − x = a (y − y )

x − x = −a (y − y )

y − y = a (x − x )

y − y = −a (x − x )

x −

B

x=x −

,y

x ,y +

x ,y −

,y

x=x +

a

y=y −

a

y=y +

a

a

=

AB =

b

=

BC =

/ . ( /a &M T&= (B&' .& Q .$ V# ^- A( (, ) / . GB *; !VO .$C ( (x , y ) & F 0 RV+ & B&' &6 0 *O&0 ([a $, a *M ) % *; !VO

=

B − A = C − B =

(A + B + C + D) = (, )

q0 B$&U .$

(x − )

+

(y − )

=

A QC *(

a

a

+

90&0 . O $ $ & q0 &M B$&U

a

(y − )

a

X =

a

(x − )

R# 0&0 ( U-.& Q y q0 cM ?U0

&+ 0 $. & !VO .& Q 5 0&6 >. T 0 9$.F (0

x +

, )

/ g. &0 `?-_ E .$ p& q0 B$&U A .8 . D = (, ) C = (, ) B = (, ) A = (, ) *0&0 &F O&0 D C B A X 0 . O $ $ p&\ S *!' 9&0 0 0 &Y1 Z % *; !VO t0&Y

: (p, ) (x, y) = (x − (−p)) , y (x − p) + y = x + p : x = : p

5 &M

A=(

C :

@&\ 5&V + G#U t0&Y O * * T&= 0 0 x y N# & & F T&= M ( T E 5 &M . Y\ B . y *( x Y\ N# & & F &M T&= . 5 &M & B T&= Gj * & + + 5 &M ZM x= d0 .$ (B 0 * &

1 5 M *O&0 x = −p y 5F B (p, ) Y\ & X T&= M ( e1 C + 0 X = (x, y) 1 & GB *; !VO 0C O&0 x = −p & X T&= 0 0 (p, ) 9A$ O )

+ B$&U

, )

!VO &#F *M oL6 F f& E ]3 . C (U- ( q0 !T&'

C

B=(

)

0 & *

x + y =

) x + y = x + y − )

x + y = x + y −

) x + y + x = y −

.

4 ( , ) (, ) & &M &0 q0 B$&U

AH

*0&0 Z X = (x , y ) cM 0 q0 M $ 5&6 A8 9$M .&3 5 #E >. j0 . b a &Y1 C : x = x + a cos t, y = y + b sin t; ≤ t < π

}

. #E &+ E N# &M T&= 5 &M / . 9M oL6

)

x + y = y

)

x=

)

+ y + y

x + y +

) y = x

0 ([" . y = x − + ( . x

= − (y − )

=

Y\ (U- A *M oL6 Z

,− )

+ y = x y (U- AI *M oL6 Z 0 (["

( y = x + C M x= A F (x, y) = (x + y + , x − y)

( + &#F *M Z. . F (C) !VO E&"+ . −y Ǳ [ 0 ([" y = x + ]VU A; *0&0 . 4 e0. M M oL6 C : y − y = P (x − x ) M x= A *M v Q !VO x v &0 M M oL6 C : y − y = P (x − x ) M x= A *M v Q !VO y v &0 M M oL6 C : y − y = P (x − x ) M x= AH *M v Q !VO P v &0 E , + B Bn G#U 0&0 V=VI5 * 0 0 oL6 Y\ $ E & F !T = !-& M ( p&\ . O oL6 p&\ *AH*; !VOC ( oL6 $, * & !HH] Y . h $. (0&f $, L

}

B Bn G#U 9H*; !VO

+ G#U AGB 9*; !VO $. & + ` A% ( ) 0 . ( , − ) .$ / . &0 + A& '() * B$&U . $ / &M T&= &0 . −y

y − (− ) =

− (x − ) ( / )

*( y + = − (x − ) &# ([a ( ) 0 . (, ) Y\ .$ / . &0 + A, .8 B$&U . $ / &M T&= &0 . −x x− =

( /)

(y − )

*( x − = (y − ) &# .$ / . &0 + N# y = x + x A .8 ! &M e0 E ]3 #E ( / &M T&= (− , ) 0 . + R# *y − = (x + ) (O 5 5$M *O&0 . −y ([a ( ) (, ) .$ / . &0 + N# y + x = y A .8 $. $ !#&+ . −x ( ) 0 M ( / &M T&= *x − = − (y − ) (O 5 #E . #E &+ E N# B$&U 0 * 9"# 0 0 ( 5F &M T&= ( (, ) 5F / . A *( . ) . −x ([a (+ 0 ( / 5F &M T&= ( ( , ) 5F / . A; *( . ) . −x (+ 0 ( 5F &M T&= ( (− , ) 5F / . A *( . ) . −y ([a (+ 0 ( / 5F &M T&= ( ( , ) 5F / . A *( . ) . −y (+

&Y1 Z cM RU R+0 * 9M Z. . & F #E &B Bn E N# & [&

) (x − ) −

(y + )

) x − y =

=

) −y = ) (x −

) y − (x −

)

M ( e1 C 0 X = (x, y) 1 & 1 &F ( R# 0&0 (−c, ) (x, y) − (c, ) (x, y) = a , , (x + c) + y − (x − c) + y = a

=

& C W Q+ R OU!U) * x= 5 M *Z $3 4$ ).$ & 0 \[@ 0 &# M O $0 . c = C : ax + by + cxy + dx + ey + f =

a 4 (−c, ) (c, ) & &M &0 B Bn C ZM x= S

A*; C

E ]3 Z$ 5&6 M ( 5F m *Z#S h .$ . N# 0 . C X& 5 .$ _z&+ ' &\ N# *$ + !#[ 5 $. & K+ X Q+ R U!U) : ') $ *

B$&U .$ xy +) X#- 5 &. 5$ $ 5 .$ &0 '+ x= #&0 (B&' R# .$ *$ + T . A*; C 4$ ).$

9$M

α=

arctan

c

a−b

=

arccot

a−b c

T uv X#- Z"# 0 v u X"' 0 . A*; C S &' R# + *$ O S E& & .&M R# 0 *$ 0 & F 5.&\ &. M ( 5F ($ R# E & S# *O&0 &j L &. E

*#0 h .$ . xy = A& '() % * 9Z#. $ H*7*; 0&0 >. T R# .$

π π α = arccot = = ⎧ π √ π ⎪ ⎪ ⎨ x = u cos − v sin = (u − v) √ π π ⎪ (u + v) ⎪ ⎩ y = u sin + v cos =

x= 5$M $& E ]3

x y − = a b

C :

a $ , *( y = ± ab x X& $ . $ . Mn B Bn * & B Bn &Y1 Z . 9Z#. $ B Bn $ $ M (B&' .$ .$ b a &Y1 Z X = (x , y ) cM 0 B Bn AGB 9A$ O ) GB 8*; !VO 0C . −x $

b

(y − y ) (x − x ) − = a b

.$ b a &Y1 Z X = (x , y ) cM 0 B Bn A% 9A$ O ) % 8*; !VO 0C . −y $

(y − y ) (x − x ) − = b a

E .&[, & [& (B&' $ .$

x = u cos α − v sin α , y = u sin α + v cos α

5F .$ M

+ c − a

9Z#. $ b =

b y − y = ± (x − x ) a

Z (

cM 0 B Bn A& '() * 9( #E B$&U 0 . −x $ .$ &Y1 ,− )

(x − )

−

(y + )

=

cM 0 (y − ) / − (x − ) / = B Bn A, .8 *( . −y $ .$ &Y1 Z (, ) #E ( B Bn N# x − y + y = A .8 5 x / − (y − ) / = >. T 0 . 5F B$&U

* " u#&Y1 Z (, ) B Bn R# cM *(O }

0 B Bn N# B$&U M u − v = Z#. $ . n#&) E ]3 T .$ . −v $ .$ &Y1 Z (, ) cM

*A$ O ) J*; !VO 0C O&0 uv

$. & B Bn ` 98*; !VO

}

}

q0 N# 5$ + & 97*; !VO U!U) < Y 1 =!)

+ 4L) 0 *

9>_$&U & $ A*; C 4$ ).$ 0 &

B Bn N# 5$ + & 9J*; !VO Q+ R

*#0 h .$ . x +

xy+y − = A, .8

9>. j# .$

'

ax + cy + d = , by + cx + e =

9>. j# .$ *ZM h . ( & 0 ) E y N# . $ & $ &F = S AGB *( F (0 !VO 5.&\ . y R# M & $ % ) *$. $ % ) N# & $ &F = S A% E V# & $ >_&U E N# *( C B Bn &# q0 cM

*M oL6 . 5.&\ &.

h .$ . xy = B$&U 0 A& '() * B Bn !VO nB ( ([a = / >. j# .$ *#0 ]3 {x = , y = } E (".&[, 0 h & $ *( &. . −y . −x ( (, ) B Bn R# cM

* " 5F 5.&\ *#0 h .$ . x + y = xy + x B$&U 0 A, .8 *( q0 !VO nB ( = −/ >. j# .$ {x − y − = , y − x = } 0 h & $ cM &0 q0 R# ]3 *( (/, /) 5F % ) M ( *O&0 x = y + x = y 5.&\ &. (/, /) . x + y + xy = x + y + B$&U 0 A .8 ( T = − = >. j# .$ *#0 h .$ (".&[, R# 0 h & $ *O&0 + !VO nB (#& 0 . $ M {x + y − = , y + x − = } E . B$&U R# *( x + y = B$&U .$ $&T % ) *O&0 + 5.&\ 5.&\ &. ` $. .$ 0 * 9M oL6 . 5.&\ cM ) x + y = xy + x

√

# √ α=

arctan

"

π

=

=

− ⎧ π π ⎪ ⎨ x = u cos − v sin = π π ⎪ ⎩ y = u sin + v cos =

π

√

u − v

u+

√

v

5$M $& O $ $ B$&U .$ y x . n#&) E ]3 0 q0 N# M Z. u/ + v / = B$&U 0 √ 0C O&0 T−uv .$ &Y1 Z Ǳ [ cM

*A$ O ) 7*; !VO $ $ 4$ ).$ & E N# ) xy =

)

xy +

) =x+y

0 & *

9M & . O

x + y =

√

xy +

√

) x + xy = y +

) x + xy + y = )

x + y = xy +

)

x + y + x = xy + y +

)

xy + y =

Q+ R U!U) Z= 0 Y=):[ * *

([" . = c − ab R[ A*; C 4$ ).$ 0 9Z b i& & B&' E h mT >. j# .$ *Z$

*( + C &F = S AGB *( B Bn C &F > S A% *( q0 C &F < S A:

5.&\ ]#& N# M A = ac bc ZM x= O A # #$&\ β α S 8*7* q1 0&0 *O&0 × 9$ . M ( RV+ (B&' $ &F O&0 0 h # &. $0 >. T R# .$ M α = β AGB V# &U . $0 $ 5 R# 0&0 $ +, Z 0 & F v M (=&# S 0 v = [a , a] v = [a , a ] *O&0 β α 0 h # . $0 X 0 v &U . $0 $ 5 Z E&0 >. T R# .$ M α = β A% M (=&# S 0 v = [a , a] v = [a , a ] V# *$ !V6α 0 h # &. $0 0 #&3 $ M a 5 . B = aa &U ]#& (B&' $ .$ a *B AB = α β B = B >. T R# .$ *$ + x &F y = Xv + Y v $ O x= S .$ t

t

t

−

t

ax + by + cxy =

x y

t

t

t

A

x y

C : αX + βY + γX + ηY + θ =

θ η γ β α G L #$&\ 0 "0 X R# 0 9( #E $. E V# 0 $ 0 ( ?, Z T GB&L β α S A& #

φ := θ −

η γ + β α

φ := θ −

η γ + β α

φ := θ −

) x + xy + y + x = x + y = xy + x + y

)

O $ $ .&3 !VO 0 4$ ).$ N#$. .$ 5.&\ &. ` 5F B$&U : L R+- *( 9M oL6 . 5.&\ cM )

x = sin t − , y = cos t + ,

)

x = t,

)

x = t,

η γ + β α

≤ t < π

y = t − t, + y= − t ,

−∞ < t < ∞

x = − cosh t, y = sinh t, + ) x = t , y = t + ,

−∞ < t < ∞

) )

− ≤t≤

≤t

x = sinh t + , y = cosh t + , −∞ < t < ∞

9M .&3 . #E 4$ ).$ & E N# ) ) )

xy = x

x

− (y − ) = +

y

)

xy + y = x + y

)

x + y + xy = x − y √

) x + xy = y

=

M $ 5&6 A (a x + b y + c ) (a x + b y + c ) = a

"

x= &0 >. T R# .$ O&0 ( ?, Z $ 5F &0 c X"' 0 C B$&U R O v = Y + η/β u = X + γ/α S M Z. αu + βv + φ !VO 0 B$&U 0 v u *( C 4$ ).$ (B&' R# .$ U# *$. 0 ) $ 0 ( ?, Z T GB&L β α S A

#

√

) x + xy = y

"

u = X + γ/α x= &0 >. T R# .$ O&0 T B$&U 0 v u X"' 0 C B$&U R O v = Y + η/β Y\ N# 5F % ) & M Z. αu + βv = !VO 0 C 4$ ).$ (B&' R# .$ U# *u = v = 9( *( Y\ N# $ 0 ( ?, Z T GB&L β α S A, #

xy + y = x + y

)

= αX + βY

!T&' B$&U &F Z"# 0 Y X X"' 0 . C S R# 0&0 9$ 0 & C

) x + y + y = xy + x

"

& [& &0 B Bn N#

a x + b y + c = *ab = ab VF 0 p6 ( &F a b = a b S M $ 5&6 A;I

a x + b y + c =

∗

(a x + b y + c ) = (a x + b y + c ) + a

*( B Bn N#

K+ X Q+ R 5U!U) :DV) *

M ( XY R# >&[f uL0 R# E m

Q+

0 &VO E V# 4$ ).$ * e@&\ y $ + B Bn q0 # $ s 9( #E KO r* , + &# Y\ N# y N# !VO 0 . A*; C B$&U XY R# >&[f 0 "#&

x

y

a c

c b

x y

+ dx + ey + f =

A;*; C

0 v u X"' 0 C B$&U R O v = Y + ηβ u = X , v = ± − u y $ M Z. βv +φ = !VO 0 B$&U

C 4$ ).$ (B&' R# .$ U# *( T−uv .$ *( E y $ E ,&+ ) φ := O&0 T GB&L α β = γ = S A&& 0 [O ?z &M (U- &F O&0 ?UB G L α θ − γ β *( AIC O&0 T GB&L β α = γ = S A&, u = X x= &0 >. T R# .$ φ := θ − η β = 0 B$&U 0 v u X"' 0 C B$&U R O v = Y + ηβ T−uv .$ v = y N# M Z. βv = !VO *( ( . y C 4$ ).$ (B&' R# .$ U# *( O&0 T GB&L α β = γ = S A&

*( A;C 0 [O ?z &M (U- &F φ := θ − γα = C >. T R# .$ θ = α = β = γ = S A& 2 *( 4&+ 5&0 X R# 0 *( .$ . x + y = xy + x A& '() * 9(O 5 *#0 h φ β

x y

−

−

x y

9]#& R# jL6 B$&U *A = = |λI − A| = λ −

−x=

−

−

.$

= λ − λ −

λ−

. $0 *β = α = − E .&[, 5F # #$&\ ]3 *( E (".&[, α = − 0 h # V# v =

$√

%t

√

/, /

E (".&[, c β = 0 h # V# . $0 v =

$√

%t

√

/, − /

9ZM x= R# 0&0

x y

⎡ √ ⎢ = Xv + Y v = ⎢ ⎣

. 1 E ]3 M y =

√

√

X + Y √ √ X − Y

⎤ ⎥ ⎥ ⎦

√

(X − Y ) x = (X + Y ) R# 0&0

X − Y −

√

Z#. $ O $ $ B$&U .$ 5$ $ √

X − Y

=

x= &0 >. T R# .$ O&0 ?UB G L & F &0 u X"' 0 C B$&U R O v = Y +η/β u = X +γ/α q0 N# M Z. αφ u + βφ v = !VO 0 B$&U 0 v C 4$ ).$ (B&' R# .$ U# *( T−uv .$ *( q0 N# $ 0 ?UB G L T GB&L β α S A #

φ := θ −

η γ + β α

"

u = X + γ/α x= &0 >. T R# .$ O&0 T 0 B$&U 0 v u X"' 0 C B$&U R O v = Y + η/β α β α v = ± u !VO 0 . 5F M Z. u + v = !VO β φ φ *( T−uv .$ e@&\ y $ E >.&[, (O 5

y $ E ,&+ ) C 4$ ).$ (B&' R# .$ U# *( e&\

$ 0 ?UB G L T GB&L β α S A #

φ := θ −

η γ + β α

"

u = X + γ/α x= &0 >. T R# .$ O&0 T GB&L

B$&U 0 v u X"' 0 C B$&U R O v = Y + η/β !VO 0 . 5F M Z. αφ u + βφ v + = !VO 0 T−uv .$ B Bn N# M (O 5 Au − Bv = ± B Bn N# C 4$ ).$ (B&' R# .$ U# *( *( >. T R# .$ O&0 T GB&L γ β α = S A4 R O v = Y + η/β u = X φ := θ − η /β x= &0 γu + βv + φ = !VO 0 B$&U 0 v u X"' 0 C B$&U

M (O 5 u = − φγ − βγ v !VO 0 . 5F M Z.

(B&' R# .$ U# *( T−uv .$ + N# *( + N# C 4$ ).$ (U- &F O&0 T GB&L γ α β = S A5 *( A8C 0 [O ?z &M η ( ?, Z φ := θ − β β = α = γ = S A0 v = Y + ηβ u = X x= &0 >. T R# .$ O&0 β &0 βv + φ = !VO 0 B$&U 0 v u X"' 0 C B$&U R O (B&' R# .$ U# *$. 0 ) S M Z.

*( C 4$ ).$ γ φ := θ − O&0 T GB&L α β = γ = S A2 α *( A7C 0 [O ?z &M (U- &F O&0 α &0 ( ?, Z O&0 T GB&L β α = γ = S A&^ x= &0 >. T R# .$ O&0 ?UB G L β φ := θ − η β

, + N# 5$ + .&3 E . h *( e0& Q &# N# $0 >. T 0 , + 5F 5&0 # $ M !VO 0 ) &0

9

C : (x − x ) + (y − y ) = R

.$ *y − y

x − x = R cos t $ + x= 5

9(O 5 #E e0& $0 >. T 0 . C

= R sin t

−−−−−−−−−−−−−−−−−−−→! r(t) = x + R cos t, y + R sin t ;

≤ t ≤ π

q0 M !VO 0 ) &0 C :

.$ *y − y

2

(x − x ) (y − y ) + = a b

√

.$ *v = Y + u = X − ZM x= 5 M M /u − v/ = &# u − v = 9(O $ Z *( T−uv .$ B Bn N# B$&U

h .$ . x = y + xy + y + A, .8 9(O 5 *#0

x

y

Zg&1 + M !VO 0 ) &0

O

C : y − y = a(x − x )

$0 >. T 0 . C .$ x − x

$ + x= 5

9(O 5 #E e0&

\= + 0 0&6 >. T 0 C : x − x = b(y − y )

$0 >. T 0 . C .$ y − y

$ + x= 5

9(O 5 #E e0&

=t

−−−−−−−−→! r(t) = x + bt , t ; −∞ < t < ∞

\= B Bn M !VO 0 ) &0 C :

V=VI5 $

(x − x ) (y − y ) − = a b

− −

x y

λ−

y − =

−

λ+

−

− −

.$

= λ −

. $0 *β = − α = E .&[, 5F # #$&\ ]3 *( E (".&[, α = 0 h # V# %t $ √ √ v = − /, /

E (".&[, c β = − 0 h # V# . $0 v =

$√

√

%t

/, / ⎡

√

9ZM x= R# 0&0 √

⎤

X + Y ⎢ − ⎥ x ⎢ ⎥ √ √ = Xv + Y v = ⎣ y X + Y ⎦ √ √ E ]3 M y = (X + Y ) x = (−X + Y ) R# 0&0

Z#. $ O $ $ B$&U .$ 5$ $ . 1 √

√

X = y + X + Y

=t

−−−−−−−−→! r(t) = t, y + at ; −∞ < t < ∞

−

= |λI − A| =

x − x = a cos t $ + x= 5

9(O 5 #E e0& $0 >. T 0 . C ≤ t ≤ π

9]#& R# jL6 B$&U *A =

= b sin t

−−−−−−−−−−−−−−−−−−→! r(t) = x + a cos t, y + b sin t ;

√

√

+

√

*v = Y − / u = X − / ZM x= 5 M y $ B$&U M v = ±u &# u = v 9(O $ Z .$ *( T−uv .$ e@&\

& E N# $. .$ . _&0 . 0 $ * 9M ) #E 4$ ).$ ) xy =

)

x − y + xy =

√

) x + xy = y )

xy + y = x + y

A E 3**' )G>F /' 7 (6 !j= UB&Y 0 4E_ E $&# uL0 R# E m 4& 5F & . uL0 R# UB&Y *( !j= c *M G1

}

.$ *y − y = b sinh t x − x = a cosh t $ + x= 5

9(O 5 #E e0& $ $0 `&+ ) >. T 0 . C r(t) = h(t) =

. c b a $ , cM . X = (x , y , z ) *( % *; !VO 0C & S 5 Sq0 &Y1 Z X 0 (x , y ± b, z ) (x ± a, y , z ) Y\ uO *A$ O ) *U1 5 Sq0 ^Y 0 (x , y , z ± c) T .$ q0 N# M x= -+ \) .$ T 5F 0 e1 b X Y\ T−xy &0 E

p&\ E .nS p Y 0 e1 p&\ 5&V *Z#. $ .& B$&U X .$ / . &0 ( @L X Y\ c q0 5F !VO 0 $ ).$ #. RQ

S :

(z − z ) (x − x ) (y − y ) = + c a b

pL 5.&\ . *A$ O ) GB ;I*; !VO 0C ( 0 &" m@ S M ( RO. *( . −z E

0 S ( . −y > E 0 5.&\ . O&0 y X"' R# 0&0 *( . −x E 5F 5.&\ . O&0 x X"' *$. $ $ ) & pL ` &zU+) eY1 pL &0 . T S O S ?z[1 M . Y&+ Az = x + y pL &0 . z = ax + b T ?za C Z$ !B$ R+ 0 *( 4$ ).$ N# F (0

* & @L e@&\ . 4$ ).$ &

4$ ).$ #. ]^3 Y=GV=VI5 $

S :

(x − x ) (y − y ) (z − z ) + − = a b c X = (x , y , z )

5.&\ . cM &0 Q.&
Q.&
>.&[, E ![1 . ( =&M . h R# 0 *$ + G#U *Z#[0 x &# y

≤ t ≤ π ≤ t ≤ π

Zg&1 B Bn 0&6 >. T 0

5 Sq0 M 9*; !VO &0 D$. 0 M #&q= !VO Y=G2 . O&0 q0 N# &# Y\ N# LB$ T 4$ ).$ #. R# M B$&U * & 5 Sq0 (x − x ) (y − y ) (z − z ) S : + + = a b c

−−−−−−−−−−−−−−−−−−−−→! x + a cosh t, y + b sinh t ; −−−−−−−−−−−−−−−−−−−−→! x − a cosh t, y + b sinh t ;

C : −

(x − x ) (y − y ) + = a b

9(O 5 #E e0& $ $0 `&+ ) !VO 0 . r(t) = h(t) =

−−−−−−−−−−−−−−−−−−−−→! x + a sinh t, y + b cosh t ; −−−−−−−−−−−−−−−−−−−−→! x + a sinh t, y − b cosh t ;

≤ t ≤ π ≤ t ≤ π

A DE H : ).$ #. . U0 (B&' 0 4$ ).$ & Z+U E *( Ǳ&O R# 0 $ uL0 R# E m * & 4$ *$ O $& .&"0 $ .$ &#. R# Q B$&U N# & 0 ) , + &q= .$ z y x X"' 0 4$ ).$ +) S

: a x + a y + a z + a xy +a xz + a yz

A*; C

+a x + a y + a z + a =

Zg S & . T .$ . #. * & 4$ ).$ #. . (+"1 .$ *O&0 T 5F .$ yz xzxy >?+) X# - M $ $ 5&6 A*; C x #. E& & D. *I*; ZM x= d0 .$ (B $&# 0 ?zU= *O $ 0 $ .$ *a = a = a = 9O&0 & S #. *Z#E $3 4$ ).$ &#. 0 + , + R `&UO X .$ cM 0 M 9L B$&U * " R T&= 0 X Y\ E M ( &q= E @&\ !VO 0 $ ).$ #. RQ

S : (x − x ) + (y − y ) + (z − z ) = R

0C O&0 5F 5.&\ Y\ X

M cM *( *A$ O ) GB *; !VO

= (x , y , z )

0 . S $ ZM x , . ( . (+ .$ &) S *( *O . −x

4$ ).$ #. S : −

]_ + Y=GV=VI5 %

(x − x ) (y − y ) (z − z ) − + = a b c

5.&\ . X = (x , y , z ) cM &0 Q.&3 $ 5 SB Bn . &0 *A$ O ) GB ;*; !VO 0C & . −z E

&0 Q.&3$ & S B Bn 5 ( ?, 5&V 5$M x , *$ + G#U . . −y &# . −x E .

}

Q.&3 $ 5 SB Bn AGB 9;*; !VO q0 5 S+ A% . −x S 5.&\ . ZM x , . x z u\ S uO R# 0&0 *( (.$ . −y 0 XY R+ *O GB ;;*; !VO 0C $. $ $ ) & B Bn 5 S + ` *A$ O ) }

}

pL AGB 9;I*; !VO Q.&
( $ ).$ #. =2 Y=GO & Y\ N D. 0 $ +, >&T &0 5F e@&\ M + D. E >&T &0 5F e@&\ ( q0 &# X = Y\ .$ / . &0 & q0 5 S+ * + ([a ( ) 0 . . −z E 5.&\ . (x , y , z ) 9B$&U 0 . −z S : z−z =

B Bn 5 S+ AGB 9;;*; !VO q0 A% B$&U S =2 =C S :

(x − x ) (y − y ) + = a b

.$ cM &0 q0 N# &3 ZM !' T .$ . R# 0 (x, y, ) Y\ S & *$ 0 b a &Y1 Z ( LB$ z M (y, x, z) p&\ + &F O&0 e1 q0 q0 g&1 &0 N# S X R# 0 *. $ . 1 S 0 S *A$ O ) % ;;*; !VO 0C ( . −z > E 0 ZM x , z &0 . y &) U# O&[ y B$&U .$ l&Q ` U# * F (0 . −y &0 E *Z#. $ q0 (x , y )

(x − x ) (y − y ) + a b

( ) 0 . 5 S+ ( . (+ ( ?, # U &0 *( ).$ >.&[, V# &0 S *Q . −z

.& .$ y X"' 0 N# ).$ .&[, ZO&0 O $ z X"' 0 XY R+ *$ 0 . −y > E 0 5 S+ S O&0 5 S+ ` uO &zU+) .$ *( (.$ . −x 0 *A$ O ) % ;*; !VO 0C $. $ $ ) & q0 ( $ ).$ #. =V=VI5 Y=GO * O&0 B Bn D. 0 $ +, >&T &0 5F e@&\ M * + + D. &0 E >&T &0 5F e@&\ X = (x , y , z ) Y\ .$ / . &0 & q0 5 S+ 0 . −y > E 0 $ . −z E 5.&\ . &0 9B$&U

S : z−z =

(x − x ) (y − y ) − a b

4$ ).$ #.

m Q+ R 5+ : ') LB$ 4$ ).$ #. N# B$&U E& & uL0 R# E !VO 0 X& >&j L v N# %&L U# ( ⎧ ⎨ x = α u + α v + α w + α y = α u + α v + α w + α ⎩ z = α u + α v + α w + α

$. & B$&U N# 0 . A*; C B$&U 0 M 0 vw uw uv >?+) X#- 5F .$ M B$&U U#C >&j L v MC >&j L v ` R# *M !#[ AT V0 $+ v . #. M !VO A O & Y *$$S !VO R =&# ]& &# /&VU 5 .$ &\ d,&0 . A = [a ] 5.&\ × ]#& A*; C B$&U 0 ?za a = a ]#& R# .$ M $ O ) C Z$ ([" Z"# 5F &0 M ( a xy 0 0 xy +) X#. \ . $ A ]#& 8*7* q1 0&0 *Aaxy+a yx # &. $0 M Z $ *( A. V &0 _z&+ ' C \\' # # . \ S ?U0 $ +, Z 0 >& # #$&\ 0 h 0 #&3 5 . + $ O . V .&0 N# E u0 T jL0 ZV' .$ *$ + %&L 5F 0 h # &. $0 &q= 9Z#. $ . #E B$&U 0 h ]#& A ZM x= A 0 h # #$&\ λ ≤ λ ≤ λ O&0 A*; C 4$ ).$ . @ v , v, v &,( &. $0 >. T R# .$ *O&0 *( λ 0 h # . $0 v M . $ $ ) q1 &+ & v & λ ZM x= $ O&0 *I*; ij

ij

ji

i

i

=V=VI5 =C

(x − x ) (y − y ) − = a b

S :

T .$ . B$&U R# S * & B Bn . R# ( =&M S Z 0 $ 0 B Bn ZM Z. . &) S *Z$ (M' & z . $ .$ . B Bn . −y $ .$ &0 !VO $ ZM x , S B$&U .$ $ .$ B Bn S U# *$ 0 . −x ;*; !VO 0C ( . −y $ .$ $ &0 . −z & B Bn ` uO &zU+) R# 0&0 *A$ O ) GB *$. $ $ ) B$&U 0 4$ ).$ #. =O =C S : y − y = a (x − x )

( ) . −z > E 0 + . a > M S *A$ O ) % ;*; !VO 0C Z & . −x ([a

. −x ( ) 0 . + &F a < > E 0 + ZM x , z &0 . y u\ S *(O $ ` uO &zU+) R# 0&0 * F (0 . −y *$. $ $ ) & +

i

i

t x y z = Xv + Y v + Zv

X"' 0 #$&\ z y x &0 A*; C .$ S >. T R# .$ U# *$ 0 & !T&' B$&U Z$ . 1 . Z Y X !VO 0 λ X + λ Y + λ Z + αX + βY + γZ + η =

v N# N+M 0 4$ ).$ #. % 9( #E &#. E V# 0 !#[ !0&1 Y >&j L

Q.&3$ 5 SB Bn Q.&
e@&\ T $ B Bn + q0 E y $ e@&\ y $ E T $ T N# *rY\ N# &# Y\ $ y N#

}

B Bn AGB 9;*; !VO + A% #E 4$ ).$ &#. E N# ` 0 9M Z. . & F $ + oL6 . ) x + y + z =

)

x + y − z =

)

x = z − y

)

x = y + z

)

y = z

)

z = y −

)

x + z = y + y +

x + y = z ) z = − x + y

)

) x + y =

R# 0&0 *( λ − λ = 0 0 ]#& R# " B$&U

&. $0 *λ = λ = λ = E .&[, A # #$&\

v = [, , ] v = [ , , ] E .&[, 0 h # E ( >.&[, c 0 h # . $0 $ +, Z 0 M ZM x= 5 M *v = [− , , ] t

t

t

4$ ).$ #. A& S :

x + y + z + yz = xz + xy +

9E (".&[, S 0 h A ]#& *#0 h .$ . ⎡

)

x + y − xy + xz + yz =

)

x − y + z + x + y − z =

)

(x + y) + y + z =

)

x − y + xz + x − z + =

)

xy + xz + yz =

)

⎦

0 0 5.&\ ]#& R# " B$&U

λ = λ = E .&[, A # #$&\ R# 0&0 *( v = [, , − ] E .&[, 0 h # &. $0 *λ = 0 h # . $0 $ +, Z 0 M v = [, , ] ZM x= 5 M *v = [− , , ] E ( >.&[, c x = Y − Z >. T R# .$ [x, y, z] = Xv + Y v + Zv R# . c#&) E ]3 *z = −X + Y + Z y = X + Y + Z M *Z. X + Y + Z = 0 S B$&U .$ #$&\

*( &q=−XY Z .$ 5 Sq0 N# 4$ ).$ #. A, .8 t

t

t

t

⎡

=

yz + x = z =

(6 !j= UB&Y 0 4E_ E $&# uL0 R# E m 4& 5F & . uL0 R# UB&Y *( !j= c *M G1

, + N# 5$ + .&3 E . h *( e0& Q &# N# $0 >. T 0 , + 5F 5&0 M M !VO 0 ) &0 9L S : (x − x ) + (y − y ) + (z − z ) = R

= R sin t cos s $ + x= 5

$0 >. T 0 . S .$ *z − z = R cos t R sin t sin s 9(O 5 #E e0& = x − x

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→! r(s, t) = x + R sin t cos s, y + R sin t sin s, z + R cos t

≤ t ≤ π, ≤ s ≤ π

⎤

i

j

9E (".&[, S 0 h A ]#& *#0 h .$ .

A E H )G>F /'

y − y

−

λ − λ + λ − =

A=⎣

) x + y + z + xy + xz −

−

S : y + xy − xz − yz − x + y − z − =

xy − yz = x

) x − y + z + xz + x − y − z +

A=⎣ − −

[x, y, z]t = Xv + Y v + Zv

E ]3 *z = X + Z y = Y x = X − Z >. T R# .$ R# *Z. Z = 0 S B$&U .$ #$&\ R# . c#&) E ,&+ ) M (O 5 Z = ± / >. T 0 . B$&U

*( T−XY E T $ . #E 4$ ).$ &#. E N# 0 * 9M oL6 F ` $.F.$ & !VO 0

'() &

− −

⎤ − − ⎦

0 0 5.&\ ]#& R# " B$&U

λ = λ = − E .&[, A # #$&\ R# 0&0 *( 9&0 0 0 X 0 R# 0 h # &. $0 *λ = *v = [− , − , ]t v = [− , , ]t v = [ , , ]t λ − λ −

λ

=

R# .$ [x, y, z] = Xv + Y v + Zv ZM x= 5 M *z = X + Y + Z y = Y − Z x = X − Y − Z >. T 0 S B$&U .$ #$&\ R# . c#&) E ]3 t

Z − X − X + Y

−Z − =

>. T 0 . B$&U R# *Z.

Y −

=

(X + /) (Z − / ) − (/) (/)

. −Y & . .$ B Bn 5 S+ N# M (O 5

*( h .$ . S : x + z + xz + 4$ ).$ #. A .8 9E (".&[, S 0 h A ]#& *#0 ⎡

A=⎣

−

−

⎤ ⎦

pL M !VO 0 ) &0 S :

(x − x ) (y − y ) (z − z ) + = a b c

&F z − z

*y − y

-+ \) %

= cs

5 Sq0 M !VO 0 ) &0 S :

ZM x= l&Q

(x − x ) (y − y ) (z − z ) + + = a b c

$ + x= 5

(x − x ) (y − y ) + = s a b

x − x = as cos t ZM x= R# 0&0 9(O 5 #E e0& $0 >. T 0 . S .$

= bs sin t

−−−−−−−−−−−−−−−−−−−−−−−−−−→! r(s, t) = x + as cos t, y + bs sin t, z + cs

≤ t ≤ π, ≤ s ≤ ∞

M !VO 0 ) &0 S :

*y − y

= a sin t cos s,

y−y

= b sin t sin s,

z−z

= c cos t.

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→! r(s, t) = x + a sin t cos s, y + b sin t sin s, z + c cos t

q0 5 S+

(x − x ) z−z (y − y ) = + c a b

&F

x−x

9(O 5 #E e0& $0 >. T 0 . S .$ ≤ t ≤ π, ≤ s ≤ π

=2 Y=GO &

z − z = cs

Y=G2

M !VO 0 ) &0

ZM x= l&Q S :

(x − x ) (y − y ) + = s a b

]^3 Y=GV=VI5

Q.&
(x − x ) (y − y ) (z − z ) + − = a b c

$ + x= 5

x − x = as cos t ZM x= R# 0&0 9(O 5 #E e0& $0 >. T 0 . S .$

= bs sin t

−−−−−−−−−−−−−−−−−−−−−−−−−−−→! r(s, t) = x + as cos t, y + bs sin t, z + cs

≤ t ≤ π, ≤ s ≤ ∞

M !VO 0 ) &0 S :

x−x

=

a cosh t cos s,

y−y

=

b cosh t sin s,

z−z

=

c sinh t.

9(O 5 #E e0& $0 >. T 0 . S .$

=V=VI5 Y=GO *

B Bn 5 S+

(x − x ) z−z (y − y ) = − c a b

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→! r(s, t) = x + a cosh t cos s, y + b cosh t sin s, z + c sinh t −∞ ≤ t ≤ ∞,

y − y = b(s − t) x − x = a(s + t) ZM x= l&Q e0& $0 `&+ ) >. T 0 . S .$ *z − z = cst &F 9(O 5 #E −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→! r(s, t) = x + a(s + t), y + b(s − t), z + cst)

M !VO 0 ) &0 S : −

≤ s ≤ π

]_ + Y=GV=VI5 $

Q.&3 $ 5 SB Bn

(x − x ) (y − y ) (z − z ) − + = a b c

$ + x= 5

−∞ ≤ t ≤ ∞, −∞ ≤ s ≤ ∞

M !VO 0 ) &0 S :

=2 =C

q0

(x − x ) (y − y ) + = a b

*z = s y − y = b sin t x − x = a cos t $ + x= 5

9(O 5 #E e0& $0 >. T 0 . S .$ r(s, t)

=

−−−−−−−−−−−−−−−−−−−−→! x + a cos t, y + b sin t, s

≤ t ≤ π,

−∞ < s < ∞

x−x

= a sinh t cos s,

y−y

= b sinh t sin s,

z−z

= c cos t.

9(O 5 #E e0& $ $0 `&+ ) >. T 0 . S .$ −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→! r(s, t) = x + a sinh t cos s, y + b sinh t sin s, z + c cosh t −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→! h(s, t) = x + a sinh t cos s, y − b sinh t sin s, z − c cosh t −∞ ≤ t ≤ ∞,

≤ s ≤ π

!<

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . u @ !<

evalm(a ∗ u) −−−−→ . . . . . . . . . . . . . . . . u .$ a %qT&' !<

dotprod(u, v) −−−−→ . . . . . . . . v .$ u $ %qT&' !<

crossprod(u, v) −−−−→ . . . . . v .$ u ).& %qT&' 0 #E &0 E B 2 2 2 =P 9$ + $& 5 v . $0 0 u . $0 # j [&

norm(u) −−−−→

proj := proc(u, v) dotprod(u, v), dotprod(v, v) ∗ v end;

. $ E ( =&M v 0 u . $0 # j [& 0 5 M *$ O $& proj(u,v) R0 # E [& 0 2 + 02 +: *$ O $& angle(u,v) . $ E v u &. $0 ZM $ $. 1 ?1 + 02 E#+ 2 .$ P:=[a,b,c] >. T 0 . (a,b,c) >&j L 0 P Y\ M &0 . Q P Y\ $ R0 !T −→ P Q . $0 *ZM $. !< y

*$.F (0 5 evalm(vector(P)-vector(Q)) . $ D. &+, 0 78 QG K+ $ *** v_2 v_1 x &. $0 , + 0 (+O 4 S $& GramSchmidt([v_1,v_2,...,v_n]) . $ E v_n *ZM

. c= 4 "0 E -&' uL0 5&#&3 & U0 0 R# E ( =&M 5F 5$ 0 / $ .$ 0 *ZM $& Geom3d *ZM ) !< y .$ . with(geom3d) . $ >&j L 0 P Y\ 5$ + $. 0 ?1 % point(P,[a,b,c]) . $ E !< y .$ (a,b,c) ?U0 *ZM $& !<

coordinates(P) −−−−→ . . . . . . . . . . . . . P Y\ >&j L

!<

xcoord(P) −−−−→ . . . . . . . . . . . . . . . . . . P Y\ x o L

!<

ycoord(P) −−−−→ . . . . . . . . . . . . . . . . . . P Y\ y o L

!<

zcoord(P) −−−−→ . . . . . . . . . . . . . . . . . . P Y\ z o L

E !< y .$ l y 5$ + $. 0 & 9$ + $& 5 #E > . $ E V# line(l,[P1,P2]) . . . . . . . . . . . P2 P1 p&\ E .nS y line(l,[P,v]) . . . . . . . . . . P &S [V v $& . $0 &0 y line(l,[p1,p2]) . . . . . p2 p1 T $ 1? ! y

M !VO 0 ) &0 S :

=V=VI5 =C

B Bn

(x − x ) (y − y ) − = a b

y − y = b sinh t x − x = ±a cosh t $ + x= 5

5 #E e0& $ $0 `&+ ) >. T 0 . S .$ *z = s 9(O r(s, t) = h(s, t) =

−−−−−−−−−−−−−−−−−−−−−−→! x + a cosh t, y + b sinh t, s −−−−−−−−−−−−−−−−−−−−−−→! x − a cosh t, y + b sinh t, s −∞ ≤ t ≤ ∞, −∞ < s < ∞

M !VO 0 ) &0

=O =C

+

S : y − y = a(x − x )

.$ *z = s y − y = at x − x = t $ + x= 5

9(O 5 #E e0& $0 >. T 0 . S r(s, t) =

−−−−−−−−−−−−−−→! x + t, y + at , s −∞ ≤ t ≤ ∞, −∞ < s < ∞

58' 9 / ;H7 T 0 !< . c= 4 E $& >& \ &6 0 *$ O U) doc.pdf !#&= &# 0 ;*I* .$ & Y[ 2 '6 + 2 >. T !< −−−−→! vector([a, b, c]) −−−−→ a, b, c . $0 4 > . $ *$ + G#U !< y .$ . . $0 5

4 "0 .$ &. $0 ).& %- $ %- ` +

5 with(linalg) . $ &0 M $. $ . 1 linalg . c= #E !VO 0 . &. $0 0 &+, *$ + $& F h=&' .$ . 5F .$ O&0 $, a . $0 v u M x= 9$ $ 4& 5

>. T R# !<

evalm(u − v) −−−−→ . . . . . . . . . . . . . . . . . . . . u E v !-& !<

evalm(u + v) −−−−→ . . . . . . . . . . . . . . . . . . . . v u ` +

A M x= /5 : Ǳ8 + # ǱO $ R# T&= >. T R# .$ O&0 T &# y Y\ *$.F (0 5 distance(A,B) . $ &0 . Z E B

y B A M x= Ǳ8 + 02 +: . $ &0 . ǱO $ R# R0 # E >. T R# .$ O&0 T &# *$.F (0 5 FindAngle(A,B) x= B Ǳ8 2 Ǳ8 < =P # j >. T R# .$ O&0 T &# y Y\ B A M 5 projection(C,A,B) . $ &0 . B ǱO 0 A O C *$.F (0 x= 2 < 9: 2 Ǳ8 < '1C .$ O&0 . $0 N# v O&0 T y Y\ N# A M . $ &0 . v . $0 E &0 A ǱO C =&# &\ >. T R# *$.F (0 5 translation(C,A,v) # /.$F .$

78

http://webpages.iust.ac.ir/m_nadjafikhah/r2.htm

*( O $.F MnB = d'&[ 0 p 0 e0& & B&a

* . . . . . . . . . z0+ct y0+bt x0+at .&3 >_$&U &0 y $ + 5&0 .&3 E b 0 & O. E V# 0 . l y l&Q . 5F .&3 !VO Equation(l) . $ &0 O&0 FixedVector(L) ParallelVector(l) > . $ *0&0 *. $ 4?, . l 0 &S V N# $& . $0 N# X 0 y .$ P T 5$ + $. 0 !"# * 9$ + $& 5 #E > . $ E V# E !<

plane(P,[X,Y,Z]) . . . . . . . Z Y X p&\ E .nS T plane(P,[X,v]) . . . . . . X &S V v & . $0 &0 T plane(P,[l1,l2]) . . . . . . l2 l1 y $ E .nS T plane(P,eq) . . . . . . . . . . . . . . . . . . . . . . . eq B$&U 0 T $ + 5&0 $. & b & O. E V# 0 . P T l&Q . 5F $. & !VO Equation(P) . $ &0 O&0 4?, . P T & . $0 NormalVector(P) . $ *0&0 *$. $

y Y\ B A M x= Ǳ8 + =2 . ǱO $ R# C $. 0 ! >. T R# .$ O&0 T &# *$.F (0 5 intersection(C,A,B) . $ &0 line(l,[a*t+x0,b*t+y0,c*t+z0],t) . . . . . . . . . . . . . . .

0 0 Rr := &r(t) t ∈ Dr ' 5F $0 Dr = [−

. $0

5&0 R+- *( R 0 R E e0 &0 #&OF !j= R# E m RU & t 6 ' U#C e0 S R# cB&F . N#c= cB&F .$ e0& R# $0.&M ARU *$ + Z KY

.&3 . $0 B$&U Rr .$ O! KY B$&U & *( $& . $0 X = , , &SV &0 ( . y $ m@ $ E t .&3 5 Q *O&0 v = −−−, −−−−,−→! E y .&3 U# *O&0 y .&3 N# r(t) $0 ]3 ( 0C r ! = − −−, −−−−,−−→! & r − ! = −−−−−−, −−, −−−→! *A$ O ) * !VO }

IG& JK FL $

; ]

−−−−−−−−−−−−−−−−−−→! + t, − t, t + − ≤ t ≤ = −−−−−→! −−−−−−−→! = , , + t , −, − ≤ t ≤

RU & t 6 ' E ( >.&[, uL0 R# ` -

$. .$ B&" M $ $ Z 5&6 . $0 e0 RU N# M $$S !V6 $& B&" Q E . $0 e0& N# -&#. .$ d0 $. e0 C R 0 R E e0 i j .$ & F E *O&0 AN#

+, !VO 0 ( U0& R r E 0 &F Z &0 Dr . r $ S *r : R −→ R 9(O 5 R# 0&0 r(t) ∈ R t ∈ Dr −−−−−−−−−−−→! r(t) = x(t), y(t), z(t) = x(t)i + y(t)j + z(t)k

( . $0 e0& N# $0 y .&3 9* !VO M x= A, .8 r(t) =

e0 R# * & r(t) )6H* . z(t) y(t) x(t) e0 !0&1 R &# R .$ . $0 e0& 0&6 >. j0 * " R 0 R E *O&0 LB$ U[@ $, n M ( G#U e0 $ w O 0 0 . $0 e0& $ M $ O ) *Dr = D ∩ D ∩ D 9( 5F B

9M x= A& '() n

−−−−−−−−−−−−−−−−→! + sin t, cos t, ; −π ≤ t ≤ π

9(O 5 >. T R# .$ r(t) =

! + sin t i +

sin t 5F B e0 U# [−π; π] 0 0 r $ M $$S h'? *O&0 z(t) = *z(t) = cos t = y(t) sin t = x(t)− ! ?U0 ( y(t) =

cos t x(t)

x

! cos t j + k

=

+

!j= 0 r(t) $0 .$

cos t + sin t =

r(t) =

y

z

−−−−−−−−−−−−−−−−−−→! + t, − t, t + ; − ≤ t ≤

x(t) = t + 5F B e0 >. T R# .$ e0& R# $ * " z(t) = t + y(t) = −t +

=@ E

8

9>. T R# .$ lim r(t)

=

t→

()

=

−−−−−−−−−−−−−−−−−−−−−−−−−→ " # et − sin t , lim cos t, lim lim sin t t→ t t→ t→ −−−−−−−−−−−−−−−−→ # " −−−−−→ et , cos( lim ), = (, , ) t→ cos t

(x − )

y

q0 z = T w 6

.$ e1 q0 N# U# A$ O ) ;* !VO 0C *( e1 q0 0 0 T−xy 0 5F # j M ( z = T 9O&0 (x − ) + y =

+

Rr :

*( O $& #F.$ 0 & 3 ,&1 E AC .$ M ZM x= A, .8

=

(x − )

+

y

,z=

=

*A$ O ) ;* !VO 0C

−−−−−−−−−→! −−−−−−−−→! r(t) = t , −t, t + , h(t) = t, t − , t

r(t) × h(t) =

lim

t→−

9>. T R# .$

i j k t −t t+ t t− t −−−−−−−−−−−−−−−−−−−−−−−→! = − t + , −t + t + t, t ! −−−−−−−−−→! r(t) × h(t) = − , , −

9Z#. $ Z\ " [& &0 =@ E

−−−−−→! −−−−−−−−−−→! , , × − , − , − lim r(t) × lim h(t) =

t→−

t→−

=

−−−−−−−−−→! − , , −

(.$ . $0 e0 0 & . \ q1 &#F

A .8 ( −−−−−−−−→! M r(t) = cos t, sin t ZM x= S ?za #E *!' p&\ R0 !T y .&3 0 C = (, ) 5 M * ≤ t ≤ π ! 0 M B&' .$ $. $ . 1 r π! = −−−−−−,−→! r() = −−,−→ yb R# U# *$ O+ r(t) = C t ∈ [, π] E 9Zg 0 M ( ! ! e1 r a r b y .&3 0 C O&0 3 [a; b] 0 r(t) S *r(t) = C M $. $ $ ) t ∈ [a; b] &F O&0 9M [& . #E $' E N# 0 % − −−−−−−−− !−−−−−−−−−−−−→ ) lim t sin πt , t − t + , et t→

−

−−−−−−−−−t−−−−−−→ sin t − t e − cos t ) lim , sin t t→ t ! π! ) lim ti − sin k ) lim i + tj + t k te/t t t→ t→ − −−−−−−→! − − − − − −→! h(t) = t, , t r(t) = t, −t, t

.$ y0 . *2

∈ {−, +,

•

}

M x= A , ×} Rl+ t = a = *M t\ . **

( . $0 e0& N# $0 q0 9;* !VO & .$ . (S 5 . $0 e0 cB&F $. .$ M lF 4&+ 9$ + T? $ O +) N# M x= 2 a2= 4V[ E# .$ *O&0 AR 0 R E U#C B +U e0 i j .$ T& P M ( P (T& . $ r(t) . $0 e0& Zg S . T *O&0 O $ . P (T& 5F r(t) B e0 E N# 9Z#. $ ** M G#U R# E 5 , 0 −−−−−−−−−→! r(t) = −−x(t), y(t), z(t) S

BC =_ + J

&F

−−−−−−−−−−−−−−−−−−−−−→

lim r(t) = lim x(t), lim y(t), lim z(t)

t→

t→

t→

t→

B e0 + M ( 3& t .$ r(t) 1 & 1 2 ∈ {+, −, , ×} a ∈ R S ?U0 *O&0 3 t .$ 5F 9&F •

) lim ar(t) = a lim r(t) t→t

! ) lim r2h = t→t

t→t

! ! lim r(t) 2 lim h(t)

t→t

t→t

ZM x= A&

'() $

−−−−−−−−−−−−−−−−→ " # sin t et − , cos t, r(t) = t t

9Z#. $ B&" >&- 0 ) &0 *!' r(t)

≈ r

!

+ tr

!

+ −−−−−−−−−−−→! = et , cos t, t sin t

t

r +

t

!

r

t→t

t=

−−−−−−−−−−−−−−−−−−→! +t et , − sin t, sin t + t cos t

( 3 t = t .$ r(t) M x= bCc) & ' M ( #n3t 6 t = t Y\ .$ r(t) Zg S . T .$ t 6 . ' R# . \ *O&0 $ ) lim t − t r(t) − r t !! >&[f S$& 0 *Z$ 5&6 r t ! &0 & t = t .$ r(t) −−−−−−−−−→! A** 0&0C &F r(t) = −−x(t), y(t), z(t) S M $ O

t=

t −−−−−−−−−−−−−−−−−−−→!

+ e , − cos t, cos t − t sin t t= t −−t−−−−−−−−−−−−−−−−−→! e , sin t, − sin t − t cos t +

r t

t

t=

−−−−−→! −−−−−→! = , , + t , , +

t −−−−−−−→! t −−−−−→! ,− , + , ,

−−−−−−−−−−−−−−−−−−−−−−−→ # " t t t = +t+ + , − ,t

e0 4 [ . y"0 ** !T #& .$ M $ O ) x(t) = e ≈ + t + t + t 9E .&[, r(t) B

*z(t) ≈ t y(t) = cos t ≈ − t (0&f &) + .$ . $0 e0& N# @ S M M (0&f A .8 $ +, D$ 0 . $0 e0& 5F t 6 &) + .$ &F O&0 *( t E 0 O&0 (0&f $, A ZM x= *!' r(t) r(t) = r(t) = A Z#. $ >. T R# .$ *r(t) = A "#&0 ]3 *r (t) r(t)+r(t) r (t) = Z#. $ \ 6 &0 $ +, Z 0 r (t) r(t) U# *r (t) ⊥ r(t) &# r(t) r (t) = * " .$ K6 . $0 e0 E N# t 6 0 9M [& . #E t

−−−−−!−−−−−− !−−−−−→ !! = x t , y t , z t

&B M ( #n<\ 6 . T .$ . $0 e0& .$ *O&0 #n<\ 6 5F O&0 #n<\ 6 . $0 e0 l h r S * >. T R# .$ a ∈ R )

!

)

! ar = ar ! r • h = r • h + r • h

•

r = (r • r ) /r ) r, h, l = r , h, l + r, h , l + r, h, l

3 4 r(t) =

k+

!

[ t 6 . $ r(t) S Z' )*;: A5 9&F O&0 t = t .$

! ! ! ! t−t r t + t−t r t + r t + · · · !k ! t−t r(k) t + h(t) ··· + k! !

) r(t) =

−−−−−−−−−−→

t+ , cos t t−

•

ti − t j r(t) = + + t

)

)

−−−−−−−−−−−−−−→! r(t) = tet , cos t, arctan t

)

t + r(t) = t −

! i − et j + tan t k

−−−−−−−−−−→! h(t) = sin t, + t, t

M x= A

−−−−−−−−→! r(t) = et , sin t, t

*M t\ $. R# .$ . 7** E H ; y0 . *M (0&f . 7** q1 A4 h(t) Zg S . T .$ 0) 'BC e0 + , + h (t) = r(t) M ( r(t) B e0& N#

* lim

•

)

! r + h = r + h ! r × h = r × h + r × h

)

•

)

t→t

h(t) !k+ = t−t

5F .$ M

−−−−−−−−−−−→! r(t) = et , cos t, t sin t

ZM x= A& '() −−−−−−−−→ . 7** E AC Y0 . >. T R# .$ *h(t) = t, t + , t ! *M \\ 9Z#. $ B&" >&- 0 ) &0 *!' −−−− −−−−−−→ !−−−−−−!− ! ! et , cos t , t sin t −−−−−−−−−−−−−−−−−−→! = et , − sin t, sin t + t cos t −−−−−−−−−− −−−−−−→! !−−−−!→ ! , t = , , t h (t) = (t) , t + ! (r(t) • h(t)) = tet + + t cos t + t sin t ! = et + tet + cos t − + t sin t + t sin t + t cos t ! r (t) • h(t) + r(t) • h (t) = tet − + t sin t r (t)

=

+t sin t + t cos t + et + cos t + t sin t

. $0 e0& At = .$ . C 5. B N y"0 A, .8 −−−−−−−→! t, t sin t *M [& 4 [ & . r(t) = −−e−,−cos t

Ǳ E 0 ( [a; b] E&0 E E = M & ` + ' S *( LB$ . \ 0 ( #n
&F O&0 r(t) B e0& N# h(t) S % ζi ∈ Ii = [ti− ; ti ] i n / ! ti → r ζi ti i=

b

a

0

b

r(t)dt = h(b) − h(a) a

9Z#. $ _&0 q1 0 ) &0

'() &

−−0−−−−−−−−0 −−−−−−−→ # " 0 −−−−−−→ ! t t [t], te dt = ) [t] dt, te dt −− −−−−−−−− −−−−−−−−−−−−0 −−−−−−→ # " 0−− 0−− / t t = dt, te − e dt dt + /

−−−−−−−−−−−→ −−−−−→

t ,e − e , = =

)

0

0

)

= i −

−−−−−−−−→ 0

, ln(t ) dt ) t 0 i + tj + ) dt − t

) )

π

0 *

9M [&

−−−−−−−t−−−−−−−−→ !! t sin t, e , ln + t dt

#&. $0 v r . $0 U0& r : R → R M x= AH B&" *O&0 \\' $, ω (0&f rr () = v r() = r ddtr = −ωrs $ ) >. T .$ $. $ $ ) % ) &z& &#F *M !' . ( & V# % ) &#F % ) #&. $0 v r . $0 U0& r : R → R M x= A8 B&" *O&0 \\' $ , c g (0&f s rr () = v r() = r ddtr = −gk − c dr dt $ ) >. T .$ $. $ $ ) % ) &z& &#F *M !' . ( & V# % ) &#F % ) *M (0&f . 8** q1 AJ

0 r(t) dt +

h(t) dt

M . T .$ M (S 5 ** !T $. .$ −−−−−−−−−→! y(t), z(t) 9&F r(t) = −−x(t), −−−−−−−−−−

0 0−−−−−−−− 0−−−−−−→ x(t) dt, y(t) dt, z(t) dt

'()

0 et i − tet j + t k dt −

0

x(t) e0 #n

0

r(t) dt

! r(t) + h(t) dt =

r(t) dt =

−−−−−−−−−→ " # −π j = , ,

. #E & B E N# . \

0

0

π

0

ar(t) dt = a

)

π

{sin ti − tj + cos tk} dt = ( )π π π t j + sin t k = − cos t i −

5&6 1 r(t) dt $&+ &0 & r(t) RU & . r(t) B *Z$

$ h h O&0 AQ.&
−−−−−−−−−−−−−−−−→

0 −−−−−−−−−−−−−→ ! t − , sin t, et dt = t − t, − cos t, et + c 0 ! t+ t sin t i − ) j dt = t− 0 0 t+ dt = i t sin t dt − j t + 0 0 t + = i −t cos t+ cos dt −j dt t + t + = i {−t cos t + sin t + c } −j ln |t + | + arctan t + c )

. #E . $0 e0 E N#

0

9M [&

−−−−−−−−−−−−−→

−−−−−−−−−−−→ ! , ln t, sin t ) r(t) = t − t, t ln t ) r(t) = t

)

ti − t j r(t) = + t +

M x=

)

r(t) = t i − sin tj + et k

0) 'BC $

P : a = t < t < · · · < tn = b

!

# $ C R > X = x , y ! ZM x= _ A, .8 !VO 0 C B$&U *( R `&UO X .$ cM 0 x−x

x−x

= R cos t

!

+ y−y

!

= R

M ZM x= Z R# 0&0 *( R# 0&0 *y − y = R sin t

−−−−−−−−−−−−−−−−−−−→! r(t) = x + R cos t, y + R sin t

t\ . & y# O 5 S$& 0 * ≤ t ≤ π M #E r = 0 9Z#. $ pO $. .$ *$ +

* * *−−−−−−−−−−−−−→!* *r (t)* = * − R sin t, R cos t * = R =

M $ O < R r(t) = r(s) x= E ?U0 x= t0&Y 5 Q *cos t = cos s sin t = sin !s *( N# C ]3 *s = t "#&0 t, s ∈ ; π ! b a &Y1 Z X = x , y cM 0 q0 A .8 9Z#. $ A;C 0 [O d0 &0 #E ( N# −−−−−−−−−−−−−−−−−−→! C : r(t) = x + a cos t, y + b sin t ;

Γf =

! x, f (x) a ≤ x ≤ b ⊆ R

Z $0 0 0 #E ( N# $ O G#U −−−−−→! r(t) = t, f (t) ; a ≤ t ≤ b

M y = x − x + B$&U 0 + &a 5 , 0 *O&0

>. j0 . − ≤ x ≤ 5F .$ −−−−−−−−−→! ; − ≤t≤ C : r(t) = t, t − t +

*$ + .&3 5

Y\ X ZM x= *&q= .$ =&# Z+U q0 A .8 * !VO 0 * &U V# v u &. $0 ( &q= E R# .$ * [a $ , b a ZM x= Rl+ *$ O ) % u > E 0 b a &Y1 Z X cM 0 C q0 >. T 0 . n = u × v Zg&1 . $0 X &SV &0 T .$ e1 v Z N+M ! ! −→ r(t) = X + a cos t u + b sin t v ;

UB&Y .$ . $0 e0 E $& uL0 R# E m ( R# T B&" *( &q= T .$ &

&0 5 . T Q .$ - C C & M *$ + t[Y #$ 0 . V# A(M'C 1 !#[ N# 5F M{ 0 & & ( VV .&"0 B&" R# &3 [B *$ O M !" #$ /.$ 0 5F >&[f Z#E $3

−−−−−−−−−→! y(t), z(t) . $0 e0& . r(t) = −−x(t), 9M Zg S I &# `# . T .$ *O&0 [a; b] !VO 0 r $ A *O&0 3 [a; b]! 0 r A; *O&0 #n3t 6 a; b! 0 r A *O&0 T GB&L a; b !0 r A *O&0 N[V# a; b 0 r AH

≤ t ≤ π

0 y = f x! ZM x= *T .$ x E U0&! . $ + A .8 f . $ + >. T R# .$ *O&0 #n3t 6 a; b 0 3 [a; b] >. j0 M [a; b] 0

3**' 5 F *H $

≤ t ≤ π

}

y .&3 AGB 9* !VO =&# Z+U q0 A% M Zg S a: &# !( . T .$ . C ⊆ R , + #E M #& + RU *O&0 0 0 Z Q &# N# $0 E ,&+ ) &0 I &# L I $ O C , + 0 0 & F $0 `&+ ) *$ O S C $ B A ZM x= _-#I A& '() . $0 e0& *O&0 AB y .&3 C $ 0 R E >&! Y\ R# .$ *#0 h .$ . [; ] ! $ &0 r(t) = − t A + tB 9( #n3t 6 ; 0 3 [; ] 0 r >. T AB

r (t) = −A + B = 0

r(t)

x= &0 ?U0 *. 10 pO .& Q ]3 ! ! R# 0&0 * − t A + tB = − s A + sB Z#. $

= r(s)

! ! s−t B−A =0

R# 0&0 *( N[V# ; ! 0 r nB s = t ]3 A = B & *A$ O ) GB * !VO 0C ( N# AB

0 *M oL6 . 3.& $' LB$ $ , β α M *$ O ) % * !VO N# $ & $ U `&+ ) 5 Q A0 .8 #) (#& 0 5 _&0 & &0 R# 0&0 ( z a *(& ⊕ !VO &# ( y .&3 `&+ ) M da ? *( y .&3 $ # $ N# `&+ ) M ?za *( N# $ N# E UY1 A2 .8 *q0 E e0. N# &# ( N# # $ Z $ y M ( 4$ e0. E +"1 D M x= A&^ .8 . 5F C E *( O ) y = − x y = x + !VO 0 *C = C ∪ C ∪ C ∪ C ZM x= *M .&3 .$ *( y .&3 C *$ O ) GB H* C : r(t)

−−−→! −−−→! = ( − t) , + t , ; −−−−→! = , t ; ≤ t ≤

≤t≤

*M eY1 ( , ) (− , ) p&\ .$ O $ $ + $ y = − x + E +"1 C *− ≤ x ≤ $ .$ ]3 y = − t .$ *x = t $M x= 5 ]3 *( C : r(t) =

−−−−−−−→! t, − t ; − ≤ t ≤

0&6 >. T 0 C : r(t) =

−−−→! t, t ; − ≤ t ≤

*$ + .&3 5

9Z#. $ 4.& Q pO $. .$ * " #0 pO r (t) = 0

⇒ ⇒

r •u = r •v =

! − a cos ! t u •u = b sin t v • v =

⇒ cos t = sin t =

9M $ O r(t) = r(s) x= E ?U0 *( & M

! r(t) • u = r s! • u r(t) • v = r s • v

⇒ ⇒

a cos t = a cos s b sin t = b sin s cos t = cos s sin t = sin s

*s = t R# 0&0 < t s < π & R# B +U E V# *v e0& $ . $ + $. 0 A4 .8 5& Y0 . $ E $& &q= .$ & G#U 0 & O. h O&0 z y x ! ! C : f x, y, z = a , g x, y, z = b

X R# 0 ( >_$&U & $ !' $M #&0 M .&M & *ZM x= e0& . #$ &$ $M x= v . V# M &F C : x + y = x + y + z = S ?za

Z M x + y = x= 0&0 & *z = −x − y z = − cos t − sin t R# 0&0 *Z#0 y = sin t x = cos t .$ −−−−−−−−−−−−−−−−−−−→! C : r(t) = cos t, sin t, − cos t − sin t ;

≤ t ≤ π

*$ O ) GB * !VO 0 }

pL ^Y E +"1 S M x= A&& .8 z = >&T y $. $ . 1 Z 6 N# .$ M ( ZM x= *M .&3 . 5F C E *( O ) z = *$ O ) % H* !VO 0 *C = C ∪ C ∪ C ∪ C z = x +y

#. $ $. 0 E !T&' AGB 9* !VO =&# Z+U 3.& A% M M (M' 5&Q M M x= *3.& A5 .8 &0 ( AHC &a .$ q0 0 0 N . $0 $ .$ 5F # j >. T 0 . R# *$. _&0 n $ .$ c (0&f (, 9$ + .&3 5 #E

} ;*;* &a E I & +"1 9H* !VO

! ! −→ r(t) = X + a cos t u + b sin t v + ctn ; α ≤ t ≤ β

q0 5 S+ $ $. 0 E !T&' C A& .8 !VO 0 *#0 h .$ . z = − x + y z = x + y & $ N# .$ #. $ R# B$&U !' E *$ O ) GB J* *x +y = &# − x +y = x + y M $$S h'?

>. T R# .$ *y = sin t x = cos t $ + x= 5 ]3 R# 0&0 z = x + y = + cos t −−−−−−−−−−−−−−−−−−→! C : r(t) = cos t, sin t, + cos t ;

.$ " y .&3 C C C : r(t) = = C : r(t) = =

:

: C

: :

;*;* &a E H & +"1 9J* !VO

M $. 0 E !T&' C

A& .8 ) % J* !VO 0 *#0 h .$ . z = x + y T $$S h'? & $ N# .$ #. $ R# B$&U !' E *$ O *x + y + xy = &# x + y + (x + y) = M &# x + y + xy = >. T 0 . R#

x + y + z =

≤t≤ ≤t≤

.$ * " # $ e0. C C

≤ t ≤ π C

}

−−−−−→! −−−−−→! ( − t) , , + t , , ; −−−−−−−−−−−→! , + t, + t ; ≤ t ≤ −−−−−→! −−−−−→! + t , , ; ( − t) , , −−−−−−−−−−−→! + t, , + t ; ≤ t ≤

⎧ ⎧ ⎨ x + y = ⎨ x + y = z : z= z= ⎩ ⎩ ≤ x, y ≤ x, y −−−−−−−−−−→! π r(t) = cos t, sin t, ; ≤t≤ ⎧ ⎧ ⎨ x + y = ⎨ x + y = z : z= z= ⎩ ⎩ ≤ x, y ≤ x, y −−−−−−−−−−−−−→! π r(t) = cos t, sin t, ; ≤ t ≤

C : r = a( + cos θ) ;

≤ θ ≤ π &+B$ A&, .8 C #0 h .$ . y L T .$

8* !VO 0 *Aa > x = r cos θ # 0 ) &0 t = θ x= &0 *$ O ) GB 9M $$S h'? x = r sin θ −−−−−−−−→! C : r(t) = a( + cos t). cos t, sin t ;

≤ t ≤ π

}

√ (x + y) + ( y) =

x + y = cos t $ + x= 5 R# 0&0

*(O 5

.$ * y = sin t √

y

=

x = z

=

√

sin t √ ( cos t − y) = cos t − sin t √ x + y = cos t + sin t

;*;* &a E ; & +"1 98* !VO >&V"+B A& .8

Z#. $ ` + .$ C : r(t)

=

√

√

!

!

cos t − sin t i + sin t j √ ! + cos t + sin t k ; ≤ t ≤ π

5$ + .&3 Q *; uL0 .$ 78 *( O $ $ ^- 4$ ).$ &

9M .&3 . & E N# *T .$

, −

!

0

!

,

y .&3 A

C : r = a cos(θ) ;

≤ θ ≤ π

8* !VO 0 *Aa > C #0 h .$ . y L T .$ R# 0&0 *cos(θ) > #&0 M ( RO. *$ O ) % &0 .$ *k ∈ N M kπ − π/ ≤ θ ≤ kπ + π/ −π/ ≤ θ ≤ π/ Z#. $ X 0 k = k = x= 9( V $ h $. U# *π/ ≤ θ ≤ π/ x = r cos θ # 0 ) &0 t = θ x= &0 *C = C ∪ C 9M $$S h'? x = r sin θ C C

, −−−−−−−−→! π π cos(t). cos t, sin t ; − ≤ t ≤ , −−−−−−−−→! π π : r(t) = cos(t). cos t, sin t ; ≤t≤ : r(t) =

*T .$ `&UO

z = x , y = x , − ≤ x ≤ √

) x = + y + z , x =

)

, x + y = z

) x + y + z = )

∗

x + y = , x + z = ) x + y + z = R ,

(x − a) + (y − a) = r ,

√

|a| + |r| < R

*M .&3 . 7* !VO .$ ;8 & ;; &

M x= U!U) < F= '=d $

,−

!

cM 0 # $ A;

)

x + y = x

)

x + y = , − ≤ x ≤

)

x / + y =

)

y = x − , y ≤

)

x − y = , |x| ≤ ) x/ + y / =

* , , − ! , , ! y .&3 A2 . 7* !VO .$ O $ $ 5&6 & I &

*M .&3

}

C : r(t) ; a ≤ t ≤ b

0 r t + dt! & r t ! E / 1 @ *a < t < b l h $. @ &0 B&" R# 0 &3 0 ( .\Q !VO 0C ZM [& . d Y\ $ 5F R0 !T . $0 @ 9A$ O ) GB * C

*−−−− !−−−−−−→!* ≈ d = *r t r t + dt * ! ! !! !! = x t − x t + dt + y t − y t + dt !! / ! + z t − z t + dt

.$ t t t $ , Z mU i S_ q1 0&0! & 9M O (=&# 5&Q t ; t + dt E&0 l ≈ d =

! ! ! ! ! ! / x t dt + y t dt + z t dt

9Z#. $ O&0 NQ M =&M E 0 dt S ]3 ,

l

x t

≈ d ≈

!!

+ y t

!!

+ z t

!!

dt

! = r t dt

U# & / 1 @ !" #$ . F >.&[, ( =&M C a: .S [& 0 5 M *ds = r (t)dt 9Z#0 [a; b] E&00 ds E

0

=

0

b

ds =

b

r (t) dt

a

M . T .$ A& −−−−−−−−−−−−−−−−−→! C ; r(t) = cos t + , sin t − ; 0

= =

'() %

≤t≤π

&0 ( 0 0 C / 1 @

*−−−−−−−−−−−−−→!* * − sin t, cos t * dt 0 π, sin t + cos t dt = π π

;8 & ; ; & I !#&" 97* !VO * , , ! − , , ! , − , ! / g.&0 da A , − , ! .$ cM &# z = T .$ e1 # $ A * `&UO )

x = y = z, − ≤ x ≤

)

x + y = , x + y = z

}

/ t k; ≤t≤

) r(t) = t i − t j − t k ; ≤ t ≤ )

r(t) = ti +

)

! r(t) = t, cosh t ; − ln ≤ t ≤ ln

)

r(t) =

cos t

! i+

sin t

√ ! j + tk ;

≤t≤π

−−−−−−−−−−−−−−−−−−−−→! √ r(t) = t sin t + cos t, t cos t − sin t ; ≤ t ≤ √ ! ! ) r(t) = t cos t i + t sin t j + t/ k ; ≤ t ≤ π

)

/ 1 @ AGB 9* !VO 8*;* &a E (+"1 A% ! ! , , , , − y .&3 / 1 @ A, .8 p&\ R# R0 !T . $0 @ &0 (

M x=

0 0 C = .d C)_ *

*−−−−−−−→! −−−−−→!* *−−−−−−−→!* = * , , − − , , * = * , , − * =

C : r(t) ; a ≤ t ≤ b

9!VO 0 . C =@ E

>. j0 . s $ UT e0& *O&0 O .&3 N# 0

t

s(t) =

−−−−−−−→! !−−−−−→! −t , , + t , , − −−−−−−−−−−−→! , , −t + ; ≤ t ≤

C : r(t) =

r (t) dt , a ≤ t ≤ b

=

a

e0& R# *$ O ) GB I* !VO 0 *ZM G#U = r > #E ( #n< VU

0 . t 5 ]3 * ds dt C / 1 @ l ≤ s ≤ l M t = t(s) 9$.F (0 s X"' 9(O $M $& s E t &0 r .$ 5 5 M *(

R# 0&0 *$ + .&3 5

C : rn (s) = r(t(s)) ;

≤s≤l

.&3 R# R"' * & C !J$S I . r (s) &z'?YT 9( N# 0 0 &) + .$ 5F (, M ( 5F .$

=

0 0 *−−−−−−−→!* * , , − * dt = dt =

. % * .$ O $ $ 5&6 / 1 @ A .8 C C Ǳc) $ E R# *ZM [&

O !V6 ! ! , , & , , 0 &0 Y .&3 C *( R# 0&0 (

n

* * *rn (s)* = = =

*d !* * r t(s) * ds * * * dt d * ds * r(t)* = *r (t)* ÷ ds* * dt * ds * *r (t)* ÷ *r (t)* =

C : r(t) =

.$ R = `&UO X C : r(t)

}

= =

−−−−−−−−−−→! t, − t, ;

=

Ǳ [ cM 0 # $ e0. C =@ E 9R# 0&0 ( T −xz

! ! → = − o + cos t i + sin t k −−−−−−−−−−−−−→! = cos t, , sin t ; ≤ t ≤ π/

+ 0 −−−−−−−→! , −, dt 0

π/

+ =

0 √

dt +

−−−−−−−−−−−−−−−→! − sin t, , cos t dt

0

π/

√

π dt = +

. #E & E N# / 1 @ = kM A% U[@ .&3 AGB 9I* !VO

≤t≤

−−−−−−−−−−−−−−−−−→! ) r(t) = sin t + , cos t + ;

0 &

9M [&

≤t≤π

9G#U 0&0 5 M Tn (s)

rn (s)

=

cos s − π i − sin s − π j + k s s − sin −π i− cos −π j

= rn (s)

=

Nn (s)

=

Bn (s)

=

s s rn (s) = − sin − π i − cos − π j rn (s) Tn (s) × Nn (s) i j k ! ! s s −π − sin −π cos ! ! s s −π − cos −π − sin s s cos − π i − sin −π j− k

=

=

.&3 E $& 50 . N# &a = kM A, .8 9ZM [& U[@ r (t) =

cos t

! ! i + − sin t j + k

r (t) × r (t) =

cos t

! i+ −

r (t) =

sin t

! j − k

N(t)

r (t) × r (t) =

cos t!i + − sin t!j + r (t) = r (t) r (t) × r (t) = r (t) × r (t) ! ! = cos t i + − sin t j − k ! ! = B(t) × N(t) = − sin t i + cos t j =

k

= kM A .8 C : x + y =

*0&0 X

√

r

π

N# 0 0 5F @ ( /&+ C 0 r (s) Y\ .$ ?U0 *Z & !( aF )*? . 5F O&0

I* !VO 0C O&0 s .&3 $&#$E & . .$ 5F ( ) *A$ O ) % *O&0 $ +, u\ 6 0 ( N# 0 0 T (s) @ 5 Q V# . $0 n

n

Nn (s) :=

√

n

n

! Bn (s) := Tn s × Nn (s)

X #& *Z & r

Y\ .$ Y `b: .

kM N# kM R# *Z & r (s) Y\ .$ C )* < c . N# 0 0 @ M U R# 0 ( $ . V# &U

*( ([a & F #& %qT&' $ +, $ 0 $ ( (0 . = kM 5 c U[@ .&3 E $& 50 *ZM >&[f ** .$ . q1 R# 9$.F O&0 [a; b] $ &0 LB$ Z N# r S &F t ∈ (a; b) r (t) × r (t) r (t) , B(t) = , r (t) r (t) × r (t) N(t) = B(t) × T(t)

T(t) =

M x= A&

C : r(t) =

sin t

! i+

cos t

'()

! j + tk ; −π ≤ t ≤ π

9E ( >.&[, C / 1 @ e0& >. T R# .$ t

= −π t

0

/, /, /

−−−−−−−−−−−−−→ = (− sin t, cos t, cos t) π

Tn (s) r (s) = n rn (s) Tn (s)

.$ . $0 R# *Z & r (s) Y\ .$ C !( !'= `b: . Ǳ& cM (+ 0 ( $ +, 0 r (s) Y\ 9).& %qT&' *$. $ m

s(t)

Y\ .$ . * 9ZM .&3 . C 0 *!' . & F E V# & . Mn

.&3 ( e@&\ # $ $ C [B π 9R# 0&0 *X = r M ( RO. ?U0 *M .&3 = −−−−−−−−−−−→ r(t) = (cos t, sin t, sin t)

Tn (s) := rn (s)

0

, x + z = √

. $0 *ZM

n

9Z#. $ @ 0 & B = N+M 0

B(t)

eUL

C n (s) Fn (s) : = Tn (s), Nn (s), Bn (s)

! ! r (t) = − sin t i + − cos t j

T(t)

$& 7*;* (+"1 >&- E

= −π

*−−−−−−−−−−−−−−→!* * cos t, − sin t, *dt

dt =

9 .$ * ≤ s ≤ C : rn (s)

s

=

r

=

sin

−π

s

t+π

!

π M t = s/ − π R# 0&0

s − π i + cos −π j s + −π k

−−−− " # √−−−− √−−→ √−−−−

r

π

=

−−−−−−−−−−−−−−−−→ (− cos t, − sin t, − sin t) π

'?T >&T 9* !VO /&+ y Zg&1 /&+

R# *( T(t) $& r(t) &SV &0 y 9/&+ y *( r(t) Y\ .$ C 0 y R# V#$c y *( B(t) & r(t) &SV &0 T 9/&+ T 0 *( r(t) Y\ .$ C 0 T R# V#$c T R# T 5F 0 T N# .$ r(t) L0 S M U R# c . /&+ T *O&0 /&+ T 5&+ * &

. $0 *( T(t) & r(t) &SV &0 T 9Zg&1 T Zg&1 T .$ M *( $ +, C 0 r(t) .$ 1 & 1 v *O&0 O $ . 1 *( N(t) & r(t) &SV &0 T 9'?T T M x= A& '()

r

π

−

× r

π

+ cos

!! ! πt i+ sin πt j+πtk ; − ≤ t ≤

! Z#. $ X R# 0 *X = r( ) = −−,−−,−−→ π 5F .$ M ! −−−−−−−−−−−→ Rl+ N( ) = −−,−−,−→ T( ) = (, −/, /) −−−−−−−→ 9R# 0&0 *B( ) = −− (, /, /) /&+ y : x − = −y −/ = z −/π

Zg&1 T

: x = , y + z = π ! ! ! : x− − / y − ! ! + / z − π = :

/&+ T

:

:

'?T T

:

z = y + π ! ! ! x − + / y − ! ! + / z − π = y + z = π ! ! x− + y− ! + z − π =

: x=

>&j L 0 X Y\ .$ . #E $ R0 # E A, .8! 90&0 , −, −

=

r

B

π

=

π

−−−−−−−→ = (, − , )

π* √ * π *= *r × r π π ÷ r

−

, ,

π

r

× r

π

π* * π * ÷ *r × r

− " #−−−−− √−−−− √−−→ =

N

, − , −

−−−− # " √−−−− √−−−− √−−→ =

r(t) =

π

√ * π* *= *r

T

−−−− −−−−−− # " √−−−−−√ √−−→ =

}

, ,

−

=

=

, − ,

B

π

×T

π

−−−− −−−−−− " # √−−→ √−−−−−√ =

−

, − , −

!VO 0 . #E & E N# 0 9#.F (0 . & F = kM $M .&3 U[@ ) r(t) =

−−−−−−−−−−−−−− √−−→! cos t, sin t , t ;

≤ t ≤ π

−−−−−−−−−−−−→! t + , − t ; − ≤ t ≤

)

r(t) =

)

C : x + y + z =

)

C : x + y + x + y =

, z=

√

y

90&0 X Y\ .$ . O $ $ = kM !

)

C : x = y = z, X =

)

−−−−−−−−−−−−−−−−→! C : r(t) = t − , t + t, t − , X = r

, , −

!

E Y\ .$ = kM $ ) !B$ 0 E Y\ 0 #E KO 0 y N# T N# 9A$ O ) * !VO 0C $ O $ $ (["

}

− −−−−−−−−−−−−−−−−→ C : r (t) = t + , t − , t + −−−−−−−−−−−−−→ C : r (t) = t , t − , − t

0 0 $ 5F 0 /&+ &. $0 R0! # E 0 0! # E R# _> # E R# 0&0 r − = r = X B *( &0 ( 0 0 α O

( ?, A% /&+ # $ AGB 9;* !VO Ǳ&

κ(t) =

- x x

x y + y x

y z + z y !/ (x ) + (y ) + (z )

z z

9(O 5 Y" & $. .$

x y − x y κ(t) = !/ (x ) + (y )

&0 M ( ([a . T .$ κ ( ?, ( ( ?, . $ M b .$ *l0 (,& &0\, ( ) .$ T(t) . $0 t $&#$E *A$ O ) % ;* !VO 0C $ 0 κ >. T R# Ǳ& `&UO! Ǳ& cM Ǳ& A& '() & X = , Y\ .$ . O $ $ 0 /&+ # $ B$&U

9#.F (0 −−−−−−−−−−−→! C : r(t) = t + , − t ; − ≤ t ≤

V# 0 ) &0 *!'

9Z#. $ X

= (, ) = r( ) −−−−−−−→! −−−−−−−→! r ( ) = , −t, = , −, −−−−−−−→! r ( ) = , −, #√ √ T( ) =

t=

,− ,

r ( ) × r ( ) =

"

−−−−−−−→! , , −

−−−−−−−→! B( ) = , , −

−

r ( ) × r ( ) = κ( ) = r ( )

cM ( R(

) =

/κ( )

√

√ ! =

√ = 0 0 Ǳ& `&UO ]3

−−−−−→! , ,

!

) r(t) = t +

i + tj + t k, X = r( )

)

−−−−−−−−−→! ! r(t) = cos t, sin t, t , X = − , , π

)

x = y + , z = y + x ; X =

)

x + y + z = , x + y + z = , X =

, −

!

,

! , − ,

−−−−−−−−−−−−−−−−−→! r(t) = a cos(wt), a sin(wt), bt

0 /&+ M $ 5&6 AH *$E& T−xy &0 (0&f # E . + &0 y = x + ax + b + M M RU . @ . b a A8 *O&0 /&+ x + y = # $ Q &0 . #V# x − y = b xy = a &B Bn AJ *M eY1 #&# E & .C T (s) > v 5 c S + ǱU! % &3&M $ O C κ (s) Ǳ( . / 1 @ 0 ([" A/&+

h E *κ (s) = T (s) 9Z & As .$ n ]# aM ' . $ `&UO r (s) + N (s) cM 0 # $ k 0 0 X Y\ .$ C Ǳ& S U# *( &0 /&+ `&UO 0 # $ N# 0 [O X #&"+ .$ C &F O&0 Ǳ( e . r (s) + N (s) !B$ R+ 0 *O&0

.$ C 0 aF * . . Mn # $ Ǳ( fJ@ . /κ (s) 0 *A$ O ) GB ;* !VO 0C *Z & r (s) Y\ 9$ + $& 5 #E & B = E Ǳ& . \ [&

n

n

n

κn (s)

9E ( >.&[, Ǳ&

−−−− −−−−→ " # √ √−−−−− √ −−−−−→! r( ) + R( )N( ) = , , + − ,− , =

/&+ T /&+ y B$&U 0 $ X Y\ .$ . O $ $ '?T T Zg&1 T 9#.&0 (0

n

, − ,

n

−−−− −−−−→ # " √−−−−− √ N( ) = B( ) × T( ) =

! ! ∠ r − , r () =

−−−−−−−→! −−−−−−→! , t, , − = ∠ t, , t=− t=

−−−−−−−−→! −−−−−−−→! = ∠ − , , , , , −

−

√ = cos− = cos− √ √

κn (s)

n

κ

n

κn (s)

n

n

n

κ(t) =

r (t) × r (t) r (t)

9#$ 5&0 0

9E ( >.&[, Ǳ& cM

h(t)

= r(t) + R(t)N(t) −−−−−−−−−−→ = (a cos t, b sin t) +

S

! + b − a cos t

) ) )

√ ! ! r(t) = e cos t i + et sin t j + et k, X = r() ! x = ln y, X = , e ! ! ! r(t) = cos t i + sin t j, X = r π/ √ √ ! x + y = z , z = , X = , − , t

Ǳ& cM 5&V E !T&' . $ + A8 *M Z . y = x Ǳ& cM 5&V E !T&' . $ + AJ *M Z . r(t) = sin ti + cos tj + tk Y\ .$ C 0 /&+ T M Z $ S ([" B (s) > v 5 c *O&0 B (s) & . $ r (s) s 0 ([" /&+ T > v 5 c R[ M / 1 @ 0 $ O C τ (s) = B (s) 9Z & %& . O&0

M $ O (0&f *As .$ n ]# & n

n

n

τ (t)

=

=

−−−−−−−−−−−√ −−−−−→! C : r(t) = cos t, sin t, cos t ; −π ≤ t ≤ π/

U#C ( Y" M $ O (0&f ( =&M *!' &) R# .$ & *O&0 (0&f c 5F Ǳ& `&UO A( (0&f 9Z#. $ −−−−−−−−−−−−−−−√ −−−−−−→ −−−√ −−−−−−−→ # " # " − T= sin t, cos t, − sin t B = − , ,

z z z + y y

S $. _&0 B(t) $ .$ O&0 ([a τ (t) l&Q T 0 0 Y" %& *(=. Rg&3 O&0

*O&0 (0&f B(t) U- RQ .$ #E (

Ǳ& cM ( R =

/κ =

!

=

Ǳ& `&UO ]3

9E ( >.&[, #E

−−−−−−−−−−−√ −−−−−→! r(t) + R(t)N(t) = cos t, sin t, cos t −−−−−−−−−−−−−−√ −−−−−−→ " # − − cos t + , sin t, cos t = 0

M ( $ 0 /&+ T .$ e1 # $ E UY1 C ]3 9( #E KO 0 B$&U . $ √

: −

P

! ! x − cos t + y − sin t √

z−

cos t

9#$ >.&[, 0 *( `&UO C

!

=; z= ! , , √ : x + y + z = , z = x √ : x + y = , z = x +

√

x

cM 0 # $ C

*( x /a + y/b = q0 C M x= A .8 *M oL6 . C Ǳ& cM 5&V

−−−−−−−−−−→! C : r(t) = a cos t, b sin t : ≤ t ≤ (B&' R# .$ *!' 9R# 0&0 *π T(t) =

z z

− −−−−−−→ #−−−−−−−−−−−−−−√ " − N= cos t, sin t, − cos t κ =

n

n

[r (t), r (t), r (t)] r (t) × r (t) x y x y x y x y + x z x y x z

B

( aM ' Ǳ& 1 & 1 9M (=S 5

t = h p&\ ]3 *cos t = U# O&0 !1 ' cos t M _ &0 * , ±b! 9 " Ǳ& aM ' &0 t = π/ π/ *O&0 Ǳ& !1 ' &0 ± a, ! p&\ 0&6

Ǳ& cM Ǳ& `&UO Ǳ& 0 * O Y\ .$ . O $ $ /&+ # $ B$&U

9#.F (0 )

√ !

! + y− =

9( # $ N# E 6L0 #E M $ 5&6 A, .8

Z"# 0 a < b S ,

!

: x − x + y =

−−−−−−−−−−−−→ (−b cos t, −a sin t)

ab −−−−−−−−−−−−−−−−−−−−−−→

! ! = cos t, b − sin t a− a b

κ(t) = ab ÷

x−

:

9( RQ /&+ # $ B$&U

N(t) = κ(t) = =

−−−−−−−−−−−−→! − a sin t, b cos t + a sin t + b cos t −−−−−−−−−−−−→ (−b cos t, −a sin t) + a sin t + b cos t ! ! ! ! | − a cos t b cos t − − b sin t − a sin t | + a sin t + b cos t ab + a sin t + b cos t

%& A&

?z &M %& Ǳ& N+M 0 c &q= .$ & M U R# 0 *$ O oL6

C : r (t) : a ≤ t ≤ b S pO &F O&0 &q= .$ $ C : r(s) : c ≤ s ≤ d U0& M ( 5F C 0 C 5$ 0 &[Y !0&1 0 =&M 4E_ t ∈ a; b! E 0 M $$S (=&# 5&Q h : [a; b] → [c; d] @ .$ ( ?, *τ h(t)! = ±τ (t) k h(t)! = k (t) *( (0&f a; b! 4&V' _&0 q1 $ E $0.&M 5 , 0 '() 9ZM KY . #E UY1 x N# V# 0 =&M 4E_ pO A *O&0 T Ǳ& M ( 5F O&0 ( . y E UY1 x N# V# 0 =&M 4E_ pO A; 5F Ǳ& T %& M ( 5F O&0 # $ E *O&0 (0&f E UY1 N# V# 0 =&M 4E_ pO A *O&0 (0&f %& Ǳ& M ( 5F O&0 3.&

MJ N $$ N# (M' Y0&- U 0 .&3 &# Z N&V &S#$ E *O&0 5F (M' " U 0 / 1 &# ( w

%& O (, 5 Q &,?@ 5 (M' Y0&- &# Z E 5$ 0 4 U &0 & *$.F (0 . Zg&1 %& O &+ %& O ?za * " RU !0&1 . Mn #$&\ (M' " &#

G L & , &0 ![

$ x $&) N# 0 ( RV+

*M (M' r(t) : a ≤ t ≤ b Y0&- &0 +") S N#c= E >&,?@ 0 &0 >. T R# .$ M (M' &q= .$ ) GB * !VO 0C ZM KY . #E KO 0 G#.&U *A$ O >. T 0 $ $ 5&6 v(t) $&+ &0 . )g(H 7T+ #$ 5&0 0 *ZM G#U rt 5& E 0 ([" 5&V v 5 c s *r (t) >. T 0 5 c s >. T 0 $ $ 5&6 a(t) $&+ &0 . )g(H K@ 0 #$ 5&0 0 *ZM G#U rt 5& E 0 ([" (, v *r (t) >. T

'()

C : r(t) = cos ti + sin tj + tk

9ZM [& . −−−−−−−−−−−−−−−→! r (t) = − sin t, cos t, −−−−−−−−−−−−−−−−→! r (t) = − cos t, − sin t, r (t) × r (t) =

−−−−−−−−−−−−−−−−→! sin t, − cos t,

−−−−−−−−−−−−−−→! sin t, − cos t, r (t) × r (t) = r (t), r (t), r (t) = r (t) =

.$ r %& U# *τ (t) = / ! = / R# 0&0 h $. 5$ 0 3.& R# !B$ *( (0&f &)+ *O&0

M ZM (0&f Z A, .8 C : r(t) = C

−−−−−−−−−−−−−−−−−−−−−−→! t − , t + t + , t − t

%& M $ O (0&f ( =&M . h R# 0 *( Y"

9( T &)+ .$

τ (t) = r (t) × r (t)

t +

t −

=

*O&0 r &B 5$ 0 4$ ).$ R# !B$ Y\ .$ . #E & E N# %& 0 9M [& O $ $ ) r(t) =

) r(t) =

−−t−−−−−−t−−−−−→ ! e cos t, e sin t, et , X = r()

) r(t) = ) r(t) =

−−−−−−−−−−−→! + t, t , t , X = r(− )

−−−−−−−−−−−−→! t sin t, t , t cos t , X = r(π)

−−−−−−−−−−−−−→! cosh t, − sinh t, t , X = r()

U#C . ?, Ǳ& N+M 0 Y" & R# 0 *$ O oL6 ?z &M A8*;* .$ tY .1 50 M U

C : r(t) ; a ≤ t ≤ b S pO O&0 T .$ $ C : r (s) ; c ≤ s ≤ d U0& M ( 5F C C 5$ 0 &[Y !0&1 0 =&M 4E_ Ǳ E ! 0 M $$S (=&# 5&Q h ! : [a; b] → [c; d] # $! a; b @ .$ ( ?, *κ h(t) = ±κ (t) t ∈ a; b *( (0&f

}

3.& &0 M ZM x= A .8 ! ! r(t) = a cos wt i + a sin wt j + btk

9>. T R# .$ *$. _&0 v(t)

=

a(t)

=

! ! − aw sin wt i + aw cos wt j + bk ! ! − aw cos wt i − aw sin wt j

aT (t)

=

aN (t)

=

a(t) − aT (t) = a(t)

U# *a(t)

+

v = aw + b ?U0 *( (0&f (, %& O E &0 M' Zg&1 %& O &+ %& O %& O (, 0 9M oL6 . #E & M E N# r(t)

= aw

! i + tj + t k ! ) r(t) = t cos t, t sin t, t ! ! ) r(t) = i − j cos t + sin t k ! ! ) r(t) = cos t i + sin t j ) r(t) = t +

−−−−−−−−−→ ) r(t) = t /, t / ! ) r(t) = ln t + i + t − arctan(t)j

3**' DOF P QR 6' ($ 5 c RU & B = R# E m OV=) #$ Z $ .$ *O&0 D$ 0 ([" = kM > v kM 5 Q *$M E& E&0 . 5 y0 . R# R O $ &0 9(O 5 R# 0&0 $ O !V6 !\ " . $0 E = ⎧ ⎨ Tn = a Tn + a Nn + a Bn N = a Tn + a Nn + a Bn ⎩ n Bn = a Tn + a Nn + a Bn

*$.F (0 . a X# - & ( 4E_ 5 M ij

Tn • Tn = Tn =

⇒ Tn • Tn = ⇒ a =

Nn • Nn = Nn =

⇒ Nn • Nn = ⇒ a =

Bn • Bn = Bn =

⇒ Bn • Bn = ⇒ a =

Tn • Nn =

•

Nn + Tn • Nn = ⇒ a = −a

Tn •

•

Bn + Tn • Bn = ⇒ a = −a

•

Bn + Nn • Bn = ⇒ a = −a

Nn •

⇒ Tn Bn = ⇒ Tn Bn = ⇒ Nn

%& O &B AGB 9* !VO # $ (M' A% # js >. T 0 $ $ 5&6 a (t) $&+ &0 . !+F K@ >. T 0 #$ 5&0 0 *ZM G#U r(M' $ .$ %& O T

a •v v projv a = v •v

# js >. T 0 $ $ 5&6 a (t) $&+ &0 . `b: K@ 0 #$ 5&0 0 *ZM G#U r(M' 0 $ +, $ .$ %& O *a − a >. T 9Y0&- &0 M ZM x= A& '() N

T

r(t) =

−−−−−−−−−−−−−−−−−−−−−−→ t − , t + , t − t + ;

≤t≤

9Z#. $ t = hB .$ >. T R# .$ *M (M'

−−−−−−−−−−→! −−−−−→! t, , t − |t= = , , −−−−−→! a( ) = r ( ) = , , −−−→! + + v = −− aT ( ) = , , + + −−−−−−−−−−→! , − , aN ( ) = a( ) − aT ( ) =

v( ) = r ( ) =

−−−−−→! a = , ,

(0&f %& O &0 N# &a .$ (M' A, .8 O&0 aM ' z(t) M ( & 5F !1 ' Y\ *O&0

[; ] & E E&0 0 e0& R# 4 + & *z(t) = t − t + U# .$ `&. !1 ' &0 hB R# 0&0 *( t = / hB .$ *O&0 z ( /) = / 5F `&. ( t = / (M' *( $ 0 v ! = −−,−−,−→! (M' R# B (, # $ (M' Zg&1 %& O M $ 5&6 A .8 −−−−−−→! t, sin t *( 0 0 −r(t) &0 . + r(t) = −−cos % * !VO 0C M ZM ) R# &6 0 *!' 9A$ O ) a(t)

aN (t)

d −−−−−−−−−−→! − sin t, cos t dt −−−−−−−−−−−→! = − cos t, − sin t = −r(t) a •v v = a(t) − v •v cos t sin t − cos t sin t = −r(t) − v = −r(t) sin t − cos t = r (t) =

Z#. $ _&0 y"0 0 ) &0 >. T R# .$ ) x(t) = s + O( )

)

y(t) =

)

z(t) =

s

s

κn () + O() κn ()τn () + O()

R# # j = kM 0 ([" (U- Z" 0 5 M *Z0&# . = kM y O & >&T .$ .

$ R0 . s / 1 @ .&3 ZM U . h R# 0 9ZM mn' z(s) y(s) x(s) B$&U E %&L &0

Nn (

¼) + Tn (¼) E)8 !"# 2 =P

V# 0 )

x≈s , y≈

κn () s

!VO 0 *( + N# M y ≈ &0

Bn (

κn () x

Z#S *$ O ) *

¼) + Tn (¼) E)8 !"# 2 =P κn ()τn () s

4 ).$ N# M z ≈ κn()τn () x Z#S

*$ O ) * !VO 0 *(

&0

Bn (

¼) + Nn (¼) E)8 !"# 2 =P

M

Tn = (rn ) = rn = κn = κn Nn

Z#. $ ?U0 *a = a = κ R# 0&0 n

Bn = a ⇒ τn = Bn = |a |

$. $ $ ) !+, $ EF N# &# .$ *a = ±τ R# 0&0 R# +U *ZM %&L LB$ 0 . &# ([a Z

[U+) 0 5 ` + .$ ]3 *$ O %&L M ( 9. #E .&3 0 ([" = kM >&\ 6 $. .$ Z#. $ / 1 @ n

⎧ ⎨ Tn = κn Nn N = −κn Tn + τn Bn ⎩ n Bn = −τn Nn

. h R# 0 OU!U) T=) 5 M *O&0 X = r () ∈ C & h $. Y\ ZM x= 9Z"# s = T Y\ B ' .$ . r (s) e0& . y"0 n

n

rn (s) =

rn () + srn () +

s

rn () +

s

r n () + O()

X"' 0 . r () 4 & [ >&\ 6 M ( 4E_ ]3 9ZM [& = kM n

%

V# 0 )

κn () κn ()τn () y≈ s , z≈ s y ≈

ij

$

V# 0 )

x≈s , z≈

U# *( 5.&\ $&3 [a ] h $. X# - ]#& R# 0&0 #$ O 5&0 0 *( ]#& $ 0 0 u V# 0 0 c Y1 _&0 T&, T 5F T Y1 T&, & X R# 0 * " T Y1 Rg&3 .$ h T&, 5 Q =@ E *ZM oL6 #&0 . a a a X# -

κn () &F τ () = S M Z#S z n τn ()

*$ O ) * !VO 0 *( [UV N#

rn ()

=

X

rn ()

=

Tn ()

rn ()

=

κn ()Nn ()

r n ()

=

−κn ()Nn () − κn ()Tn () +κn ()τn ()Bn ()

M Z#S . y"0 .$ 5$ $ . 1 &0 R# 0&0

9$ + T? 5 #E q1 .$ . _&0 XB&Y

OU!U) T=) U5 U2 &

( O .&3 U[@ >. T 0 C ZM x= ZM x= *( C E LB$ Zh Y\ X = r() 0 X .$ = kM 0 ([" X Y\ B ' .$ .

>. T rn (s) = x(s)Tn () + y(s)Nn () + z(s)Bn ()

rn (s)

= X + sTn () + +

s

s

(κn ()Nn ())

− κn ()Tn () + κn ()Nn () +κn ()τn ()Bn () + O()

ZM x= S ]3 rn (s) = x(s)Tn () + y(s)Nn () + z(s)Bn ()

O .&3 r(t) y C l&Q r (t) ) T(t) = r (t)

&F O&0

)

N(t) = B × T

)

κ(t) =

)

B(t) =

r (t) × r (t) r (t) × r (t)

)

τ (t) =

(r (t) × r (t)) • r (t) r (t) × r (t)

= =

y≈

κn () x

z≈

κn ()τn () x

y ≈

κn () z τn ()

}

r (t) × r (t) r (t)

Z#. $ v(t) = r (t) V# 0 ) &0 T(t)

>. T R# .$ *Z# O

d d dt rn (s) = × rn (s) ds ds dt d r(t) v(t) × r(t) = v(t) dt r(t)

Tn (s) =

E ( >.&[, 5F E i& N# *O >&[f AC = R# 0&0 *r = vT

r = (vT) = v T + vT = v T + κv N r × r = (vT) × (v T + κv N) = κv B

AC = .$ r × r = κv B = |κ|v R# 0&0 AC = _&0 >.&[, .$ . \ R# 5$ $ . 1 &0 *#$S >&[f ?U0 *$$S c

r

=

(r ) = (v T + κv N)

=

v T + v T + κ v N + κvN + κv N

=

v T + v (vκN) + κ v N + κvN

& U- u#&+ 9* !VO kM >&\ 6 $. .$ C / * Z#. $ LB$ .&3 0 ([" = ⎧ T (t) = v(t)κ(t)N(t) ⎪ ⎪ ⎨ N (t) = −v(t)κ(t)T(t) + v(t)τ (t)B(t) ⎪ ⎪ ⎩ B (t) = −v(t)τ (t)N(t)

*v(t) = r (t) = ds &# .$ M dt M Z#. $ )

+κv (−vκT + vτ B)

T (t) =

(r (t) × r (t)) • r (t) = κ v τ = r (t) × r (t)τ

=

*( #0 A;C = *#$S >&[f A8C =

2

.&3 &0 C Y" $. .$ ! x (t), y (t)

=

MC

Z#. $ r(t) =

! x(t), y(t)

)

N(t) = sgn(κ(t)) +

)

κ(t) =

= =

−v(t)κ(t)T(t) + v(t)τ (t)B(t)

=

! − y (t), x (t)

d d N(t) = Nn (s) dt dt d ds × Nn (s) = v(t) × Nn (s) dt ds v(t) (−κn (s)Tn (s) + τn (s)Bn (s))

N (t) =

) T(t) = + x (t) + y (t)

x (t) + y (t)

x (t)y (t) − y (t)x (t) + ( x (t) + y (t) )

O&0 O $ $ C x Y" N# = kM S .$ O&0 oL6 5F >&hB + .$ R# Ǳ& S c *(=&# $0 j >. T 0 . C 5 >. T R#

B (t) = = = 2

d d T(t) = Tn (s) dt dt d ds × Tn (s) = v(t) × Tn (s) dt ds v(t) (κn (s)Nn (s)) = v(t)κ(t)N(t)

d d B(t) = Bn (s) dt dt ds d × Bn (s) = v(t) × Bn (s) dt ds v(t) (−τn (s)Nn (s)) = −v(t)τ (t)N(t)

*( 4&+ 5&0 X R# 0

58' 9 / +$ T 0 !< . c= 4 E $& >& \ &6 0 4 "0 E $ .$ *$ O U) doc.pdf !#&= &# ;H7 . $ &0 . . c= 4 R# *ZM $& linalg . c= *$ + -&' !< y .$ 5 with(linalg) . $0 e0& 2 a2 $ −−−−−−−−−−−→! r(t) = f (t), g(t), h(t)

.$ 5

. $ &0 . *$ + $. !< y

r:=vector([f(f),g(t),h(t)])

&+ (.$ 2 a2= 2 V=) '6 $ e0 &0 5 O $ $ 5F KO ![1 !j= & .$ M &. $0 *$M !+, . $0 2 a2 'BC + bCc) . !) $

*O&0 nO mU !< y .$ ?z[1 r . $0 e0& M x= >. T R# .$ !<

map(diﬀ, r, t) −−−−→ t 0 ([" r . $0 e0& t 6

!<

map(int, r, t) −−−−→ t 0 ([" r . $0 e0& RU & !<

map(int, r, t = a..b) −−−−→ b & a E r . $0 e0& . $0 e0& Z L0 M x= U!U) / $ R# .$ *ZM Z [a; b] E&0 0 . r(t) = f(t), g(t), h(t)! . $ E >. T plots[spacecurve]([f(t),g(t),h(t)],t=a..b)

numpoints color > . $ &0 *ZM $& Y1 p&\ $ U . 5 X 0 thickness *$ + M . O Z # /.$F .$ 78 $ $ http://webpages.iust.ac.ir/m_nadjafikhah/r2.htm

*( O $.F MnB = d'&[ 0 p 0 e0& & B&a

^Y" N# Ǳ& S M ( 5F XY R# N# A1 !#[ N# E h mTC 5F O&0 O $ $ *( oL6

OU!U)

G

U5

U2

$ 0 I ⊆ R E&0 0 U0& f (t) M x= 2 !?D)3 u = v = U#C O&0 R=F kM N# X , u, v N# >. T R# .$ *( LB$ $, t ∈ I Au ⊥ v M $. $ $ ) I $ &0 r(t) Y" N# & ) ∀t ∈ I : κ(t) = f (t)

) T(t

) r(t ) N(t

)=u

)=X )=v

& . @ Y" $ S MC 5F >. T R# .$ 0 0 Ǳ& . $ & p&\ .$ M O&0 *1&[Y !0&1 $ kM 0 ?, M ( 4E_ ^Y" b & $. .$ V# RL & M *$ O $ $ c %& Ǳ& = OU!U)

G

U5

U2

E&0 0 U0 g(t) ≤ f (t) M x= U#C O&0 $ . R=F kM N# < X , u, v, w > " O&0 $ +, Z 0 $ 0 $ $ 0 N# 0 0 . $0 R# @ R# .$ *( LB$ $, t ∈ I Aw = u × v ?U0 M $. $ $ ) I $ &0 r(t) N# & N# >. T

I ⊆R

) ∀t ∈ I : κ(t) = f (t)

) r(t ) N(t

) ∀t ∈ I

: τ (t) = g(t)

)=X

) T(t

)=u

)=v

) B(t

)=w

M O&0 & . @ $ S MC $ 5F >. T R# .$ 0 0 %& Ǳ& . $ & p&\ .$ *1&[Y !0&1

i j R# .$ 60 UB&Y 0 78 $ + $& !" #$ s %& M 0

*M U) r7 5 5 6# !'&

! " #$ % *f ( , /a) = a .$ *y = /a Z#. $ x = x= &0 ) * !VO 0C R = R − {} E ( >.&[, f $0 U# *A$ O f ( , − ) = − M $ O h'? e0& R# $. .$ *f ( /x, x) = f (x, /x) = f (, ) = / }

5F ' v Q e0& 4 =U !j= R# E m e0 S R# UB&Y 0 %& M U0 XB&Y +, *O&0

M . -&' !j= UB&Y /& R+ 0 $. $ i&j Z+U e0 S R# *$ + \ 5 j= #& Z = 0 * " R → R B +U e0 U[@

f

/S' *T I & U& ( O& 0 R 0 R E M e 0& U0& f : R → R S * & −n . & F E 0 f M #& X ∈ R + , + O&0 v −n Z$ 5&6 D $&+ &0 & f )* . $$S G#U R $&+ &0 & f . X ∈ D M & f (X) + , +

*D := X ∈ R f (X) ∈ R 9Z$ 5&6 n

n

n

v $ e0& 9* !VO .$ *#0 h .$ . f (x, y) = log x v $ e0& A, .8 E ( >.&[, f $ >. T R#

f

z = f (x, y)

f

y

Df

=

f

n

a

a

f(

, ) = log

f ( , y) = logy

= log

n

K?YT D#n3 . h Y\ E 78 S R# ( 0 e1 .$ ( . O$ +M rv Q e0&s f s U# *( . $0 N# f : R → R e0& v M $ O S *r( B −n . $0 v &0 U0& . f (x, y) = /xy v $ e0& A& '() T GB&L xy #&0 f (x, y) 5O G#U 0 *#0 h .$ E ( >.&[, f $ R# 0&0 O&0

(x, y) x > , y > , y = − {y = }

e0. y ) f $0 RU 0 ?U0 *A$ O ) ;* !VO 0C y = e x= &0 *x = y &F f (x, y) = a S M ZM

R !M 0 0 f $0 U# *f (e , e) = a nB x = e Z#. $ z y x S M $$S h'? &a 5 , 0 &# .$ *( &F O&0 N# GB&L ([a \\' $ , =

f

a

=

Df

=

(x, y) xy = = R − (x, y) xy = R − (x, y) x = y = & ' & ' −y ∪ −x R −

&# .

.

= O&0 . 1 S M ZM ) f 0 RU 0 ?U0 *a = &z+ ' ]3 *xy = /a #&0 &F f (x, y) = a

=

=

f (x, x) = logx x = f ( /x, y) = logy /x = − logy x = −f (x, y) f (xy, z) = logz xy = logz x + logz y = f (x, z) + f (y, z)

J

oL6 . #E e0 E N# $0 $ , x + y ) f = arcsin x − y

)

f = ln ( − xy)

)

f=

f = ln x + y

)

) f=

0

9M

. f (x, y, z) = Df

√ f = x+ y ) f = ln x + y + z

)

v e0& A .8 0 0 f $ >. T R# .$ *#0 h .$ − x − y − z

(x, y, z) − x − y − z ≥ (x, y, z) x + y + z ≤

=

)

√ √ x + yz

,

=

}

f = ln (xyz)

)

f = arccos (x + y + z) , ) f = ( − x) ( − y) ( − z)

M . T .$ A&& 9[& ("0 Y v = (, ) %) lim f (X + tv)

X = (− , ) f (x, y) = x/y − x y

GB ) lim f (X

+ tv)

t→

:)

t→

d ) f (X + tv) dt t= d ) f X − sin(t )v dt t=−

$

lim f (X + tv)

t→−

K) dtd f (X

+ tv)

t=

. f (x, y) e0& *f (x + y , y/x) = x − y M x= A; *0&0

e0& $ 9;* !VO 5 Q ?U0 *O&0 N# `&UO [ cM 0 M M ( ≤ − x − y − z ≤ ]3 ≤ x + y + z ≤ f $0 R# 0&0 * ≤ f (x, y, z) ≤ Z#. $ R =S .c) &0 .$ *A$ O ) * !VO 0C R = [; ] E ( >.&[, Z#. $ (x, y, z) ∈ R E 0 M $$S h'? &# *f (x, y, z) = f (±x, ±y, ±z) = f (±y, ±x, ±z) } f (x, y) = logy x

f

e0& * ≤ r f (r sin θ , r cos θ) = r tan θ M x= A *0&0 . f (x, y) f

ρ sin φ cos θ, ρ sin φ sin θ, ρ cos φ = ρ

l&Q A *0&0 . f (x, y, z) e0&

#E $' f (x, y) GB )

= xy/(x + y )

lim

lim f (x, y)

x→

y→

%)

lim x→

M . T .$ AH 9M [&

lim f (x, y)

y→x

}

&a e0& . $ + 9* !VO 9>. T R# .$ f (x, y, z) = z lim f

t→ +

t , t , t

= lim

t→ +

v $ e0& . $ + AGB 9* !VO e0& . $ + A% D , + 0 z = f (x, y) D : x + y ≤ , + 0 f = x + y

+

, x + y

= lim

t→ +

ZM x= A .8

+ t

t + t

+ t =

d f ( − t , t + , t − ) = dt t=− , d = ( − t) + (t + ) dt t − t=− 4 t +

+ = (t − ) (t − ) t + t + 5 + √ − t + t + = t=−

ZM x= 9fC) + a2 = $ D 0 f . $ + E . h *D ⊆ D ( v $ U0& 9z = f (x, y) M ( &q= E (x, y, z) 5 Q @&\ + , +

z = f (x, y)

f

(x, y, z) (x, y) ∈ D , z = f (x, y)

Γf,D :=

S *Γ : (x, y) ∈ D , z = f (x, y) j L u#&+ .$ &# *AGB * !VOC ZM . $ $ D M{ E D = D '() % f (x, y) = x + y M x= A& S (x, y, z) ∈ Γ >. T R# .$ *D : x + y ≤ +"1 D 0 f . $ + X R# 0 z = x + y S y\= T−xy 0 5F # j M ( q0 5 S+ N# E 9M ) % * !VO 0 *( x + y ≤ f,D

f

} v $ e0& N# E 98* !VO . D 0 z = f (x, y) . $ + $. .$ 0 & 9M Z

f,D

) f = − x − y , D : x + y ≤ , ≤ y , ) f = − − x − y , D : x + y ≤ , ) f = x + y , D : ≤ x + y ≤ , ) f = + x + y , D : x + y ≤

, ) f = x + y − , D : ≤ x + y ≤

Γf,D : x + y ≤ , z = x + y

D : ≤x≤ , ≤y≤ , ) f = y − x , D : x ≤ y ≤

}

) f = ,

) f = x, ) f =

D :

− x − y, D :

) f = sin x,

D :

≤x≤

, ≤ y ≤

+ x + y

e0& . $ + AGB 9H* !VO ≤ x , ≤ y , x + y ≤ , +

+ 0 − x − y v $ e0& . $ + A% x + y ≤ , +

, f (x, y) = x + y M x= A, .8

−x

≤ x ≤ π , ≤ y ≤

) f = cos(x + y ), D : x + y ≤

) f =

0

≤ x ≤ , ≤ y ≤

π

− |x| − |y|, D : |x| + |y| ≤

D :

*a ∈ R z = f (x, y) M x= : U!U) * . ( a 0 0 f & F E 0 M #& (x, y) + , +

.$ *Z$ 5&6 C $&+ &0 & a 0 h f E

0C O&0 f e0& . $ + &0 z = a T $. 0 ! C e1 & N+M 0 *C : f (x, y) = a 9A$ O ) 8* !VO *$M Z" . e0& . $ + 5 E h .$ . f (x, y) = x − y e0& A& '() 9$ . ( RV+ (B&' a ∈ R S *#0 B Bn N# C : x − y = a >. T R# .$ 9 < a AGB *( . −x & . .$ y$ `&+ ) C : x = y >. T R# .$ 9a = A% f

a

a

,

*z = x + y M (x, y, z) ∈ Γ @O 0 >. T R# .$ & z . > E 0 pL N# M z = x +y #$ 5&0 0 ( h $. 5F #_&0 + E +"1 ]3 ≤ z 5 Q *( H* !VO 0C *$$S D 0 0 T−xy 0 5F # j M E ( >.&[, D 0 f . $ + .$ *A$ O ) GB f,D

a

a

≤ x , ≤ y , x + y ≤

Γf,D :

≤ x , ≤ y , x + y ≤ , z = ,

, x + y

R# .$ *f (x, y) = − x − y M x= A .8 (x, y, z) ∈ Γ 1 & 1 D : x + y ≤ >. T , x + y + z = M $ O ) *z = − x − y M ) % H* !VO 0C O&0 `&UO Ǳ [ cM 0 M *A$ O f

f

e0& A& '() N# S &F a > S >. T R# .$ *#0 h .$ . ( pL S &F a = S ( Q.&3 N# 5 S B Bn 0C O&0 Q.&3 $ 5 S B Bn N# S &F a < S *A$ O ) GB * !VO } f (x, y, z) = x + y − z a

a

a

*( y = −x y = x .$ B Bn N# C >. T R# .$ 9a < A: *A$ O ) J* !VO 0C ( . −y & . } a

: x − y = a

e0& E & 9J* !VO S *#0 h .$ . f (x, y) = min {x , y} e0& A, .8 9&F a ∈ R & ' C : min x , y = a : a = x ≤ y &# a = y ≤ x 7* !VO 0C O&0 (a, a) Y\ E `O &0 y Z $ M *A$ O ) } f = x − y

e0& E K Y AGB 9* !VO e0& E K Y A% f = x + y − z

a

f = x + y + z

.$ *#0 h .$ . f (x, y, z) = x + y + z e0& A, .8 &F a = S *( S &F& a < 'S >. T R# &F a > S _&0 √ ( (, , ) Y\ N S % * !VO 0C O&0 a `&UO Ǳ [ cM 0 M S *A$ O ) >. T R# .$ *#0 h .$ . f = x+ y − z e0& A .8 0C O&0 x + y − z = a B$&U 0 T E >.&[, S *A$ O ) GB I* !VO } a

a a

a

E & 97* !VO . #E e0 E N# E & 0 9M Z" . & F . $ + $M Z. f = min{x, y}

) f = x + y

)

e0& E K Y AGB 9I* !VO X 0 X X X A% f = x + y − z * " A , + 0 E .$ Y\ . O $ $ e0 E N# E K Y 0 9M Z ) u=x+y+z

)

u = x − y − z

)

u = z − x − y

)

u = z − x + y

u = |x + y| + |z| ) u = z − xy ) u = arcsin x + y + z

) u = sgn sin x + y − z

)

f = x + y

)

f = |x| + |y| − |x + y| & ' ) f = max x , y

)

f = x sin y

)

f = (x + y)

)

f = y/x

√ xy

) f = sin π x + y

)

f=

)

f = |x − y |

&q= 5$ 0 U0 !B$ 0 : g=? !1 ' *$. $ ) v e0& . $ + Z 5&V R t0&Y *( 5F E K Y Z $ $ 4& 5 M .&M + , + f (x, y, z) e0& E a $, 0 h E ^Y G#U 9U# *f (x, y, z) = a M ( (x, y, z) X &#& Sa : f (x, y, z) = a

U0& f : R → R M x= $, f $ 0 ["Q Y\ X ∈ R v Q ( X f Zg S . T .$ *( \\' δ > N# ε > Ǳ E 0 M lim f (X) = Z"#

tY .1 X ∈ B (X ) ∩ D E 0 M $ O (=&# ) * !VO 0C $ O +M ε E f (X ) f (X) !-& #$ 5&0 0 *A$ O

R - 6V (

n

n

X→X

δ

f

A* C

∀ε > ∃δ > ∀X ∈ Df < X − X < δ ⇒ |f (X) − f (X ) | < ε

E ( 1 +M < X − X < δ pO .0 5 Q 9$ + $& 5 #E $&U pO |x − x | < δ , |y − y | < δ , |z − z | < δ · · ·

Y\ X U# * =

A;* C

M x= A& '() − M (0&f >. T R# .$ *( (− , ) * lim xy = − Z$ 5&6 #&0 XY R# >&[f 0 *!'

f = xy

(x,y)→(− , )

∀ε > ∃δ > ∀(x, y) ∈ R & ' |x + | < δ, |y − | < δ ⇒ |xy + | < ε

9ZM `O |xy + | tY .1 E |xy + | =

| (x + ) y + − y |

=

|y (x + ) + (y + ) (y − ) |

≤

y |x + | + |y + ||y − |

≤

y δ + |y + |δ

!j= .$ MC #&"+ `&UO NV E δ X# - mn' 0 δ ≤ ZM x= *ZM $& A( O 0 .&"0 nB − < y < ]3 *|y − | < |x + | < R# 0&0 R# 0&0 *|y + | < y < |xy + | ≤ δ + δ =

*( !" #$ %&"' .$ T 4 E V# ' uL0 R# E m *# O &OF R 0 R E e0 ' 4 &0 ?z[1 G#U 0 R $3 E ![1 *O&0 R 0 R E e0 0 5F Z+U *Z#. $ E& ["Q Y\ K?YT 0 ' E . h < r ∈ R X ∈ R S ( R E X @&\ + , + r `&UO X cM 0 3 *X − X < r U# O&0 r E +M X & & F T&= M 0 )*6+ 3 *Z$ 5&6 B (X ) $&+ &0 . , + R# X Y\ B (X ) , + E >.&[, r `&UO X cM

9( n

n

n

r

r

Br (X ) = Br (X ) − {X }

( A ⊆ R , + *$E )*"9 X Zg S . T .$ , + r `&UO X cM 0 S r > E 0 M ["Q X X p&\ % I* !VO .$ *M eY1 . A $ ) X cM 0 S N# !1 ' #E (" X B " *M+ eY1 . A M $. $ # $ 0 ["Q p&\ , + A& '() *( 0 0 # $ $ &0 x + y = x + (y − ) < E&0 S 0 ["Q p&\ , + A, .8 *O&0 x + (y − ) ≤ "0 S 0 0 &F A : ≤ x ≤ , x ≤ y < , ≤ z ≤ S A .8 E (".&[, A 0 ["Q p&\ , +

n

A :

≤x≤

,x≤y≤

,

p&\ , + &F A = & /n n ∈ N' S A .8 ) #&0 $. .$ *A = A ∪ {} E (".&[, A 0 ["Q !VO 0 $, (#& 0 ?z+, (−ε; ε) !VO 0 E&0 E&0 M $ O *n ≥ [ /ε] M $ O x= ( =&M *$. $ $ ) /n }

δ

* δ ≤ ε $ O x= ( =&M |xy + | < ε 0 ]3 M ZM x= Z R# 0&0 *δ ≤ ε/ #$ 5&0 0 (=. 5 B&" 0 0 5 M *δ = min { , ε/ } *$ + >&[f . h $. @O Y0 . X = (, ) f (x, y) = xx+−yy M x= A, .8 #$ >.&[, 0 * = / M M (0&f >. T R# .$

≤z≤

' >&[f 9* !VO

R# 0&0 ∀ε > ∃δ > ∀x ∈ [− ; ] " # √ π < x + < δ ⇒ arcsin x + < ε

ZM (0&f Z M lF &0 . cS R# XM &0 $ O (0&f #&0 M Z#S

∀ε > ∃δ > ∀ (x, y, z) ∈ Df < |x + | < δ , √ < y − < δ , < |z − | < δ √ < ε ⇒ xyz +

9M ZM ) R# >&[f 0

√

yz (x + xyz + = ) − z y− ≤ |yz||x + | + |z| y − = |yz| + |z| + δ

−

+

z−

√

√ z −

9Z#. $ δ <

/

B

∗ δ

(X ) ∈ Df √ xyz + / < { + + /} δ < δ

R# 0&0 *δ

x= &0

&# δ

R# 0&0

$ O x= ( =&M ]3 *δ = min { / , ε/} sin x + y + z f (x, y, z) = x + y + z ZM x= A .8 U# * = M $ 5&6 >. T R# .$ X = (, , ) ≤ ε/

≤ ε

sin x + y + z lim

(x,y,z)→( , , )

x + y + z

=

9M Z$ 5&6 #&0 *!' ∀ε > ∃δ > ∀ (x, y, z) ∈ Df

lim

(x,y)→(,)

x + y = x − y

M $ O (0&f #&0 *!'

∀ε > ∃δ > ∀(x, y) ∈ Df

x + y − (|x − | < δ, |y − | < δ) ⇒ x − y X = (, )

< ε

D = R − {y = −x} ∪ {y = x} &# .$ M >. T R# .$ *O&0 D 0 ["Q Y\ N# f

f

x + y x − y −

= x + y − x + y ! x − y

( x + ) (x − ) + ( y + ) (y − ) |x − y | ≤ | x + ||x − | + | y + ||y − | |x − y | x + | + | y + | δ | ≤ |x − y |

=

M O&0 . @ δ x= #&0 #EC δ ≤ / ZM x= &' >. T R# .$ AB (X ) ⊆ D ∗ δ

⎧ ⎧ <x< ⎪ ⎪ ⎪ |x + | < ⎪ ⎪ ⎪ ⎪ ⎪ ⎪

⎨ ⎨

⎪ ⎪ ⎪ ⎪ √ √ √ ⎪ ⎪ ⎩ ⎪ ⎩ z − < − +

⎧ √ ⎪ ⎪ − ≤

− < xyz ⎪ ⎪ ⎪ ⎪ ⎪ √ + < ⎪ ⎨ < ⇒ √ − < yz < √ + < ⎪ ⎪ ⎪ ⎪ ⎪ √ ⎪ √ ⎪ ⎪ ⎩

−

*

⎧ ⎪ ⎨ ⎪ ⎩

f

⎧ ⎪ ⎨

<x< ⇒ ⎪ ⎩ < |y − | < < x + < ⇒ ⎪ |x + | < ⇒ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ < y + < ⎩ |y + | < < |x − | <

x + y x − y −

R# 0&0 *δ

<

≤

× (/) δ ≤ δ = ε/ &# δ < ε $ O

R# 0&0

+×

δ

x= ( =&M ]3

*δ = min { / , ε/ } Rl+ f (x, y, z) = arcsin (xyz) Mx= A .8 √ #$ >.&[, 0 * = π/ M (0&f *X = − , /, * lim arcsin(xyz) = −π/ ( ) M $ O (0&f #&0 *!' (x,y,z)→ − , / ,

√

∀ε > ∃δ > ∀(x, y, z) ∈ Df √ |x + | < δ , |y − /| < δ , z − < δ ⇒ |arcsin(xyz) + π/| < ε

0 X D = {(x, y, z) | − ≤ xyz ≤ } &# .$ M lim arcsin x = −π/ 5 Q *A QC ( ["Q D f

x→−

√

f

/

)

lim

(x,y,z)→(x ,y ,z )

O&0 ÷ × − + &+, E V# 2 S * 9&F a ∈ R O&0 v Q e0 g f )

X→X

X→X

lim f (x) =

X→X

X→X

g : R → R f : R

n

→R

S A

lim g (f (X)) = lim g(t)

X→X

lim g(t) = X g t→t

t→

: R→ R

n

f

S A &F S AH

: Rn → R

lim f (g(t)) = lim f (X)

t→t

X→X

, + E X E 0 M $ O (=&# r > O&0 . 10 f (X) ≤ g(X) &" & D ∩D ∩B (X ) lim f (X) ≤ lim g(X) >. T R# .$ 9Z#. $ 7*;* 0 ) &0 '() f

X→X

g

∗ r

X→X

lim

(x,y)→(,)

x y =

lim

(x,y)→(,)

lim

x

lim

(x,y)→(,)

y

lim y =

x→ y→

arcsin x + y + z =

= lim x (x,y,z)→( ,,−)

π = lim arcsin(t) = t→ x + y z = ) lim (x,y,z)→( ,,−)

= lim x + lim y lim z = y→

x→

)

)

sin x + y + z x + y + z ≤

X→X

lim (f 2g) (x) = lim f (X)2 lim g(X)

&F

)

E 0 ≤ sin α ≤ α M Z $ *D

lim af (X) = a lim f (X)

X→X

)

)

⎞ sin x + y + z ⎠ ⎝{|x| < δ, |y| < δ , |z| < δ} ⇒ x + y + z < ε ⎛

k=k

z→−

x − y = lim (x + y) (x,y)→(,−) x − y (x,y)→(,−) lim

= lim x + lim y = x→ y→ sin x + y + z

lim

(x,y,z)→(,,−)

x + y + z

= lim t→

sin t = t

* lim x + y + z = + − = #E 0&0 >. T R# .$ u = max &|x|, |y|' ZM x= A5 .8 9Z#. $ 7*;* E H (+"1 (x,y,z)→( , ,− )

≤

lim

(x,y)→( ,

x y u u ≤ lim = lim u = ) |x| + |y| u→ + u u→

R# 0&0 *|y| = u &# |x| = u . + #E

lim

(x,y)→( ,

x y = ) |x| + |y|

f

&# .$ M 9R# 0&0 ≤ α

= R

x + y + z x + y + z

*u < δ x= #&[ ]3 u = max {|x|, |y|, |z|} ZM x= &' 9?U0 sin x + y + z x + y + z ,

≤

u + u + u = u < δ + + u

*δ = ε/ &# δ = ε $ O x= ( =&M ]3 M (0&f ' G#U N+M 0 0 $ 9M @O 0 lim f (X) = X→X

) f = xy − y/x, X = (, −) , = /

) )

f = sin (x − y) , X = (π/, π) , = x + y f= , X = (, ) , = / x + y

f = x + y + xyz, X = ( , , − ) , = , √ ) f = x + y + z , X = ( , , ) , = ) f = tan (x + y) − tan z, X = π , , π , =

)

E e0 [O (.$ G J .R K+ % $' [& 0 5E D. N# c (B&' R# .$ R 0 T * & [) D. . D. R# *$ + K@ 5

C $ O KY [ 5 , 0 ' $ U 9( RQ 5F N+M 0 M q1 $ U ]< A$$S >&[f ε − δ >. j0 $ + 5&0 5 $& $' X"' 0 . l3 $' & F ( =&M LB$ ' N# [& 0 5 M *$ O KY

AB x $'C ) 0 & ZM $& &q1 E .\F *Z0 f v N# e0& Ǳ E 0 + J '+ R & 9Z#. $ k (0&f $, R

)

) )

lim

f (x) = lim f (x)

lim

f (y) = lim f (y)

lim

f (z) = lim f (z)

(x,y,z)→(x ,y ,z ) (x,y,z)→(x ,y ,z ) (x,y,z)→(x ,y ,z )

x→x

y→y

z→z

= lim r cos θ = r→ ⎞z ⎛ sin x + y ⎝ ⎠ lim (x,y,z)→( , ,−) x + y # "z sin r = lim = − = (r,z)→( ,−) r xyz lim (x,y,z)→( , , ) x + y + z (ρ cos φ cos θ) (ρ cos φ sin θ) (ρ sin φ) = lim ρ→ (ρ cos φ cos θ) + (ρ cos φ sin θ) + (ρ sin φ) = lim ρ cos φ sin φ cos θ sin θ =

9M [& . $' E N# . \

)

)

)

) ) ) )

lim

(x,y)→( ,

xy + ) x + y

lim

(x,y,z)→( ,

lim

)

x y

)

(x,y)→( , )

9M

x + y

(x,y,z)→( , ,)

#

tan

x + y x−y

"

x − y + z

lim

(x,y,z)→(,−,)

xy − y − x + x− (x,y)→(,) lim

lim

(x,y,z)→(π, ,)

ze−(x+y) cos (z)

x y (x,y)→( , ) x + y ) lim ln x + y + z

)

x y z ) |x| + |y| + |z| , arctan x + y + z +

(x,y,z)→( , , )

lim

0

lim

x sin (πy)

x + y − z

zxy x+ ) lim y (x,y,z)→(,,−) # "xy+ x + y ) lim x+y+ (x,y)→( , )

x sin y − y cos z ,) |x| + |y| + |z|

(x,y,z)→( , ,

lim

)

lim

(x,y)→(π, )

)

ρ→

[& *;* N+M 0 . #E $'

lim

(x,y)→(,−)

0

lim

(x,y,z)→(, ,−)

√ √ x−y+ x− y √ √ x− y (x,y)→( , ) ) lim sin arctan x − y z

)

x+y+z x + y + z

lim

(x,y,z)→( , , )

R ! , $( $ ) $' A&Y0 U#C $ ) 4, >&[f 0 Z D. $ ' 4 Z+U " ' *&B&[$ " ' *$. $ *( v Q e0 $. .$ =YV#

)

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+ =C ,.? @PC\) 2 E .

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lim

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)

}

lim f (x, y, z) =

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(r,z)→( ,z )

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)

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ρ→

n

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lim

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n

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=

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.$ f " ' &F O&0 AD .$ &z &+ C X .$ "

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limit(f(x(t), y(t), z(t)), t = 0)

# /.$F .$

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http://webpages.iust.ac.ir/m_nadjafikhah/r2.htm

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X

x y + lim

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>. T 0 v = y − u = x + x= &0 . |Q (+ ' 9(O 5 #E

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f (x ) = lim

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x→x

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(x,y,z)→(−,−,)

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r(t) = −(−−,→t) r (t) = −(t,−−→) " &' 0 |Q (+ e0& ' "#&0 x= 0 &0 *Z#S h .$ . .$ *O&0 T &" R# E N#

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t

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t→

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4

lim

xy

(x,y,z)→(,−,)

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.$ y 0 ([" f #c) t 6 0&6 >. j0 *Z$ 5&6 X 0 . X Y\ .$ z 0 ([" f #c) t 6 c X Y\ 9ZM G#U #E >. T 0

∂f ∂z X

∂f (X ) = fy (X ) ∂y d f (x , y, z = f (X ) = dy

=

)

y=y

∂f (X ) = fz (X ) ∂z d = f (X ) = f (x , y , z) dz z=z

=

*$. $ $ ) v . . . .& Q $ e0 0 0&6 G#.&U f (x, y, z) = xyz M x= A& '() $ .$ f gc) >&\ 6 . \ >. T R# .$ *X = (, , − ) *M [& . X Z#. $ G#U 0 &0 *!' ∂f ∂x X ∂f ∂y X

= = =

∂f ∂z X

= =

d d f (x, , − )|x= = {−x}x= = − dx dx d f (, y, − ) dy y= d −y = {−y}y= = − dy y= d f (, , z) dz z=− d z = z = dz z=− z=−

c f (x, y) = arcsin(x/y) M x= A,√.8 X .$ f gc) >&\ 6 . \ >. T R# .$ *X = ( , ) *M [& . Z#. $ G#U 0&0 *!' ∂f ∂x X

=

√ xy, X = (, ),

)

) f=

&$&+ E V# &0 A$ ) >. T .$ [B C ZM G#U

∂f ∂y X

f=

x d d f (x, ) √ = arcsin dx dx x=√ x=

X = (, ), xyz x + y + z

(x, y, z) = (, , )

(x, y, z) = (, , )

,

X = (, , )

O&0 #n<\ 6 X .$ f S A& $ $ ?U0 ( #n<\ 6 X .$ af &F a ∈ R

(af ) (X ) = af (X )

f /g

f g f − g f + g &F O&0 #n<\ 6 X .$ g f S #n<\ 6 X .$ c

)

(f + g) (X ) = f (X ) + g (X )

)

(f − g) (X ) = f (X ) − g (X )

)

(f g) (X ) = g (X ) f (X ) + f (X ) g (X )

g (X ) f (X ) − f (X ) g (X ) f ) g (X ) = g (X ) n f :R →R t r : R → Rn X = r (t )

.$ .$ S A8 ( #n<\ 6 t .$ c f (r(t)) &F O&0 #n<\ 6

* {f (r)} (t ) = f (r (t )) r (t ) ?U0 f (X ) .$ g : R → R X .$ f : R → R S AJ ( #n<\ 6 X .$ c g (f (X)) &F O&0 #n<\ 6

*{g (f )} (X ) = g (f (X )) f (X ) ?U0

•

n

&0 .&"0 H**H q1 E 8 (+"1 78 % $ uL0 .$ *$ O & t 6 E ,&1 ( (+ *( $3 Z 5F 60 UB&Y 0 >&[f . H**H q1 E (+"1 A& 0 & $ *M *M (0&f . H**H q1 E ; (+"1 A, *M (0&f . H**H q1 E (+"1 A

JE WP' + #E (

+, >&-&#. .$ Z .&"0

gc) t 6

t 6 0 v Q e0 t 6 4 (6S0 0 . ?z+,

t 6 &0 ln f ∂f ∂y

∂f = f . z . y z− . ln x ∂y ∂f f = ln ∂y X ln(ln f ) = z . ln y + ln(ln x)

Z Z#0 ln .&0 $ E S * R=@ E S t 6 &0 ∂f /∂z f

R# 0&0 (O $ Z#. $ z 0 (["

∂f = f . ln f . ln y ∂z ∂f = ∂z X

x+y f (x, y) = arctan − xy

*

= (, )

∂f ∂x X

=

= ∂f ∂y X

=

=

R# 0&0

( ) ( − xy) − (x + y) (−y) ( − xy)

x+y + − xy X = x + X ( ) ( − xy) − (x + y) (−x) ( − xy)

x+y + − xy X = y + X

f = x sin(y ), , ) f = x + y ,

)

f = xy ,

X = ( , )

)

f = arctan(x/y),

X =( , )

)

f = logy x,

X = (, )

)

)

f = xy − xz,

X = (, , )

)

f = xy ln z ,

X = (, , e )

)

f = ln(x + y) −

+ x + z , X = (, , )

f = arctan(x + y + z ), X = (, , − )

) f =e

xyz

,

X = (− , , )

d dy

4

# √ "5

arcsin

=−

y=

y

y=

f (x, y) = y √x M x= A .8 *(" $ ) ∂f /∂x M (0&f >. T M ( 5F R# !B$ *!'

R# .$ *X

= (, )

∂f ∂x X

f (x, ) − f (, ) x− x→ √ x = lim √ lim = +∞ x x→ x→ x

=

lim

=

5 $& . #c) >&\ 6

78 $ $ ) ' ∂f /∂x [& .$ e1 .$ *$M %&"' 9$. $ z y . nS&) A x 0 ([" S t 6 A; *x . nS&) A 9$ $ u&M 5 ' $ 0 . ' R# *Z#S t 6 x 0 ([" z y 5$ 0 (0&f x= &0 AGB *ZM [& X .$ . !T&' e0& A% X

0 $ $

X = (, − ) √ X = ( , π) √ X = ( , )

) f = xy − x/y,

)

=

=

X

S A .8 &F

Y\ .$ . f e0& #c) >&\ 6

X

=

= ln y =⇒

ln f

X

∂f ∂y X

√

/

=

X

= z . y z− . ln x =⇒

f

, √ − (x/) x= d √ f , y = dy y= √ − /y , √ − ( /y)

0 ) &0 * ∂f = R# 0&0 ∂x Z#. $ y 0 ([" Y0 . R# R=@ E S

= y z ln x

90&0

f (x, y, z)

= xz − y/x ∂f ∂x X ∂f ∂y X ∂f ∂z

S A& '() $ &F X = ( , , − )

y z+ = x X − = =− x X

=

= {x}X = .

X

&F X

= (π, /)

∂f ∂x X ∂f ∂y

=

f (x, y) = y sin(xy) S A, .8

y cos (xy) X

=

=

sin (xy) + xy cos (xy)

= X

X

&F X

= (, , ) z

∂x(y ∂f = ∂x ∂x

)

f (x, y, z) = x S A .8 yz

= y z . x(y

z

−)

( RV+ A2 h) 4R @1Cc) $ 5 #c) t 6 z $ 5F E nB O&0 #n<\ 6 U0& 9

#E & #. nS $&+ *(=S

f = x sin(y + cos z), X = ( , π/, π/) 0 y ) f = t dt, X = (, )

)

∂f ∂x

∂f ∂x

∂ ∂f ∂f = fyx = f = ∂x∂y ∂x ∂y " # ∂ ∂f ∂f = fz x = f = ∂x ∂z ∂x∂z ∂f ∂ = fxx = f = ∂x ∂x

∂f

X

f (x, y, z) = arcsin

B&' .$ ( −

Z#. $ ` + .$ R# 0&0

∂f (x + θ h, y + θ k). ω = kh . ∂x∂y

A*H C

E&0 &F ψ(x, y) = f (x, y + k) − f (x, y) ZM x= &' $ , 0&6 >. T 0 ω = ψ(x + h, y) − ψ(x, y) Z M . $ $ ) θ, θ ∈ (; ) ω

∂ψ (x + θ h, y) ∂x ∂f ∂f (x + θ h, y + k) − (x + θ h, y) = h. ∂x ∂x ∂f (x + θ h, y + θ k). = kh . ∂y∂x = h.

A;*H C

S A&

−

'() & $

&F

xyz , ∂f = + ∂y − x y z

0 0 f $ M $ O ) 0 0 !T&' #c) t 6 e0& $ M *O&0

≤ xy z ≤

< xy z <

&F f (x, y) =

M $. $ $ ) θ ∈ (; ) S_ q1 E $& &0 Z E&0 ∂f (x + θ h, y + θ k). ∂x∂y

xy z

∂f xy z = + ∂z − x y z

∂ϕ (x, y + θ k) ∂y ∂f ∂f (x + h, y + θ k) − (x, y + θ k) . k. ∂y ∂y

∂f ∂f (x + h, y + θ k) − (x, y + θ k) = ∂y ∂y

y z ∂f = + , ∂x − x y z

k.

= h.

∂f D X → ∈R ∂x X

0&6 >. j0 * & x 0 ([" f #c) t 6 . e0& R# ( =&M ∂f [& 0 * " G#U !0&1 ∂f ∂f ∂x ∂z ∂y ∂f $ 4& . ∂x [& .$ 4$ '

=

X =( , , )

∂f : R → R , ∂x

E #&"+ N# .$ ∂y∂x ∂x∂y S $ *( 0 0 X .$ $ 5F . \ &F O&0 3 X #&"+ N# .$ ∂ f /∂x∂y ∂ f /∂y∂x ZM x= (x + h, y + k) (x, y + k) (x + h, y) (x, y) p&\ ! &O ϕ(x, y) = f (x + h, y) − f (x, y) ZM x= *O&0 3 *ω = ϕ(x, y + k) − ϕ(x, y) M $. $ $ ) θ ∈ (; ) S_ q1 0&0 =

dt , z+t

f

?za U# ( Z #c) >&\ 6 X M (B&' .$ ∂f ∂f = 0 =&M pO N# *A$ O ) H R#+ 0C ∂x∂y ∂y∂x *( #E KO 0 #c) >&\ 6 X 5$M (+ 0

ω

f=

e0& #c) t 6 S 4R @1Cc) a2= % $ $ ) D ⊆ D .$ X p&\ M .$ x 0 ([" u = f (x, y, z) 9ZM G#U #E KO 0 #) e0& Z O&0

***

∂f

)

x xy

0

x ∂f = + ∂x x + y

,

+ x + y

∂f = ∂y

y

+

S A, .8

x + y

$ M B&' .$ ( ≤ x + y 0 0 f $ (B&' R# .$ *( < x + y 0 0 !T&' e0& $ . #E e0 E N# #c) >&\ 6

) f = x sin(x + y)

)

f = y cos x

)

f = arctan (y/x) , ) f = x + y + z

)

f=

√ x

yz ) f = ln x + y + z

)

0 * $

9M [&

f = x + y − x y

)

f = xy + y/x , ) f = arcsin x + y

)

f = xy + yz + zx

)

f = zex + xey + yez

)

f = x cos y + z ln(x + y)

) )

M Z#S A;*H C A*H C "#&\ &0

∂f =? ∂x∂y∂z ∂ f =? ∂x∂y

f = x ln (y + z) , f = xy ,

∂f ∂f (x + θ h, y + θ k) = (x + θ h, y + θ k). ∂x∂y ∂y∂x

(,−)

xy z

∂f =? ∂x ∂y ∂z

)

f =e

)

f = (x − a) (y − b) ,

∂ p+q f =? ∂xp ∂y q

)

f = xyz + ex+y+z ,

∂ m+n+k f =? ∂xm ∂y n ∂z k

, p

q

>. T 0 . f : R → R e0& A5 ⎧ ⎨

x − y xy f (x, y) = x + y ⎩

S (x, y) = (, ) S M $ 5&6 >. T R# .$ *ZM G#U * ∂f = x x E 0 AGB ∂y * ∂f = −y y E 0 A% ∂x ∂ f ∂ f * ∂y∂x = A: ∂x∂y (x, y) = (, )

(x, )

( ,y)

( , )

( , )

5F N+M 0 M Z#. $ . q1 5&0 5&V 5 M *$ + t\ 5 ' . 0 . v Q e0 #n<\ 6

*$ O Z = c t 6 e# [& 5&V ?U0 >&\ 6 4&+ u = f (x, y, z) e0& S $ $ .$ e0 R# &F O&0 3 X Y\ .$ 5F [ #c) ?U0 ( #n<\ 6 X −−−−− # " −−−−−−−−−−−−−−−→ ∂f ∂f ∂f f (X ) = , , ∂x X ∂y X ∂z X

e0& 0 S_ q1 E $& &0 $ ) 5&Q t ∈ (x ; x) M Z#S [x ; x] E&0 0 &F X = (t , y, z) S M $. $ h(x) = f (x, y, z)

f (x, y, z) − f (x , y, z) = h(x) − h(x ) ∂f = h (t ) . (x − x ) = . (x − x ) ∂x X

A*H C

$ ) 5&Q t ∈ (z ; z) t ∈ (y ; y) $ , 0&6 >. T 0 M . $ ∂f . (y − y ) ∂y X ∂f f (x , y , z) − f (x , y , z ) = . (z − z ) ∂z X f (x , y, z) − f (x , y , z) =

A*H C AH*H C

(x, y) Y\ .$ gc) >&\ 6 3 pO 0 ) &0 5 M (h, k) → 1 m@ $ ' "#&0 M Z#S 2 *( q1 ZV' 5&+ R# M O&0 0 0 (, ) 9$ $ Z+U 5 #E M q1 0 ' . 0 . q1 R# .$ _&0 X gc) >&\ 6 S $ S t 6 X &F O&0 3 Y\ #&"+ *$. + pO $. E .&"0 .$ 5 Q 78 $ & F X 0 + ( R & #c) >&\ 6 3 *Z$+ R# .$ f (x, y) = x y ZM x= A& '() $ >. T ∂f ∂f = xy = x y ∂x ∂y ∂f ∂ = y = xy ∂x ∂x ∂ ∂ f = x y = xy ∂x∂y ∂x ∂ ∂f = xy = xy ∂y∂x ∂y ∂ ∂ f = x y = x y ∂y ∂y ∂f ∂ = y xy = ∂x ∂x ∂y ∂ ∂f = xy = y ∂x∂y∂x ∂x x+y f (x, y) = x−y 4 5 ∂f ∂ ∂f ∂f = = ∂x ∂y∂x∂y ∂ x∂ y ∂ x ∂y x ∂ − ( x + ) = = ∂ x (x − y) (x − y)

ZM x= A, .8

>. T R# .$

ZM x= A .8 0 >. T R# .$ f (x, y, z) = e * ∂x∂ ∂y ∂zf = a b c e n m l E . O #c) t 6 $. .$ 0 $ 9M [&

ax+by+cz

l+m+n l

m

l m n ax+by+cz

n

) f = x − xy + y + y ,

∂f =? ∂x∂y

.$ .$ * O G#U X .$ " & \

?U0 *( #n<\ 6 X .$ f R# 0&0 3 X f (X ) =

=

R# .$ *X

&0 .$ *X = (x , y , t) X = (t , y, z) & F .$ M Z#. $ AH*H C A*H C A*H C E $& ! f (x, y, z) − f (x , y , z ) = f (x, y, z) − f (x , y, z) ! + f (x , y, z) − f (x , y , z) ! + f (x , y , z) − f (x , y , z ) ∂f ∂f = . (x − x ) + . (y − y ) ∂x (t ,y,z) ∂y (x ,t ,z ) ∂f + . (z − z ) ∂z (x ,y ,t )

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→ x −x + ,+ + x − y X + x − y X − −−−→ ,

= ( , , − )

f = xyz M x= A, .8 5F gc) >&\ 6 e0 f >. T

∂f ∂f ∂f = yz , = xyz , = xy z ∂x ∂y ∂z

.& Q .$ * O G#U X .$ " & \

?U0 *( #n<\ 6 X .$ f R# 0&0 3 X .$ e0& −−−−−− −−−−−−−−−−−−−−−−−−−→ y z , xyz , xy z

f (X ) =

X

X

X

−−−−−−−−−−→! − , − ,

=

e0 f >. T R# .$ *f = ln( ∂f = ∂x

+ xy)

y ∂f , = + xy ∂y

x + xy

f

f

−−−−y−−−−−−− x−−→ , + xy + xy

*( D $ &0 f

M $ O ) >. T R# .$ *f = arcsin(x + y + z) M x= A .8 5F gc) >&\ 6 f f

M Z#S X Y\ .$ ∂f 3 E $& &0 5 M ∂x &F |x − x | < δ S M $. $ $ ) 5&Q δ > ∂f ε ∂f < − ∂x X ∂x X

: R → R

∂f ∂f − ∂x X ∂x X ∂f ∂f − ∂x X ∂x X

f

f (x, y, z) =

+

− (x + y + z)

*( D $ &0 f

.

−−−−−→! , ,

: R → R

M $ O )

A7*H C A*H C

* ∂f * *f (X) − f (X ) − . (x − x ) ∂y X * ∂f ∂f * − . (y − y ) − . (z − z )* < ∂y X ∂z X ∂f ∂f < − . |x − x | ∂x X ∂x X ∂f ∂f + − . |y − y | ∂y X ∂y X ∂f ∂f + − . |z − z | ∂y X ∂z X ε < |x − x | + |y − y | + |z − z |

<

f

ε < ε <

X − X < δ δ = min{δ, δ, δ } S >. T R# .$ Z#. $ AH*H C .$ A*H C A7*H C AJ*H C 5$ $ . 1 &0 &F

∂f ∂f ∂f = = =+ ∂x ∂y ∂z − (x + y + z)

( D : − ≤ x + y + z ≤ 0 0 f $ * " & \

U# ( D E 0 0 5F gc) >&\ 6 e0 $ M B&' .$ D , + 0 e0& .& Q .$ *D : − < x+y+z < R# 0&0 * 3 , + R# 0 + nB O G#U (x, y, z) ∈ D E 0 ( #n<\ 6 , + R# 0 f

AJ*H C

S M . $ $ ) δ > δ > 0&6 >. T 0 &F |z − z | < δ |y − y | < δ

M x= A .8 5F gc) >&\ 6

* O G#U D : +xy > , + 0 " & \

R# 0 f R# 0&0 * 3 , + R# 0 .$ (x, y) ∈ D E 0 ( #n<\ 6 , +

f (x, y) =

A8*H C

ε

× × X − X = ε . X − X

*$$S t 6 G#U E ZV'

2

$ U &0 e0& 0 q1 R# 78 % $ *$$S >&[f 0&6 >. T 0 ( (.$ c v

+

f = + x − y M x= A& '() & $ 5F gc) >&\ 6 e0 f >. T R# .$ *X = (, − ) x −y ∂f ∂f = + = + , . ∂x ∂y +x −y + x − y

&0 & F E !T&' e0 M + #.&B #&+ &aa "M .$ &+, R# *$$S !V6 & \ !+, & $ U * O 5&0 **

f

e0& $ 5&6 8 & >&#+ .$ 0 * $ *M [& . f (X ) ]< ( #n<\ 6 X Y\ .$

) f = x − y sin(πx),

X = ( ,− )

O&0 i& & \ u = f (X) e0& S $ u = f (X) &F O&0 5F G#U $ E .$ Y\ X Int (D ) 0 u = f (X) #$ 5&0 0 *( #n<\ 6 X .$ *( #n<\ 6

) f = ln(xy),

X = (−, − )

) f = arctan(x + y),

X = (, )

*f (x, y) = arcsin (x/y) ZM x= A&

) f = x − y + z , x+y−z

f

Df

'() $

9>. T R# .$

(x, y) − ≤ x/y ≤ (x, y) < y , −y ≤ x ≤ y ∪ (x, y) y < , y ≤ x ≤ −y

= =

( i& & \ f 5 Q ( 3 D 0 f 5 Q ]3 9, + 0 ]3 f

Int (Df )

=

(x, y) < y , −y < x < y ∪ (x, y) > y , , y < x < −y

*( #n<\ 6 A$ O ) ;*H !VO 0C }

) f = √xyz, ) f = z

X = (, , ) √ − x − y , X = − , ,

,

X

= (, , )

p&\ 4 M .$ f e0& M M oL6 ; & J >&#+ .$ *M oL6 . f Y0&- ]< ( #n<\ 6

) f = yx − xy ,

) f = ln(

) f = (x − y )/ , x , )f = yz

+ x − y ),

) f = xy + yz + zx , -

) f =

x y z − − − . a b c

. T .$ *A ⊆ R M x= $ (T& &0 r > M ( A ! )*"9 N# X Zg S

, + *A$ O ) *H !VO 0C $$S (=&# B (X ) ⊆ A 5&6 Int (A) $&+ &0 & A L . A .$ p&\ + u@&\ + M Zg S B . T .$ . A , + *Z$

*Int (A) = A U# *O&0 .$ E&0 R A $ ( E&0 E&0 &, + E LB$ $, `&+ ) A; *( E&0 E&0 &, + E & $ U w O A n

r

n

i& & \ e0& #n<\ 6 $ 9;*H !VO f (x, y) = arcsin (x/y)

>. T R# .$ *

M x= A, .8 0 0 ( i& & \

f R# 0&0 *A$ O ) *H !VO 0C ( E&0 D U# *( *O&0 #n<\ 6 3 D D $ 0 f (x, y) = x ln(x − y) f (x, y) y < x Int (Df ) = Df

f

f

⎧ ⎨

xy

(x, y) = (, )

Z#S A .8 D = R !M 0 M B&' .$ *(" & \ f e0& R − {(, )} E&0 , + 0 =@ E *A QC ( 3 0 f ]3 *( 0 0 xxy+ y i& & \ e0& &0 #n<\ 6 (, ) .$ f =@ E *( #n<\ 6 R − {(, )} *f (x, y) = ⎩ f

x + y

(x, y) = (, )

} ( A .$ Y\ X 9*H !VO G#U & \ e0& ** (+"1 .$ $ e0& E u = f (X) & \ e0& (& .$ S ( O i# ! 9 . u = f (X) e0& $ 6 $& tY .1 +) Q e0 E & i& & \ e0& ]3 *Z &

.$ #E e0 E N# M M oL6

√ ) f = x sin ( xy) cos x ln x

)

f=

)

f = logz (x + y)

)

y

. f=

x+y+z x−y−z

0 $

9#n<\ 6 p&\ 4 M

)

f = arctan(x + y)

)

x+y f= x + y

)

f= + x + y − z

)

4E_ pO N# X .$ f 5$ 0 3 V# &0 ]3 *A QC (" ?U0 *(" =&M pO & ( X .$ f #n<\ 6 0 ∂f ∂x ( ∂f ∂y (

(, )

, )

, )

f (x, ) − f (, ) = lim x− x→ x→ f (, y) − f (, ) = lim = lim y− y→ y→ = lim

− =

x − = y

.$ f B $ ) (, ) .$ f gc) >&\ 6 U# *(" #n<\ 6

f = arcsin (xyz)

f = |x − y | ) f = sgn (x + y + z) , ) f= − x − y − z ) f = −z ln + x + z

)

3 (, ) Y\ .$ (, ) $ ) ∂f ∂y

+ f (x, y) = |xy| ∂f (, ) ∂x (, )

e0& $ 5&6 A #c) >&\ 6 ( *(" #n<\ 6

.$ f B

e0 ( 3 (, ) .$ O $ $ e0& $ 5&6 A ∂f ∂f (, ) .$ f & . M #c) >&\ 6

∂y ∂x 9(" #n<\ 6

⎧ ⎨ + xy *f (x, y) = ⎩ x + y (x, y) = (, )

(x, y) = (, )

& ( #n<\ 6 (, ) .$ O $ $ e0& $ 5&6 AH 9 " 3 (, ) .$ ∂f ∂f #c) t 6 e0 ∂y ∂x

∗

f (x, y) = (x + y ) sin(x + y )−/

(x, y) = (, ) (x, y) = (, )

n

y y ∂z ∂z n− n −y +x + y = x nx x f f ∂x ∂y x x x

y y = nz +y xn f = nxn f x x x

e0& #n<\ 6 $ 9*H !VO f (x, y) = x ln(x − y)

R# .$ *f (x, y) = |x + y| M x= A .8 R !M 0 f ]3 D = R ( & \ f >. T E&0 , + 0 f *(" i& & \

B ( 3 , + 0 x+y i& & \ e0&&0 (x,y) < x + y 0 0 −x − y i&& \ e0& &0 (x, y) x + y < E&0 E *( #n<\ 6 (x, y) x + y = 0 f R# 0&0 *( → 9 "#&0 &F f (X ) = −− (a, b) X = (x , −x ) S =@ f

lim

(x,y)→(x ,−x

−−→ −−−−−−−−−−−→ |x + y| − − (a, b) • (x − x , y + x ) * * = *−−−−−−−−−−−→* ) *(x − x , y + x )* * *

" $ 5 M −−−−−−−−−−−→ −−−−−−−−−−−→ , r (t) = x − t , −x r (t) = x + t , −x

R# 0&0 *Z#S h .$ .

U0& f z = x f y/x S A& '() $ $ ∂z ∂z + y = nz M (0&f O&0 #n<\ 6

*x ∂x ∂y 9ZM [& . _&0 &" |Q (+ ( =&M *!'

}

B$&U .$ z = yf x − y e0& $ 5&6 A, .8 ∂z ∂z + xy = xz #n<\ 6 U0& f 5F .$ M M T y ∂x ∂y

=

|x − t − x | − (a, b) * !* lim * t , * t→

=

lim t→

= =

t − at = t

−a

t + at = t

+a

•

t ,

|x + t − x | − (a, b) • −t , * !* lim * −t , * t→ lim t→

0 y\= y\=

f

` + .$ ]3 *( 1& M *( #n3 t 6 (x, y) x + y =

9Z#. $ y

∂z ∂x

=

∂z ∂y

=

x x x x + f + g y y y y

x x x x x − f +g − g y y y y y f

∂z ∂z e+) . & F $ + %- y .$ . ∂y x .$ . ∂x 5 M ∂z ∂z +y = z M Z#S ZM

*x ∂x ∂y !" #$ U0 g f V# x= &0 0 % $ 9M (0&f . #E &. cS E N# #n3 &F z = yx + f (xy) S M $ 5&6 A ∂z ∂z − xy = −y ∂x ∂y z = ey f yex /y ∂z ∂z x − y + xy = xyz ∂x ∂y

*( 9ZM [& . _&0 &" |Q (+ ( =&M *!' y y × xf x − y + xy f x − y = +y × (−y) f x − y = xyf x − y = xz

+- !V60 z = z(x, y) e0& M x= A .8 ∂z . \ *O&0 O G#U x + y + z = xyz Y\ .$ . ∂x *M [& X = (− , , − ) Z#S t 6 x 0 ([" O $ $ Y0 . R=@ E *!' x + + z

x

&F

S M $ 5&6 A;

&F u = f (y − x) + g (y − x) S M $ 5&6 A ∂u ∂u ∂u + = + ∂x∂y ∂x ∂y

∂ f ∂x∂y

S M $ 5&6 A ∂z ∂z ∂z + abz = b +a &F ∂x∂y ∂x ∂y &F z = xg(x + y) + yf (x + y) S M $ 5&6 AH =

z

= f (x, y)eax+by

∂z x − yz = X Y\ .$ [& &0 ∂x R# 0&0 xy − z ∂z * ∂x = − M Z#S

?U0 O&0 #n<\ 6 e0 g f S M (0&f A .8 * ∂∂tu = a ∂∂xu &F u = f (x − at) + g (x + at) 9Z#. $ x= 0 ) &0 *!' X

a

∂z ∂ z ∂z ∂ z = ∂x ∂x∂y ∂y ∂x y y + x−n g z = xn f x x ∂z ∂z ∂z +y x + xy = n (n − ) z ∂x∂y ∂x ∂y

&F

S M $ 5&6 A7

.$ z = y (f (ax + y) + g (ax − y)) S M $ 5&6 A

∂z ∂z a ∂ = >. T R# y ∂y ∂x y ∂y 50 Y0 . 3&3 S t 6 &0 #E >&#+ E N# .$ 90&0 z y x R0 g f . q' )

z = f (x) + g(y)

, ) z = f x + y

) z = f (x + y) + f (x − y)

x ) z = f (xy) + g y

=

= ∂u ∂t

S M $ 5&6 A8

&F z = f (x + g(y)) S M $ 5&6 AJ

∂u ∂x

=

∂z ∂z ∂z + = ∂x∂y ∂x ∂y y y + xg z = yf x x ∂ z z ∂z ∂ + y = x + xy ∂x∂y ∂x ∂y

&F

∂z ∂z = yz + xy ∂x ∂x

= = =

∂ ∂u ∂x ∂x ∂ { f (x − at) + g (x + at)} a ∂x a {f (x − at) + g (x + at)} ∂ ∂u ∂t ∂t ∂ {−af (x − at) + ag (x + at)} ∂t a f (x − at) + a g (x + at) a

z = x + f (xy) x= &0 3&3 S t 6 &0 A .8 *0&0 z y x R0 f . q' 50 Y0 . x 0 ([" O $ $ &" m@ $ E S t 6 &0 *!' Z#. $ y ∂z = ∂x

+ yf (xy) ,

∂z = + xf (xy) ∂y

Z#.F (0 $M mn' Y0 . $ R# R0 . f (xy) &' ∂z ∂z =x+y *x ∂x ∂y x= &0 3&3 S t 6 &0 A4 .8

x x z = xf + yg y y

*0&0 z y x R0 g f . q' 50 Y0 . x 0 ([" O $ $ &" m@ $ E S t 6 &0 *!'

}

( , − , −)

∂z Y\ .$ . ∂x . \ xyz = x + y S A *0&0

>.&[, . \ x +y +z = xy+yz+ M . T .$ AH ∂z ∂z + x + y + z (, , −) Y\ .$ . xyz ∂x ∂y *M [&

t 6 E ,&1 9*H !VO x = sin t f = zyx M x= A& '() $ Z\ " !VO 0 >. T R# .$ *z = ln t y = cos t df dt

d (ln t) (cos t) (sin t) dt

= =

t

cos t sin t − ln t cos t sin t + ln t cos t sin t = A

A$ O ) % *H !VO 0C AI*H C = N+M 0 9Z#. $ df dt

=

∂f dx ∂f dy ∂f dz + + ∂x dt ∂y dt ∂z dt zy x (cos t) + zyx (− sin t) + y x ( /t)

=

ln t sin t cos t − ln t cos t sin t

=

+ cos t sin t( /t) = A

*y =

/t

x = ln t f = arcsin (x/y) M x= A, .8 >. T R# .$

d ln t + df = {arcsin (t ln t)} = , =A dt dt − (t ln t)

9Z#. $ t 6 E ,&1 N+M 0 M B&' .$ df dt

= =

∂f dx ∂f dy + ∂x dt ∂y dt " # /y , t − (x/y) "

# − −x/y =A + , t − (x/y)

y = s/t x = st f =

∂f ∂t

=

+ x + y + z

M x= A .8 >. T R# .$ *z = s ln t

∂f ∂x ∂f ∂y ∂f ∂z + + ∂x ∂t ∂y ∂t ∂z ∂t

∂ z [& . ∂x∂y >.&[, xyz = x + z M . T .$ A8 *M

x + y + z

=

M M x= AJ ∂ z ∂z * ∂x ∂x ("0 Y

x + y + z

f

∗

M ax + by + cz = f x + y + z M x= A7 9M (0&f * 0&f $ , c b a ( #n<\ 6 U0& (cy − bz)

∂z ∂z + (az − cx) = bx − ay ∂x ∂y

WP' /XF9 /DY $+ .$ O&0 R 0 R E e0 g f S M Z#. $ $&0 ) E >. T uL0 R# .$ f g(x)!! = f g(x)! g (x) >. T R# (+"1 *ZM KY v Q e0 0 . = R# M *$ + "# E&0 5 #E >. T 0 . H**H q1 E A8C

x x &v E #n<\ 6 U0& f S $ t &v E #n<\ 6 U0& & x E N# $ 0 x · · · t t &v E N# 0 ([" f &F O&0 t · · · t ?U0 ( #n<\ 6 t · · · i

n

m

m

∂f ∂f ∂x ∂f ∂x ∂f ∂xn = + + ···+ ∂ti ∂x ∂ti ∂x ∂ti ∂xn ∂ti

AI*H C

(B 0 *Z & C B *T: . = R# 0 ZM $& Z /&

. $ + E = R# R O .$ *$ O ) GB *H !VO S MC $ #n<\ 6 c t 0 ([" f &F O&0 #n<\ 6 z = θ(t) ?U0 (

y = ψ(t) x = ϕ(t) u = f (x, y, z)

∂f dx ∂f dy ∂f dz df = + + dt ∂x dt ∂y dt ∂z dt

A*H C

z

y x 0 ([" O $ $ Y0 . R=@ E S t 6 &0 *!' 9Z#. $ s = y/z t = x/y x= &0 ∂u ∂x ∂u ∂y ∂u ∂z

= = =

∂f ∂t ∂f ∂t ∂f ∂t

∂f ∂s ∂f ∂t ∂f ∂f + = + = ∂x ∂s ∂x y ∂t ∂s y ∂t ∂t ∂f ∂s x ∂f ∂f + =− + ∂y ∂s ∂y z ∂s y ∂t ∂t ∂f ∂s y ∂f ∂f ∂f + = − =y ∂z ∂s ∂z ∂t ∂s z ∂s

Z#. $ 4 Y0 . E Z#. $ Y0 . E R# 0&0 z ∂u =− 4$ B$&U .$ >.&[, R# . nS&) &0 * ∂f ∂s y ∂z Z#S

∂f ∂u = y ∂t ∂x

# "

x z ∂u x ∂u z ∂u ∂u ∂u =− y − =− − + ∂y ∂x z y ∂z y ∂x y ∂z y

V# x

∂u ∂u ∂u + +z = ∂x ∂y ∂z

E N# 0 . t 6 E ,&1

0 $

9M t\ #E $.

df ) f = x sin (yz) , x = t , y = t , z = t , =? dt √ ) f = y x + arctan z , x = sin t, y = cos t , x df z = x , =? dt s ∂f =? ) f = x + y , x = st , y = , z = s ln t , x+z t ∂s , ) f = x + y + z , x = uvw , y = u , v w ∂f z= , =? u ∂u , ) f = arctan x + y , x = sin t , y = cos t , df =? dt

)

f = x/ + y / , x = ln t , y = et ,

df =? dt

s ∂f =? f = x sin y , x = st , y = , t ∂t √

) f = x √ y + y x , x = s cos t , y = t cos s , ∂f =? ∂s y z , u = (xy ln x) + xf z x x ∂u ∂u xy ∂u +y +z =u+ x ∂x ∂y ∂z z y z u = xn f , xα xβ ∂u ∂u ∂u + αy + βz = nu x ∂x ∂y ∂z

)

M (0&f

M (0&f

x= &0 A

x= &0 AI

=

∂f ∂s

= =

x y + (s) + + x + y + z x + y + z s z ++ x + y + z t ∂f ∂x ∂f ∂y ∂f ∂z + + ∂x ∂s ∂y ∂s ∂z ∂s x

y

+ (t) + + x + y + z x + y + z

−s t

t

z (ln t) ++ x + y + z

*y = r sin θ x = r cos θ z = f (x, y) M x= A .8 >. T R# .$

∂z ∂θ

∂z ∂x ∂z ∂y + + = ∂x ∂r ∂y ∂r r

∂z ∂x ∂z ∂y + + ∂x ∂θ ∂y ∂θ r

∂z ∂z cos θ + sin θ = ∂x ∂y

∂z ∂z r cos θ + − r sin θ + ∂x ∂y r

∂z ∂z = + ∂x ∂y

∂z ∂r

x = st O&0 z y x E U0& f M . T .$ A .8 50 . s 0 ([" f gc) t 6 z = t/s y = s/t 9$ + [& 5 f 5$ 0 oL6

∂f ∂s

= =

∂f ∂x ∂f ∂y ∂f ∂z . + . + . ∂x ∂s ∂y ∂s ∂z ∂s ∂f ∂f ∂f −t . (t) + . + . ∂x ∂y t ∂z s

$ 5&6 u = f (y − z, z − x, x − y) M x= A4 .8 ∂u ∂u + + = * ∂u ∂x ∂y ∂z R# .$ *c = x − y b = z − x a = y − z ZM x= *!' 9>. T

∂u ∂a ∂u ∂b ∂u ∂c ∂u ∂u ∂u + + = + + ∂x ∂y ∂z ∂a ∂x ∂b ∂x ∂c ∂x

∂u ∂a ∂u ∂b ∂u ∂c + + + ∂a ∂y ∂b ∂y ∂c ∂y

∂u ∂a ∂u ∂b ∂u ∂c + + + ∂a ∂z ∂b ∂z ∂c ∂z

∂u ∂u ∂u ∂u ∂u ∂u + − − + + = ∂a ∂b ∂c ∂a ∂b ∂c

∂u ∂u ∂u + + + − ∂a ∂b ∂c =

Y0 . u = f (x/y, y/z) E 3&3 S t 6 &0 A5 .8 *0&0 u z y x R0 f . q' 50

(f , f , · · · , f ) *Z$ 5&6 ΔΔ (x $&+ &0 $ + G#U , x , · · · , x ) m

n

0 R E O& F = (f , · · · , f ) S $ · · · f f U#C 5F B e0 E N# O&0 R .$ F 0 M W 0 0 F (X ) >. T R# .$ *O&0 #n<\ 6 Af *( X n

m

m

m

R0 M W M $ O ) 78 $ L &F n = m S *O&0 m × n ]#& N# (f , f , · · · , f ) *Z$ 5&6 ∂∂ (x $&+ &0 . F jk , x , · · · , x )

r

n

z = t y = s x = st ZM x= A& ⎡ ∂x/∂s Δ (x, y, z) ⎣ = ∂y/∂s Δ (s, t) ∂z/∂s

'() $

>. T R# .$

⎤ ⎡ ∂x/∂t t ∂y/∂t ⎦ = ⎣ s ∂z/∂t

s

t

⎤ ⎦

.$ *z = uvw y = u − w x = uv ZM x= A, .8 >. T R# ∂ (x, y, z) ∂ (u, v, w)

=

=

∂x/∂w ∂y/∂w ∂z/∂w −w = −u v uv

∂x/∂u ∂x/∂v ∂y/∂u ∂y/∂v ∂z/∂u ∂z/∂v v u vw

u

uw

c x = uvw S M x= A& 0 $ $ v u 0 ([" y x R0 M W ("0 Y y = u − v + w *w *c = wu v b = vw u a = uvw M x= A, *w v u 0 ([" c b a R0 M W 5& $ ("0 Y

Z#. $ & \ 6 $ ) x= 0 % $ ! Δ (x , · · · , x ) =I n × n&+ ]#&

Δ (x , · · · , x ) n

)

n

n

Δ (z , · · · , zl ) Δ (y , · · · , ym ) Δ (z , · · · , zl ) = • Δ (x , · · · , xn ) Δ (y , · · · , ym ) Δ (x , · · · , xn ) − Δ (y , · · · , yn ) ) Δ (x , · · · , xn ) = Δ (y , · · · , yn ) Δ (x , · · · , xn )

)

=&# Z+U E ,&1 . A;C Y0 . V# ^- & y 0 ([" & z S ,&1 R# t0&Y *# S t 6

&F O&0 #n<\ 6 & x 0 ([" c & y O&0 #n<\ 6

i

i

i

i

r

− r

M (0&f u = f (r − s, r + s) x= &0 A ∂u = ∂y∂x

F : Rn → Rm

m

= f x + y , xy

x= &0 A ∂w ∂w ∂w + 9M [&

+ ∂x∂y ∂x ∂y (0&f y = e sin θ x = e cos θ4u = f (x, y) 5x= &0 A; ∂u ∂u ∂u ∂u + = e + 9M ∂x ∂y ∂r ∂θ . O $ $ >.&[, w

#

∂ u + ∂ u − ∂ u ∂r∂s ∂r ∂s

"

v = x − y u = x + y w =f (x + y, x− y) x= &0 A ∂f ∂f ∂w ∂w × = − 9M (0&f ∂x ∂y ∂u ∂v ( #n<\ 6 U0& f V# x= &0 AH u = f (y + z − x, z + x − y, x + y − z) ∂u ∂u ∂u + + = 9M (0&f ∂x ∂y ∂z R0 g f . q' 50 Y0 . 3 .$ 3 \ 6 &0 90&0 u z y x ) u = f

x z ! , y y

) u = x f (x + y , z)

) u = f

x! y x + ,y + z z

) u = f (x + y) + g(x − y)

) u = f

x+

! y

+g y+

! z

) u = f (x − y, y − z, z − x)

GNZ (+ *F

M x= ( t 6 E ++U 0 M W !VO 0 . F (O& *Z & (O& . U0 RQ

: Rn → Rm

F (x , · · · , xn ) = f (x , · · · , xn ) , · · · , fm (x , · · · , xn )

" v −n U0 f · · · f f M (O 5

* O & F &B

O& F = f, · · · , f ZM x= $ M ( #n<\ 6 F Zg S . T .$ *O&0 R 0 R E M $$S (=&# A & m × n ]#&

m

m

m

lim

X→X

n

F(X) − F(X ) − A(X − X ) = X − X

5&6 F (X ) $&+ &0 & X .$ F . A ]#&

9>. T 0 . F !k *Z$

⎡

⎤ ⎡ f ∂f /∂x ⎢ f ⎥ ⎢ ∂f /∂x ⎢ ⎥ ⎢ ⎢ ⎥=⎢ ⎣ ⎦ ⎣ fm ∂fm /∂x

***

***

∂f /∂x ∂f /∂x

***

∂fm /∂x

⎤ ∂f /∂xn ∂f /∂xn ⎥ ⎥ ⎥ ⎦ · · · ∂fm /∂xn ··· ···

***

=

×

−xy (x − y) xy − y − s

=

− t xy − x

*Z#. $ . A;C &" #n<\ 6 & x 0 ([" c & z q1 R# t0&Y * & 5. (O& q1 . AC &" O&0 T GB&L & x 0 ([" & y R0 M W 5& $ S 5 & y X"' 0 . & x AU- !VO 0 !1 'C &F (O& #$ 5&0 0 *Z#. $ . AC &" ?U0 $ + 5&0 *( #n< VU (x , · · · , x ) → (y, · · · , y ) Z#. $ & \ 6 $ ) x= 0 MC & $ i

s −t s t

×

i

i

i

−xy (x − y)

sxy − sy − ts ty − txy − t s xy − s − s x ts + t xy + t x U# ( _&0 ]#& E (, ) #F.$ ∂y/∂t .$ ∂y −ts + t xy + t x = ∂t −xy (x − y) t s + xyt + tx = − xy (x − y)

y = ρ cos ϕ sin θ x = ρ cos ϕ cos θ M x= A .8 >. T R# .$ AM >&j L vC z = ρ sin ϕ

0 $

v = x + y + z u = x + y + z M x= A *∂y/∂v ∂x/∂v ("0 Y *w = x + y + z z = u/vw y = u − w x = uvw M . T .$ A; ∂x/∂t [& ("0 Y w = t/s v = s/t u = st *∂y/∂s x = arcsin(u/v) f = x − yz M . T .$ A w = −t v = t u = t z = uvw y = tan(u/w) *df /dt ("0 Y

*v = cos (x + y) + y/x u = x + y M x= A *u 0 ([" y x gc) >&\ 6 ("0 Y

v = sin (xy) u = x − y f = x + y M x= AH *M [& . df /dt >. T R# .$ *v = sin t u = t *A[Y1 >&j L C y = r sin θ x = r cos θ M x= A8 *M [& . θ r 0 ([" y x R0 M W 5& $ . T .$ " v u E U0 y x (S 5 &#F AJ *v = x /y − /y + x u = arcsin (x/y − x) M Q

i

n

)

∂ (x , x , · · · , xn ) = ∂ (x , x , · · · , xn )

)

∂ (x , x , · · · , xn ) = ∂ (y , y , · · · , yn )

x

= uv w

∂ (x, y, z) = ∂ (ρ, ϕ, θ) cos ϕ cos θ −ρ sin ϕ cos θ −ρ cos ϕ sin θ = cos ϕ sin θ −ρ sin ϕ sin θ ρ cos ϕ cos θ sin ϕ ρ cos ϕ = sin ϕ −ρ cos ϕ sin ϕ cos θ − ρ cos ϕ sin ϕ sin θ −ρ cos ϕ ρ cos ϕ cos θ + ρ cos ϕ sin θ = −ρ cos ϕ sin ϕ − ρ cos ϕ = −ρ cos ϕ

i

÷

n

∂ (y , y , · · · , yn ) ∂ (x , x , · · · , xn )

M M x= A& '() * $ >. T R# .$ w = t v = t u = t y = (u + v)/w

Δ (x, y) Δ (u, v, w) = Δ (u, v, w) Δ (t) ⎤ ⎡ v w uvw uv w ⎣ t ⎦ = /w /w −(u + v)/w t ⎡ ⎤ v w + tuvw + uv w Δ (x, y) = ⎣ w + tw − t u − t v ⎦ = Δ (t) w

("0 Y *v = xy − x/y u = x + y M x= A, .8 *∂x/∂u Z#. $ 8**H q1 E (+"1 0 ) &0 *!' Δ (x, y) = Δ (u, v) =

Δ (u, v) Δ (x, y)

=

x

y

−

y − /y x + x/y x + x/y −y x − y + y + x /y −y + /y x

U# ( _&0 ]#& ( ∂x ∂u

, )

B E >.&[, ∂x/∂u .$

x + x/y x − y + y + x /y

= =

v

−

xy + x x y − y + y + x

M x= A .8 *∂y/∂t ("0 Y *v = s + t u = s − t &0 *(" $ 5 v u E U0 . y x B$&U $ E *!' *(" $ 5 t s E U0 . v u c F B$&U $ 0 ) ?U0 *O&0 t s E U0 y x ]3 = yx − x u = xy − y

− Δ (x, y) Δ (u, v) Δ (u, v) Δ (x, y) Δ (u, v) = = Δ (s, t) Δ (u, v) Δ (s, t) Δ (x, y) Δ (s, t) − s −t xy − x x = s t y xy − y

z

=

5 z X"' 0 z ∂x ∂z

=

∂y ∂z

=

*Z

=

x − y

Z

E #&"+ N# .$ . y x R# 0&0 ?U0 *$ + 5&0

z y ∂ (f, g) − z − − x y xy xz ∂ (z, y) ! = ! = ∂ (f, g) z x − y x x − y ∂ (x, y) x z ∂ (f, g) − x − − y z yz xy ∂ (x, z) ! = ! = ∂ (f, g) z x − y z x − y ∂ (x, y)

= ( , − , )

&F

x + y + z =

M x= A, .8 M $ O x= S

f = x + y − z −

∂f = x = ∂x Z Z

9?U0 *(" $ 5 z y E U0& . x nB ∂x ∂y

∂f /∂y y = − =− ∂f /∂x x

∂x ∂z

= −

⎧ f (y , y , · · · , ym , x , x , · · · , xn ) = c ⎪ ⎪ ⎪ ⎨ f (y , y , · · · , ym , x , x , · · · , xn ) = c ⎪ ⎪ ⎪ ⎩

***

fm (y , y , · · · , ym , x , x , · · · , xn ) = cm

M x= A .8 ∂y ∂z =− * ∂x ∂y ∂z ∂x Z#. $ *H*H q1 0 ) &0 *!' = c

=

∂f /∂y ∂f /∂z ∂f /∂x − . − . − ∂f /∂x ∂f /∂y ∂f/∂z

=

(− ) = −

*xy + y = uv + v x + y = u + v M x= A .8 *∂x/∂u ("0 Y

ZM x= *!' f = x + y − u − v , g = xy + y − uv − v

>. T R# .$

∂x ∂u

e0& 5& y0 . E $& e0 =U k# . & O. E V# Y0 . &a 5 , 0 *O&0 &v &# v e0 &# & *M =U x E U0& 5 , 0 . y x + y = x $ O h'? H*H !VO .$ M . @ 5&+ &) + .$ U0& x E #&"+ N# .$ . y 5 + &F x = ± S Y# O &# pO RU uL0 R# E m (" $ x v E *$ + G#U . +- e0 5 0 & F @ M ( m+ n &0 B$&U m M x= UT a2 $ $

∂f /∂z z = − z =− ∂f /∂x x x

M $ 5&6 *f (x, y, z)

∂x ∂y ∂z . . ∂y ∂z ∂x

*>< I & ++

= − =

∂ (f, g) ∂ (u, y) = − ∂ (f, g) ∂ (x, y) =

−u y −v x + y =− x y y x + y

ux + uy − vy x + xy − y

∂z

[& . ∂x xy z = x + y + z M . T .$ A .8 *M z &F f = xy z − x − y − z S M ( RO. _ *!' ∂f /∂x y z − ∂z =− =− ∂x ∂f/∂z xy z −

}

+- e0& 9H*H !VO M x= *O&0 $, & c e0& & f M O&0 O $ $ Y = (y , y , · · · , y ) X = (x , x , · · · , x ) Y\ .$ f * * * f f e0 M Z = (Y , X ) V# 0 =&M 4E_ pO >. T R# .$ *#n<\ 6 Z 0 . y * * * y y 5 0 X Y\ E #&"+ N# .$ M ( 5F $ + G#U x * * * x x E U0 !VO ∂ (f , f , · · · , f ) e0 y * * * y y ?U0 * ∂ (y, y, · · · , y ) = " x * * * x x E . + i

i

m

n

m

m

n

m

m

m

Z

n

∂ (f , f , · · · , fm ) ∂yi ∂ (y , · · · , yi− , xj , yi+ , · · · , ym ) =− ∂ (f , f , · · · , fm ) ∂xj ∂ (y , y , · · · , ym )

x + y + z = M x= A& '() $ $ y x 5 >. T R# .$ *Z = (−, −, ) xyz = g = xyz f = x + y + z x= &0 #E (" $ z E U0& . Z#. $ ∂ (f, g) ∂ (x, y) Z

x = yz

xz Z

y

X

= (− , , − )

xy z = M . T .$ A ∂ z * ∂x∂y [& ("0 Y

X

∂z ∂x

∂ ∂x

=

E)8

@.6

5fC)

i= % $

ZM x= Y[ @1Cc) + 9fC) < a2 y e0& x v X"' 0 .&[, f x, y, y , y , · · · ! = S *A· · · y = y y = y C ( x 0 ([" y >&\ 6

v X"' 0 y x 5&0 X 0 y = ψ (t, u) x = ϕ (t, u) E ,&1 N+M 0 O&0 u = u (t) #) e0& t #) 9&a 5 , 0 #E (O 5 u t X"' 0 . E t 6

x

xx

yx

=

yxx

= =

xx

x

yt tt ψt + ut ψu ψt + ut ψu = = xt tt ϕt + ut ϕu ϕt + ut ϕu ψt + ut ψu (yx )x = (yx )t tx = xt ϕt + ut ϕu t (ψtt + utt ψu + ut ψut ) (ϕt + ut ϕu ) (ϕt + ut ϕu ) − (ϕtt + utt ϕu + ut ϕut ) (ψt + ut ψu )

y t tx =

!" #$ B$&U A& '() % $ "# E&0 t #) v y e0& X"' 0 x = e x= &0 . *M R# 0&0 *y = u x = e Z#. $ *8*H &0 "#&\ .$ *!' x y +xy +y =

t

t

y x = y t tx =

U# *x y

(yx )t = t xt e

yt

=

(yx )x =

=

ytt x − xt yt ytt − yt = x x x

x

t

.$ *$$S !#[ y + y = !VO 0 . Mn B$&U

+ xy + y = ytt − yt + yt + y = ytt + y tt

4

!

y z ∂z

xy z −

∂x 5 ∂z y z − − y z + xy z ∂x

*Z$ . 1 [1 &" E ∂z/∂x &0 ( =&M &' 0 $ $

("0 Y *xy+yz+zx = xyz = M x= A *dy/dz dx/dy dz/dx [&

*xyz = z + xy + yz + xz = M x= A; *dx/dz dz/dx ("0 Y

∂z/∂x ("0 Y x arcsin (y + z) = M x= A *∂y/∂x ∂x/∂z ∂x/∂y [& ("0 Y xyz + sin (xyz) = M x= A *∂ z/∂x∂y ∂ z/∂x &F f (x, y, z) = c S M M (0&f AH ∂z ∂x∂y

4

=

∂ f ∂f ∂f ∂f − ∂x∂z ∂y ∂z ∂x∂y

∂f ∂z

∂ f ∂f ∂f ∂ f ∂f ∂f + − ∂y∂z ∂x ∂z ∂z ∂x ∂y

5

÷

∂f ∂z

?U0 u + v = ln (x + y) ZO&0 O $M x= A8 *∂y/∂v ∂u/∂x ("0 Y *uv = sin x + y *u +v = x/y u +v = xy z = x +y M x= AJ *∂z/∂v ∂z/∂u ("0 Y

∂ z [& ("0 Y A7 M B&' .$ ∂x∂y

∗

f x + y + z, x + y + z =

M @O 0 ∂z/∂y ∂z/∂x [& ("0 Y A

yt yt yt = t = xt e x

nB yxx

xy z −

S' S& 0+

<

∂z ∂x −

=

B +U C !" #$ >_$&U ` - 0 E uL0 R# 5F E 5 UB&Y .$ .$ ( A#c) >&\ 6 &0 &# *O L $& U0 & 6L0 .$ #E $M h mT

.$

f (x − y, y − z, z − x) =

[& ("0 Y f (xz, yz) = Z 0 M . T .$ AI *∂ z/∂x Z 0 M 0&0 B&'.$ . dy/dz dx/dz A g xyz, x , z = f xyz, x , y =

Zg 0 M ( (.$ &#F f (x, y, z)

M x= A;

= c ∂x ∂y = ∂y ∂x

*"# 0 t = ln |x| v X"' 0 . x y

B$&U A B&' .$ t X"' 0 . − x !y − xy + y = A *x = cos t M "# 0

= y

*y = u/ cos t x = tan t ( t E U0& u M x= AH *"# 0 E&0 t u X"' 0 . + x y = y B$&U

B$&U .$ . v e0& u\ A, .8 *M x , &0 R# 0&0 y = t x = u ZM x= #&0 #$ 5&0 0 *!' 9Z#. $ *8*H 0 ) y − xy + y = x

B$&U *yt = u xt = ( t E U0& u M x= A8 *"# 0 E&0 t u X"' 0 . y − x y + xy = y

( + y ) − y y =

B$&U M M (0&f

v e0& v 0

A7

$+ !VO v

5F .$ M >. T 5&+ 0 U#C *A$ O !#[ θ r X"' 0 . (xy − y) = xy( + y ) B$&U A x= θ E U0& r y = r sin θ x = r cos θ M "# 0 *$ O

. y = f (x)

B=f

y

= yx =

zv

= zx xv + zy yv = ϕv zx + ψv zy

zuu

= (zu )u = (zu )x xu + (zu )y yu

y

= yxx =

=

t + uu (t) + (u) ut = (u) + (t) ut u + tu

(u + tu )

() + (u) u + u (u + tu )

− (u + tu + u ) ((t) + (u) u ) u − t u + tu (u + tu )

!VO 0 . x B$&U .$

( y x !\ "

(u + tu − t u )

(u + tu )

−

(t + uu) u + tu

= ut

Z#. $ .$ *(O 5

y

= zx xu + zy yu = ϕu zx + ψu zy

x = ut >&- y − y = x B$&U .$ A .8 *( O x= t v E U0& u M M $. . y = u + t "# E&0 t #) v u #) e0& X"' 0 F #$ 5&0 0 *M Z#. $ *8*H & B = 0 ' &0 *!'

" ∂z ∂z ∂ z ∂ z ∂ z , , , x, y, z, , ,··· ∂x ∂y ∂x ∂x∂y ∂y

zu

x + x (x ) + y (x ) = x (x )

9fC) U] a2 <

x X"' 0 z #c) >&\ 6 z y x X"' 0 .&[, v u #) !\ " &v X"' 0 y x M x= *O&0 R# .$ * O =U y = ψ (u, v) x = ϕ (u, v) !VO 0 u X"' 0 . B >.&[, t 6 E ,&1 N+M 0 >. T _z&a #E (O 5 v

]3 u = x t = y 5 Q *$$S !#[

E)8 .6 E1CD) 5fC) i= % $

#

− u ut + t

(ut )

utt + u (ut ) + t (ut ) = u (ut )

Ǳ& = AI *"# 0 [Y1 >&j L .$

&v E U0& z M x=

utt

.$

!VO 0 . Mn B$&U ]3

κ = y /( + y )/

=

∗

a u + b t + c a u + b t + c , y= x= au + bt + c au + bt + c a b c a b c = a b c u ( + u ) − u u =

yxx

+ ut = + ut ut (yx )t utt −utt = =− xt ut (ut ) (ut )

y − xy + y = yxx − xyx + y = −

=

Ay x= &0 . y = ]M B$&U AJ ((x − a)(x − b)) x − a y E U0& 5 U0 u .& &0 u = x − b t = ln x − b *M !#[ t

([" y

yx

t u − tu − u + (t + uu ) (u + tu ) = ut (u + tu )

0 % $

t

X"' 0 x = e x= &0 . x y t

− xy + y =

A

*"# 0 B$&U ( v t = xy e0& x V# x= &0 A; *M "# E&0 . y + x y + y =

xv = y 5F .$ M xz

x

+ x + y + z + u = z + x + y + z

+ yzy = z +

u = y + z M (x + z) z

*

x

+ (y + z) zy = x + y + z

*v = x + z

A

=

ϕu (ϕu zx + ψu zy )x + ψu (ϕu zx + ψu zy )y

=

ϕu ((ϕu )x zx + ϕu zxx + (ψu )x zy + ψu zxy ) +ψu (ϕu )y zx + ϕu zxy + (ψu )y zy + ψy zyy

AH =

+ϕu zy (ux ψuu + vx ψuv ) + ϕu ψu zxy

hDJ 2 .6 a2 + 5fC) i= & % $

>.&[, S # B=f

#) e0& v u #) &v $ 0 .& .$ z = η (u, v, w) y = ψ (u, v, w) x = ϕ (u, v, w) !VO 0 (O 5 w v u X"' 0 . B &F O&0 O KY

& $ E ?za #E

*v = x − y u = x + y M x= A& '() $ % $ . 5F ]< O v u X"' 0 . ∂z/∂x = ∂z/∂y B$&U

*M !' Z#. $ B&" >&- 0 ) &0 *!' ∂z ∂y ∂z ∂x

zx (ϕu + ϕw wu ) + zy (ψu + ψw wu ) = ηu + ηw wu zx (ϕv + ϕw wv ) + zy (ψv + ψw wv ) = ηv + ηw wv

*$M [& w w v u X"' 0 . z z 5

∂z ∂z +x = (y + x) z B$&U .$ A& y '() * % $ ∂x ∂y *z = uv + vw y = /u + /v x = u + v M x= B$&U ( #) e0& w #) &v v u V# x= &0 *"# 0 w v u X"' 0 . Z#. $ J*8*H 0 ) &0 *!' v

u

zx ((u) + ()wu ) + zy

y

x

− /u + ()wu =

zx ((v) + ()wv ) + zy

− /v + ()wv = zy

= (u + w) + (v)wv

Z#. $ "#& !VO 0 &#

u v

− /u − /v

zx zy

=

v + vu u + v + vv

=

zy

=

=

∂z ∂u ∂z ∂v ∂z + = − ∂u ∂y ∂v ∂y ∂u ∂z ∂u ∂z ∂v ∂z + = + ∂u ∂x ∂v ∂x ∂u

= zu ux + zv vx x = zu + x + y = zu uy + zv vy y = zu − x + y

x

zv = x + y

v (u + v + wv ) − u (v + vwu ) v − u u v (u + v + wv ) − u v (v + vwu ) v − u

9Z#. $ O $ $ B$&U .$ . nS&) E ]3

+ v (u + v + wv ) − u (v + vwu ) u v + u + v u v (u + v + wv ) − u v (v + vwu ) =

= u −v u +v + + (uv + vw) u v

∂z ∂v ∂z ∂v

x (zu + zv ) x + y

y y (zu − zv ) zv = x + y x + y

Z$ . 1 B$&U .$ ]3

R# 0&0

zx

=

% ) M Z. ∂z/∂v = B$&U 0 & # 5$ $ . 1 0 0 &0 U0& f M z = f (x + y) #$ 5&0 0 *( z = f (u) 5F *O&0 LB$ + ! *v = arctan(y/x) u = ln x + y M x= A, .8 ∂z ∂z − (x − y) = !" #$ B$&U

u X"' 0 . (x + y) ∂x ∂y *"# 0 v Z#. $ B&" >&- 0 ) &0 *!' zx

= (v) + (v)wu

+ψu zy (uy ψuu + vy ψuv ) + ψy zyy

"

w = w(u, v)

+ψu zx (uy ϕuu + vy ϕuv ) + ϕu ψu zxy

O[ 4R @1Cc) + a2 ,5fC)

∂z ∂z ∂ z ∂ z , , ,··· x, y, z, , ∂x ∂y ∂x ∂x∂y

ϕu zx (ux ϕuu + vx ϕuv ) + ϕu zxx

(x + y)

x (zu + zv ) y (zu − zv ) − (x − y) = x +y x + y

O&0 z = f (u −v) 5F

+, % ) M z + z = &# + *z = f ln x + y ! − arctan(y/x) U# #) &v 5 , 0 v u %&L &0 0 % % $ 9M "# E&0 . #E >_$&U E N# *v = x + y u = x M yz − xz = A *v = y/x u = x M xz + yz = z A; + c u = ln x 5F .$ M xz + + y+ z = xy A *v = ln y + + y u

x

v

x

y

x

y

y

(O 5 #E !VO 0 . O $ $ B$&U nB v

#

u

u − wu − v

u wuu v

"

# +

u wu v+ v

" =

v

e0& R# 0&0 *$ O $& w = −/u !VO 0 M M (" g e0& w = /u + f (v) M (" B$&U % ) U# *w = − /u + uf (v) + g (v)

f

uu

u

∂z ∂z + = x + yz B$&U A, .8 &0 . (xy + z) ∂x − y ∂y e0& X"' 0 w = xy − z v = xz − y u = yz − x x= *"# 0 v u #) &v w #) 9Z#. $ B&" >&- 0 ) &0 *!'

Δ (x, y, z) Δ (u, v, w)

= ⎡

− ⎣ z y

= x y z = xy + − f (x) − g(x) x y

* LB$ e0 g f M (

Δv (u, v, w) Δ (x, y, z) z − x

− ⎤− y x ⎦ −

× x + y + z + xyz − ⎡ ⎤ − x xy + z xz + y ⎣ xy + z − y yz + x ⎦ − z xz + y yz + x

=

}

9$M [& 5 t#@ $ 0 . z ?U0 v

zx xv + zy yv

= zv

= (xy − w)v

$ t 6 98*H !VO

=

X"' 0 . O $ $ >_$&U E N# 0 % $ 9"# 0 v u #) &v w #) e0& v

=

/y − /x u = x

y = ve x = ue M w

w

M x z + yz = z A *w = /z − /x x

y

(xzx ) + (yzy ) = z zx zy

*z = we

A;

w

e0& X"' 0 . xu + yu + zu = u + xy/z B$&U A M "# 0 B&' .$ γ β α #) &v w #) *w = u/z γ = z β = y/z α = x/z x

y

z

&0 B$&U *w = z/x v = y/x u = x + y M x= A ∂z ∂z ∂z + X"' 0 . ∂x − = gc) >&\ 6

∂x∂y ∂y *M "# E&0 v u #) &v w #) e0& B$&U *w = x + y + z v = x + y u = x M x= AH ∂z ∂z y ∂z + − + = ∂x∂y x ∂y ∂x

*M "# E&0 v u &v w #) e0& X"' 0 .

(xy + z) zx +

− y zy

x + y + z + xyz − − y x (xy + z) y + − wv x + y + z + xyz −

.$

− y zy = wv = (yz + x) − x + y + z + xyz −

(xy + z) zx +

]3 ( yz + x 0 0 _&0 &" |Q (+ x= t0&Y & 5&0 0 *w = f (u) M $. $ $ ) f e0& U# *w = "#&0 *( B$&U % ) xy − z = f (yz − x) #$ ∂ z ∂z = B$&U A .8 v = x u = x/y x= &0 . y + ∂y x ∂y E&0 v u #) &v w #) e0& X"' 0 w = xy − z *M "# *z = v /u y = v/u x = v B&" >&- 0 ) &0 *!' $ + [& 5 !VO $ 0 . z v

u

v v v zx xu + zy yu = − wu ⇒ zx − zy = − − wu u u u

[& #E t#@ $ 0 . z 5 M *z yu

zxy xu + zyy yu =

u w

&# − uv z

v

yy

u

=

u wu v

R# 0&0 ZM

u wuu v

R# 0&0

y

=v+

+

u wuu v

u v

wu +

u u zyy = − wu − wuu v v

0 y x &v u# c= j1 RB " O&0 X = (, ) R# ) 3 .$ M M oL6 O&0 O $ . : ([" *O $&# % >&UY1 $ U .$ v Q (& f = x y − y e0& $ t 6 E >.&[, B&" &3 *!' 9U# O&0 v = −(−,−→) $ X = (, ) Y\ .$ Dv f (X ) =

−−−−−−−−−−−→ xy, x − y

•

X

=

−−−→ −−−→ (, ) = (, ) •

v v

=

Z+j h $. RO& M (U- 0 ) &0 R# 0&0 % >&UY1 u# c= d,&0 #E ( $ [ (.$ O {&L ( O (, &0 E (MO N# 5&#E $ UB&Y .$ M x= A .8 (MO $ UB&Y E ]3 ( O $& z y x &v

M (U- S *( O =U f = x − yz >. T 0 O&0 O $ Z+j # >& O&0 X = ( , , − ) (MO $ v 5 c $ u# c= − : : ([" &0 . &v & *M oL6 . Z+j R# ) E ]3 (MO Y\ .$ f = x − yz $ t 6 E >.&[, B&" &3 *!' *O&0 v = −(−−−−,−−,−→) . $0 & . .$ X = ( , , − ) #$ >.&[, 0 Dv f (X ) = =

−−−−−−−−→ (x, −z, −y)

v v X −−−−−−−→ −−−−−−−→ (− , , ) − (, , −) • √ = √ •

*$ O (MO 5&#E X) $ 0 &[ O h $. Z+j U# .$ X Y\ .$ . f e0& $ t 6 0 & $ 9M [& v . $0 $ ) f = x arctan (y/z) , X = (, , ), −−−−−−−−→ v = (− , , −)

, −−−−−→ ) f = x + y , X = (− , ) , v = (, −)

)

f = ln |x + y + z |, X = ( , − , ) , −−−−−−−→ v = (, −, )

)

−−−→ f = xy − y , X = (, ) , v = (, )

)

f = xy + yz + zx, X = ( , − , ) ,

−−−−−−−→ v = (, , −) −−−−−→ ) f = x sin (x/y) , X = (, /π) , v = (, −)

' WP' 2+ E *X ∈ D ( v Q U0& u = f (x) M x= `O &0 y Z U#C $ O :.& `&UO (#& 0 X Y\ oL6 5 v V# $& . $0 N# &0 . `&UO *AX E 0 y Z R# .&3 B$&U *AN# @ &0 U# V#C $ + *$ O ) 8*H !VO 0 * ≤ t M ( t → X + tv !VO & u = f (X) e0& > v 5 c & $ (M' 0 `O v = o . $0 $ .$ X Y\ E X w M $&+ &0 & v & . .$ X .$ f $ t 6 . M

_&0 d0 0 h *Z$ 5&6 Dvf (X ) f

Dv f (X ) := =

v f X +t − f (X ) v t→ t

d v f X +t dt v t= lim

Z u = f (X) e0& 0 t 6 E ,&1 E $& &0 Z#. $ r (t) = X + tv v Dv f (X ) = f (X ) A;*H C v

•

M O&0 t > U#C O&0 ( + w v S R# 0&0 *0 0 w $ .$ v $ X .$ f t 6 &F Aw = tv .$ . f = x sin (yz) > v 5 c A& '() & $ −−−−−→ *M [& v = (, , ) & . .$ X = ( , , π) Y\ 9Z#. $ *J*H G#U 0 ) &0 *!' v v −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→ x sin (yz) , x z cos (yz) , x y cos (yz)

Dv f (X ) = f (X ) =

•

•

X

−−−−−→ −−−−−→ (, , ) = (, π, ) •

=

v v

Y\ .$ . f = arctan (y/x) e0& $ t 6 A, .8 *M [& v = −(−−−−,−→) . $0 $ X = (, ) 9Z#. $ G#U 0 ) &0 *!' v v −−−−−−−−−−−−−−−→

−y x , x + y x + y

Dv f (X ) = f (X ) =

•

X

−−−−−→ (−, ) √ • =

√

$& y x &v E . c= RO& N# UB&Y .$ A .8 >&UY1 $ U M ( $ 0 R# >&\\ ( O RO& M (U- S *( f = xy − y 0 0 %

&0 U# *$ 0 D = f (X ) = .&M `O .$ (=&# u# c= (, &0 (MO $ Z+j R# {&L

max

$. .$ . D v D v min

min

max

max

0 % & $

90&0

) f = x + xy + y , X = (− , ) ) f = ln x + y − + y sin z, X = (, , π) √ ) f = x √y + y x, X = (, )

)

f = ln(xy) + ln (y/z) − x, X = ( , , )

)

f = x / − y / + xy, X = (, −)

)

f = z arctan (x/y) + xyz, X = (− , , −)

Y\ .$ M f v $ e0& E 0 M $ 5&6 AJ Z#. $ ( #n3 t 6 X Dif (X ) = ∂f ∂x *Djf (X ) = ∂f ∂y ]< 5&0 v e0 0 J R#+ 0&6 ZV' A7 *M . 5F (0&f X

X

R# ZM h'? . AC ;*J*H &a &3 M & ( &[ O O %&L (& S 9M $ O KY ( 4 M (& R# 0 ( 4 M ^T (& ]3 *( #E q1 .$ u3 $ R# % ) X .$ u = f (X) e0& M x= & $ Y\ .$ f e0& > v aM ' >. T R# .$ *( #n<\ 6

$#n3 >. T v = f (X ) . $0 $ .$ X 0&6 >. T 0 *O&0 D = f (X ) &0 0 0 .&M $ .$ X Y\ .$ f e0& > v !1 ' M $$S &0 0 0 .&M $#n3 >. T v = −f (X ) . $0 *O&0 D = − f (X ) =@ E *$ O ) A;*H C = 0 ( =&M v R0 # E α M ( f (X ) cos α 0 0 5F ( . (+ (0&f c f (X ) ( (0&f f 5 Q *O&0 f (X ) M ( aM ' . T .$ %qT&' .$ ( ( ) Z f (X ) v #&0 R# 0&0 *α = &# cos α = Z (" Z $ t 6 .$ . $0 @ 5 Q *O&0 $ t 6 . \ (B&' R# .$ *Z#0 f (X ) 0 0 . v >. T 0 !1 ' (B&' *$ O f (X ) × = f (X ) 2 *$$S d0 0&6

Z+j R# 0 *AC ;*J*H &a $ A& '() $ & $ E ( >.&[, $$S % >&UY1 !1 ' X) M

max

M x= C) bCc) U5 . & & $ ( f $ E Y\ X = (x , y ) z = f (x, y) −−→ xy T 0 $ +, T X Y\ E *v = (a, b) = 0 −−→ .$C .nS 5 O&0 ! &O c . v = −− (a, b, ) . $0 M ZM x= *$ O ) 8*H !VO 0 A( v × k 5F & e1 R# .$ *Z &0 C A C . T R# f . $ + $. 0 !

!VO 0 . C >. T −−−−−−−−−−−−−−−−−−−−−−−−−−−−→ C : r (t) = (x + at, y + bt, f (x + at, y + bt))

E $ 0 >.&[, Dvf (X ) 5 M *$M .&3 5

# E ( & &0 U# r() Y\ .$ C 0 /&+ y XO *( 0 0 $E& xy T &0 r () Y\ .$ C 0 /&+ M *$ O ) J*H !VO 0 0 [U R# $ 0 .\ _&0 U0 &0 &&q= . j l&Q U0 . . . k3 .& Q &&q= .$ v . . . .& Q e0 $ 0 (.$ }

max

min

min

−−−−−−→ vmin = −f (X ) = (−, −)

.&M `O .$ (& R# &+,

√ Dmin = −f (X ) = −

$ t 6 [U 9J*H !VO

&0 % >&UY1 $ U Z+j R# {&L &0√U# *$ 0 0&# u&M − ≈ − (, X) M Z+j R# 0 *AC ;*J*H &a $ A, .8 E ( >.&[, $ O h $. (MO 0 $ aM ' (& R# &+, *( v = f (X ) = −(−−, −−, −−−→) max

d f

X

= {}X dx + + {

d f

= = X

= = X

y

G& 5 F 4+

xy}X

dx dy dxdy + x X

dy X

dx dy − dxdy + dy {}X dx + {}X dx dy + {

d f

y}X

dx dy + {

x}X + {}X

dxdy dy

dx dy − dxdy {}X dx + {}X dx dy

0 ([" e0& N# > v 5&0 0 !" #$ !" #$ G#U 0 4E_ pO *O&0 5F &v

*O&0 e0& #n<\ 6

Y\ .$ z = f (x, y) M x= * $ 4 k [ & #c) >&\ 6 . $ 3 X = (x , y ) dx = x = x − x S >. T R# .$ *O&0 X .$ 3 >. T 0 . X .$ f >E 6 dy = y = y − y df

{}X dx dy + { }X dx dy + {}X dxdy + {}X dy dx dy d f = · · · = +

d f

= = X

X

d f

d f

X

−x dy = dx − dy y X X − = {}X dx + dxdy y X x + dy y X =

X

y

dx +

= −dxdy + dy X

= {}X dx + {}X dx dy + −x dxdy dy + y X y X =

dxdy − dy

*dy = y − dx = x − &# .$ M z = f (x, y) e0& 4 k [ & !" #$ 0 * $ 90&0 X Y\ .$ . ) f = x arctan y, X = (, − ) , k = , ) f = x + y , X = (, ) , k =

)

f = x y − x/y, X = ( , − ) , k =

)

f = ln |x − y |, X = (, ) , k =

)

f = ex

)

+y

, X = (, ) , k =

X

∂f ∂f dx + dy ∂x X ∂y X

0 . X .$ f 4$ [ !" #$ $ .$ *ZM G#U >. T

*dy = y − dx = x + &# .$ M >. T R# .$ *X = ( , ) f = x/y M x= A, .8 df

:=

d f

:=

X

X

.$ f 4 k [ !" #$ M (B&' .$ *ZM G#U !VO 0 . d f k

∂ k f dxi dy k−i i ∂ i x∂ k−i y X

i=

n n! = m m! (n − m)!

M ZM G#U *O&0 ǱO n R0 v $ M U R# 0 ( v .& Q d f M $ O ) dx & " #$ 0 (x, y) v $ X# - 0 (x , y ) z a S #$ 5&0 0 *dy df : R → R Y0&- &F k = ? 9( RQ k

(x , y ; x, y) →

∂f ∂x (x

,y )

(x − x ) +

∂f ∂y (x

,y )

(y − y )

v 0 ([" d f ?U0 k

ΔX := X − X = (x, y) − (x , y ) = (dx, dy)

=

*O&0 4 k [ R+ +) Q x y M M x= A& '() * $ 9>. T R# .$ *X = (− , )

=

X

d f

= =

X

e0 0 . 4 k [ . y"0 4 k [ !" #$ A5 M (B&' .$ *M M{ B&a N# E $M G#U v

k / k

E ǱO m %&L

df

f = tan (x/y) , X = (, ), k =

:=

X

f

∂ f + ∂ f dxdy + ∂ f dy dx ∂x∂y X ∂x X ∂y X

xy dx + x y X X −dx + dy y dx + xy X

∗

dy

dxdy + x y X

X

=

dx − dxdy + dy

dy

e0& 4 [ . y"0 A& '() $ * $ *0&0 X = (− , ) Y\ .$ . 9ZM $& AC ;*7*H &a .$ >&[& E *!'

(S 5 Q

f = x y

x y

{ } + {−dx + dy} + dx − dxdy + dy +

dx dy − dxdy + dy +O (X − X )

=

− dx + dy + dx − dxdy + dy

=

+O (X − X )

e0& 4 [ . y"0 A, .8 *0&0 X = ( , ) 9ZM $& A;C *7*H &a .$ >&[& E *!'

x y

=

{ } + {dx − dy} +

−dxdy + dy

dxdy − dy

+ O (X − X )

+ dx − dy − dxdy + dy + dxdy −dy + O (X − X )

%C

≈

+ dx − dy

=

+ (

≈

+ dx − dy − dxdy + dy

) − (−

)=

f (X ) + df

+

X

··· +

k!

dk f

X

d f

X

+ ···

+ O (X − X )k

!

*Z & X .$ 4 k [ . y"0 . !T&' >.&[, R# .$ *ϕ(t) = f (x + tΔx, y + tΔy) ZM x= Z#. $ t 6 E ,&1 E $& &0 >. T

ϕ (t) =

fx (x + t Δx, y + t Δy) . Δx +fy (x + t Δx, y + t Δy) . Δy

ϕ (t) =

fxx (x + t Δx, y + t Δy) . Δx +fyy (x + t Δx, y + t Δy) . Δy

= ≈

+ dx − dy − dxdy + dy + dxdy − dy

) + (−

)

*** Z#.F (0 ZM [& t = . & F S M

+ ( ) (− ) − (− ) = [ (1$ &0 . () +() ! [#\ . \ A .8 *M [& 4$ + *k = X = (, ) f (x, y) = x ++ y &# .$ *!' &0 ( 0 0 &z[#\ x + y .$ =

X→X

+ h (X)

X

+ . fxy (x + t Δx, y + t Δy) . ΔxΔy

− (

) (−

k!

dk f

h (X) = X − X k

lim

f (X) =

9&a 5 , 0

=

X

··· +

= x/y

+ =

X

.$ * & X f Yk )*$ "# . z = h(x, y) e0& 9Z"# >. T R#

+dx dy − dxdy + dy

Y\ .$ . f

[ & #c) >&\ 6 4&+ S = * $ 3 $ ) X Y\ .$ z = f (x, y) e0& 4 (k + ) 9M $. $ $ ) z = h (x, y) U0& &F O&0 GB C f (X) = f (X ) + df + df + · · ·

ϕ()

=

f (x , y )

ϕ ()

=

fx (x , y ) . Δx + fy (x , y ) . Δy

ϕ ()

=

fxx (x , y ) . Δx + . fxy (x , y ) . ΔxΔy

***

/

# " x y + + dx+ + dy X x + y X x + y X # y + + dx ( x + y ) X " −xy x + + dxdy + + dy ( x + y ) X ( x + y ) X

+fyy (x , y ) . Δy

f (x + t Δx, y + t Δy) = ϕ(t)

.$

t + ··· = ϕ() + ϕ () . t + ϕ () . = f (x , y ) + fx (x , y ) . Δx + fy (x , y ) . Δy . t + fxx (x , y ) . Δx + . fxy (x , y ) . ΔxΔy

, x + y

t +fyy (x , y ) . Δy . + ···

2

*t = Z$ . 1 &" R# .$ ( =&M 5 M

)

f = ex

+y

, X = (, ) , k=

)

)

,−

dx + dy + dx − dxdy + dy 9 nB dx = dy = − i& $. R# .$

, (

f = ln (x + y + ) , X = ( , − ) , k= X =(

)

X = (,

= +

)

f = x/(x + y), X = (− , ) ,

, )

k = X = (−

.$ . f = sin

f

=

d ∂f f (x, y ) = g (x ) = dx ∂x X x

&0 e0& (B&' *$ + d0 5 0&6 >. T 0 y v 0 2 *( d0 !0&1 0&6 >. T 0 c v $ E u0

x + y

sin u = u −

u

u

+

+ O (X − X )

x + y (x + y) − x + y + O (X − X ) + y + x − +

=

y − x y − x y y + O (X − X )

dy + dx − +

dy − dx dy − dy + O (X − X )

dx dy

Y\ .$ f = ln (xy+ ) e0& 4.& Q [ . y"0 A .8 *0&0 . X = (, ) 9Z#. $ -&#. >&,?@ 0 ) &0 *!' f

ln (dx + ) − dy + dy − dy + · · · × =

× dx − dx + dx − · · ·

=

y

ln (x + ) =

+ dy

dx − dxdy + dx + dxdy + dydx + O (X − X )

= dx −

e0& U- 4 "M Y\ X S $ *$ 0 e0& 5F RV Y\ X &F O&0 u = f (X) *( yb XY R# ]V, X = (x , y ) ( v $ e0& ZM x= $. $ U- 4 "M X .$ f 5 Q *g(x) := f (x, y ) 0&0 & *O&0 U- 4 "M . $ x = x .$ c g #&0 ( RV+ . T .$ & R# N#

+, -&#. E q1 & *g (x ) = M

= =

r

e0& j? )*"9 X Zg S . T .$ $ .$ &# O&[ #n3t 6 X .$ u = f (X) M ( u = f (X) *f (X ) = 0 5$ 0 #n3t 6 >. T

e0& Z3 [ . y"0 A .8 *0&0 X = (, ) Y\ S *ZM $& y = sin x e0& 5. B N y"0 E *!' 9&F u = x + y

B&" !' .$ 5F E $& t 6 &$0.&M R# + E V# RO. . U- 4 "M E . h #&0 0 *O&0 4 "M *ZM #&"+ N# .$ u = f (X) ZM x= $ Y\ N# X Zg S . T .$ *$ O G#U X Y\ E E 0 r > E 0 M ( f !JO YFe ) GB 7*H !VO 0 *f (X) ≤ f (X ) X ∈ B (X ) *( G#U !0&1 !JO YF Y\ 0&6 >. T 0 *$ O U- 4 + &# 4 +#cM& U 0 !JO YE K?YT *(

) + () ≈ + + −

+ − + = − + =

.$ z = f (x, y) e0& 4 k [ . y"0 0 % * $ 9M [& . f (X) [#\ . \ ]< 0&0 . X Y\ ) f = x arctan y, X = (, − ) ,

k = X = (

)

, )

f = x y − x/y, X = ( , − ) , k= X =(

, − )

}

d0 D = (B&' _&0 q1 .$ *$ O ) % 7*H !VO 0 &#&q1 E #&0 $. R# .$ M ( R# (U1 *( 6 $ RV+ & B&' + UB&Y *$ O $& _&0 [ E R# T ' E $ !V6 . >&-&#. E $ "S &O N# *( :.& %& M *f = x + y + xy M x= A& '() $ $ RV p&\ ]3 *( #n3t 6 R !M 0 f >. T R# .$ pO R# 0&0 *Af = 0 U#C O&0 4$ ` E 9ZM t\ . f (x, y) = 0

U- 4 +#cM& AGB 97*H !VO #E Y\ A%

x + y = y + x =

⇒ ⇒

x = −y y = −x x = x ⇒ y = −x

x = ± , y = −x

X = (, ) RV Y\ . $ e0& R# ]3 t 6 5 EF &' *O&0 X = (− , ) X = ( , − ) V# 0 ) &0 *Z#S .&M 0 . 4$ A=

∂f = ∂x

x ,

B=

∂f = ∂y

y ,

C=

∂f = ∂x∂y

9A$ O ) GB *H !VO 0C Z#. $ i

A

B

C

Y\ N# X ( #E ( U- 4 +

( U- 4 +

D −

i

f (Xi )

− −

+

>. T R# .$ *f = xy − x − y M x= A, .8 T&=?0 ]3 *( D : x + y ≤ &0 0 0 f $ p&\ .$ f A]3 ( & \ f 5 QC M (=S 5

f RV p&\ C p&\ + (" #n3t 6 C : x + y = R# 0 Int(D) : x + y < 0 f = 0 pO ?U0 * " M ( U

⎧ , ⎪ x y ⎪ ⎪ − x − y − + = ⎨ y − x − y , ⎪ ⎪ − y − + xy ⎪ x − x = ⎩ − x − y y( − x − y ) = ⇒ x( − x − y ) =

*O&0 y √= ±

√

/ &#

&#

#&0 &F y = x = S #&0 &F x = y = S #&0 &F x = = y S *O&0

− y =

x = ± / − x = x + y = x + y =

O&[ U- 4 "M Y\ V Y\ ( RV+ 5 Q $ ) XY R# t\ 0 Z\ " b O. & ( 4E_ " 4 + " 4 +#cM& RV p&\ 4 M Ms O&0 O $ r4 Vl &# gc) >&\ 6 M x= Q+ bCc) Y=):[ $ $ ) X RV Y\ .$ u = f (x, y) e0& 4$ [

M x= ?U0 * 3 ∂ f A= ∂x

X

∂ f , B= ∂y

X

∂ f , C= ∂x∂y

X

>. T R# .$ *D = AB − C Y\ N# X &F < D A < B &#C < A S AGB *( u = f (X) U- 4 +

Y\ N# X &F < D A > B &#C > A S A% *( u = f (X) U- 4 +#cM&

u = f (X) e0& #E Y\ N# X &F D < S A: *( 9Z"# X Y\ .$ . f 4$ [ . y"0

f (X) ≈ f (X ) +

*C = f

xy (X

)

B =f

f (X) − f (X ) ≈

A dx + C dx dy + B dy

5F .$ M >. T 0 . 5F A = l&Q

yy (X

A

) A = fxx (X )

(A dx + C dy) + D dy

A > D > S M $ O q'? *(O 5

( ([a X E #&"+ N# .$ ( . (+ >.&[, &F D < (B&' .$ *( U- 4 + Y\ N# X R# 0&0 2 *$ ( ?, v c 3 ! $ >.&[, #&"+ .$ M ( Y\ !B )*"9 V# ^- M $. $ $ ) . @ X X p&\ 5F E LB$ f (X ) ≤ f (X) ≤ f (X )

E ( >.&[, !Y " XUV Z' X R# V (x, y) = xyz = xy

< y < x pO &0 . V

S − xy x+y

U- 4 +#cM& ( =&M 5 M 9ZM oL6 . V RV p&\ 0 *ZM [& < V

y (S − x − xy)/(x + y) = x (S − y − xy)/(x + y) = x=y x + xy = S ⇒ ⇒ x = S y + xy = S + V x = y = S/ + + X = S/, S/
pO &0C RV Y\ & U# * Y\ A & *O&0

A =

B

C

=

=

.$ <x >. T R# .$

√ S , −y (y + S) =− (x + y) X √ S , −x (x + S) =− (x + y) X √ xy(S − x − y − xy) = − S , (x + y) X

D

= AB − C =

S

aM ' U# *( U- 4 +#cM&

Y\ N# X R# 0&0 + $S >. T x = y = z = S/ $&U0 + E 0 Z' XUV [&) ('&" Gj S M ( V = ( S/) 0 0 ( % ) 5$ 0 XUV >&[& R# i jL0 *( 0 M ( y x E U0& z = f (x, y) M x= A .8 x +y +z −x+y−z− = >. T 0 +- !VO *M oL6 F U- &

"M *( O G#U Z $ *!' (x − ) ∂z =− , ∂x (z − )

√

]3 *y = ±x= ±√ / ]3 *x = / y = ±x U# X = X = , / 9E .&[, f RV p&\ ` + .$ √ √ √ X = − /, X = /, , − / √ √ √ √ X = − /, / X = /, / √ √ √ √ X = − /, − / X = /, − / .&V0 X & X 0 . 4$ t 6 5 EF &' *X = (, ) 0 E& nB ( T 0 0 C 0 f Z $ #E Z#S

V# 0 ) *(" @&\ RQ UB&Y

∂f ∂x

=

∂f ∂y

=

∂f ∂x∂y

=

xy(x + y − ) ( − x − y )/ xy(x + y − ) ( − x − y )/

− y − y + x + y + x y ( − x − y )/

Z#. $ A$ O ) % *H !VO 0C i

A

B

√

√

√ √ √ − −

√ √ √ − −

C √ − √ − √ − √ − √

√ − √ − √ − −

− − −

D

−

−

}

∂z (y + ) =− ∂y (z − )

.$ &F x + y − x + y = S U# z = S ]3 ( U R# 0 z = 0 pO ?U0 *(" #n3t 6 (x, y) z = &# z − z = .$ *y = − x = M # $ 0 p&\ z RV p&\ R# 0&0 *z = −

C : z = , (x − ) + (y + ) =

.$ * " X = ( , − , −) X = ( , − , ) Y\ $ 0 0 z . + #E ( 4&+ d0 AC 0 p&\C $ $.

V# 0 ) &0 c X X $. .$ *O&0 ∂f ∂x

= −

(z − ) + (x − ) , (z − )

S = &

"M 9*H !VO [&) ('&" &0 &!Y " XUV + R0 E A .8 ( aM ' Z' . $ N# 4 M 0 0 O&0 z y x !Y " XUV $&U0 ZM x= *!' #$ 5&0 0 O&0 S 0 0 !Y " XUV R# [&) ('&"

0 *z = (S − xy)/(x + y) R# 0&0 *xy + xz + yz = S

*M ' @ !1 ' ('&" (0&f Z' &0 L N# A . z

= f (x, y)

+- e0& U- & + "M $. .$ 90&0

) x + y + z − xy − yz + x + y + z − =

) (x + y + z ) = a (x + y − z )

i

(

B AG' N :+ l&Q *(O $ (+ e0& $ & U- 4 "M .$ 5F 0 U0& aM ' !1 ' & $ O oL6 , +

S = 4 "M B&" 4&0 #) B&" #F (0 , +

*#F $ ) 0 M x= G Q=)CDL VD) $ + RU ("0 Y *D ⊆ D ( e0& N# u = f (X) X ∈ D E 0 M S 0 D E X X 5 Q @&\ *f (X ) ≤ f (X) ≤ f (X) f

S = 4 "M B&" l&Q $ l9 X 0 . X X O&0 % ) . $ D 0 u = f (X) f (X ) f (X ) Z & D 0 u = f (X) YFe YF Z & D 0 u = f (X) `Fe YF X 0 c . * min f (X) = f (X) max f (X) = f (X) Z"#

$ ) >. T .$ AS =C 4 + 4 +#cM& M $ O ) 4 + &# 4 +#cM& p&\ M B&' .$ " $0 j

% ) $ ) B&" 0 &3 0 *O&0 V# E u0

*Z#. $ E& #E G#U 0 S = 4 "M B&" 0 X∈D

X∈D

.$ *X ∈ R D ⊆ R M x= $ E 0 M ( D B )*"9 N# X Zg S . T D D &, + r `&UO X cM 0 S r > 9A$ O ) GB I*H !VO 0C M eY1 . AD Z+ U#C n

n

c

∀r >

∃ X ∃ X

:

X ∈ Br (X ) ∩ D , X ∈ Br (X ) ∩ Dc

5&6 ∂D $&+ &0 & D B . D E p&\ + , +

p&\ + M Zg S )E . T .$ . D , + *Z$

9$ + >&[f 5 *∂D ⊆ D 9O&0 O $ 0 .$ . $ E

∂f ∂y

=

∂f ∂x∂y

(y + ) − (z − ) (z − )

=

( − x)/(z − )

Y\ N# X ( U- 4 +#cM&

( U- 4 +

, ) cM 0 M N# z . $ + V# 0 ) &0 [B ( #0 !T&' k#& O&0 `&UO

A B C − / − / / / ,−

D / /

0 % $

90&0 . z = f (x, y) e0& U- &

"M $. .$ ) f = x + xy + y + x − y +

) f = xy − x − y + x − ) f = x + xy + y − ln x − ln y ) f = x + y − x − y ) f = x + xy

) f = x + xy + y

) f = xy − x − y ) f =

x

+ xy +

y

) f = ex cos y

) f = y sin x ) f = xy ln(x + y )

) f = +

x+y+ + x + y

Y\ .$ z = f (x, y) e0& 4$ [ . y"0 N+M 0 A *M >&[f . **H q1 X RV Y\ .$ u = f (x, y, z) e0& 4$ [ . y"0 N+M 0 A 5&0 v e0 0 . **H q1 [O X RV *M (0&f ]< $ + 90&0 . #E e0 E N# U- &

"M

∗

) f = xy − x − y + x − ) f = x + y + z + x + y − z ) f = x +

z y + + x y z

) f = xy z (

− x − y − z)

x y z a + + + x y z b +

) f = xyz − x − y − z

) f =

& 0 &0 R# * " .$ RV Y\ V# &# U1 *$. $ .&SE& (U[@ E S = 4 "M B&" !' 0 #E KO 0 + #. B 9$ O $& 63 G Q=)CDL VD) EJ /C =BV * $

4 "M B&" !' 0 *D ⊆ D u = f (X) M x= 9Z$ 4& . #E !' D 0 u = f (X) S = RQ S *( $6= u = f (X) &#F M ZM t\ A *Z$ $ $ 0 S *( 3 D 0 u = f (X) &#F M ZM t\ A; *Z$ $ $ 0 RQ =&# . " e1 D .$ M u = f (X) RV p&\ A *ZM [& & F E N# .$ . u = f (X) . \

D , + E 0 . u = f (X) S = &

"M A *Z0&# A∂D U#C R# S.c0 ' .$ F (0 $ , "#&\ &0 AH e0& 4 + . & F R# VQ M 4 +#cM& . . \

*ZM 4?, D , + 0 u = f (X) x

f

e0& S = &

"M A& '() $ 0 $ D , + 0 f = x − x + y − y + *#.F (0 . y = x y = x = y *H !VO 0 ZM Z R .$ . D , + 0 *!' & \ U0& f ( . M "0 D 5 Q *$ O ) GB 5F 0 f $6= Ds ]3 $$S G#U D !M 0 ( R =&# 0 *O&0 % ) . $ B&" R# 0&0 *r( 3 9ZM t\ . f = 0 pO Int(D) 0 f RV p&\

x − = y − =

=⇒

x= =⇒ X = ( , ) y=

S = 4 "M 9*H !VO

}

O&0 E&0 D

( "0 D A& $ "0 R n E 0 A, ( "0 "0 , + $ U w O A

( "0 "0 , + & $ U `&+ ) A *∂D ∩ D = ⇔ ( E&0 D A *$ O U) H*;*H 0 E&0 , + G#U &6 0 . T .$ . D ⊆ R , + #E $ $ u@&\ T&= U# $0&) S N# .$ M Zg S #$ 5&0 0 *A$ O ) % I*H !VO 0C O&0 $

c

⇔

n

n

∃M > ∀X, Y ∈ D : X − Y ≤ M

"0 M Zg S < . T .$ . D

⊆ Rn

, + #E *O&0 . M }

*( D E Y\ X AGB 9I*H !VO *$S &) S N# .$ D . M , + A% *D : x + y ≤ ZM x= A& '() % $ D M ( #0 5 Q ∂D : x + y = >. T R# .$ *O&0 $6= D ]3 ( . M >. T R# .$ *O&0 R .$ & x . D ZM x= A, .8 T&= #E (" . M D & *( "0 D nB ∂D ⊆ D *(" $6= D ]3 *( 5 M 0 u@&\ .$ *D : − ≤ x ≤ , ≤ y < ZM x= A .8 (− , ) .$ / g. &0 !Y " E ( >.&[, D E >. T R# D 0 (− , )(, ) e- 5 Q *(, ) (, ) (− , ) *( . M D M B&' .$ *(" "0 D ]3 $. tU *(" $6= D R# 0&0 D $6= , + 0 u = f (X) e0& S & $ % ) . $ D 0 u = f (X) 4 "M B&" &F O&0 3 0 u = f (X) 4 "M B&" % ) N# X S ?U0 *( Y\ N# X ∈ Int(D) V# &# X ∈ ∂D &F O&0 D *( u = f (X) e0& RV u = f (X) e0& S = &

"M J*I*H q1 0 h D E 0 &# V0 $. . 5&V &) + .$ D , + 0

9ZM UB&Y &S ) . @ 0 . OA

: (x, ) ;

≤x≤π f (π, ) = f (, ) = ≤ f (x, ) = sin x ≤ = f (π/, ) (π, y) ; ≤ y ≤ π , f (π, y) = (x, π) ; ≤ x ≤ π f (π, π) = f (, π) = ≤ f (x, π) = sin x ≤ = f (π/, π) (, y) ; ≤ y ≤ π , f (, y) =

AB

:

BC

:

CO

:

p&\ .$ M ( 0 0 D 0 f 4 +#cM& ` + .$ ]3 − 0 0 D 0 f 4 + $ O .& (π/, π) (π/, ) *$ O .& (π/, π) Y\ .$ M ( f = + x + y − x − y e0& S = &

"M A .8 *0&0 D : x + y ≤ , + 0 . R# 0&0 ( 3 D 0 f ( . M "0 D 5 Q *!' U 0 f = 0 pcO *( % ) . $ D 0 f 4 "M B&"

− x = − y =

⇒

x= y=

⇒X =( , )

E ( >.&[, D E *f (X ) = ?U0 ( ∂D : x + y =

>. T 0 M ∂D : r(t) =

−−−−−−−−−−−→ cos t, sin t ;

≤ t ≤ π

ZM G#U X R# 0 *$ O .&3 ! g(t) := f r(t) = − + cos t + sin t

0 g (t) = pO *$ O UB&Y [; π] 0 "#&0 M &# t = ∂π/ nB√ ( − cos t + sin t = U 0 g(π/) = − + g() = g(π) = 5 Q *t = π/ 0 0 D E 0 f 4 +#cM& √]3 √g(π/) = − − √√ f 4 + $ O .& , .$ M ( − + √ √ .& − , − .$ M ( − − √ 0 0 D E 0 M Z#S f (X ) . \ &0 √"#&\ .$ ]3 *$ O

√ √ , Y\ .$ M ( − + 0 0 D 0 f Z+#cM&

M ( − − √ 0 0 D 0 f 4 + √$ O √ .& ) GB ;*H !VO 0 *$$S .& − , − .$ *$ O (; π)

E *$. c d0 0 E& $. $ . 1 D E 0 5 Q M N0 *$ + Z"\ 5 OB AB OA y.&3 0 . 9ZM UB&Y &S ) . & Y.&3 R# E * ≤ x ≤ M ( (x, x) p&\ + , + OA eM ( =&M R# 0&0 *f (x, x) = x − x + & E&0 0 . g(x) = x − x + e0& S = &

"M g() = 5 Q x = U# g = pO *Z0&0 [; ] (, ) .$ ( 0 0 AB 0 f 4 +#cM& ]3 g( ) = − ( , ) .$ ( − 0 0 OA 0 f 4 + $ O .& *$$S .& M ( (x, ) !VO 0 p&\ + , + AB e=&M R# 0&0 *f (x, ) = x − x − & * ≤ x ≤ E&0 0 . g(x) = x −x− e0& S = &

"M M ( g() = − 5 Q x = U# g = pO *Z0&0 [; ] (, ) .$ ( − 0 0 AB 0 f 4 +#cM& ]3 g( ) = − ( , ) .$ ( − 0 0 AB 0 f 4 + $ O .& *$$S .& & *( ≤ y ≤ &0 (, y) p&\ + , + OB e&

"M M ( =&M R# 0&0 *f (, y) = y − y + pO *Z0&0 [; ] E&0 0 . g(y) = y − y + e0& S = g() = − g() = 5 Q y = U# g = $ O .& (, ) .$ ( 0 0 OB 0 f 4 +#cM& ]3 *$$S .& (, ) .$ ( − 0 0 OB 0 f 4 + .& (, ) .$ ( 0 0 D 0 f 4 +#cM& V# .& ( , ) .$ ( − 0 0 D 0 f 4 + $ O

*$$S

, + 0 . f = sin x cos y e0& S = &

"M A, .8 *D : ≤ x ≤ π, ≤ y ≤ π 90&0 D O&0 R .$ 3 !Y " N# D M ZM ) 0 *!' *$ O ) % *H !VO 0 ( . M "0 R# 0&0 R# 0&0 C &aa U0& f ( . M "0 D 5 Q f $6= Ds ]3 $$S G#U D !M 0 ( A& \

0 *O&0 % ) . $ B&" R# 0&0 *r( 3 5F 0 9ZM t\ . f = 0 pO Int(D) 0 f RV p&\ R =&# D

cos x cos y = − sin x sin y =

⇒ ⇒

cos x sin x

&# cos y = &# sin y =

cos x = sin y = cos y = sin x =

& ∂D : < x < π, < y < π V# 0 ) &0 ]3 O&0 X = (π/, π) E >.&[, ∂D .$ $ ) f RV Y\ (, ) / g. &0 B& !Y " N# D E *f (X ) = − M ∂D e- .& Q E N# *O&0 (π, ) (π, π) (, π)

e0 M x= -+ c) Q=)CDL VD) $ e0& &

"M ("0 Y *#n<\ 6 f, g : R → R 4 "M . B&" % ) *S : g(X) = , + 0 u = f (X) *Z & g = pO 0 f p6

. g = M x= U5 . $ 5&+ .$ . f E & >. T R# .$ *Z# $M Z >. T R# .$ M $$S h'? *$ + Z 5 T Ǳ E 0 M " X 5 Q @&\ g = pO 0 f &

"M 0 *$$S /&+ g = 0 f E E N# & F *$ O ) % ;*H !VO B&" % ) N# X S NGA $ S : g = , + 0 g O&0 g = pO &0 f 4 "M (O $ $ ) λ $, >. T R# .$ O&0 T GB&L

*f (X ) = λg (X ) M 0 . f = xy e0& &

"M A& '() $ *0&0 x + y = pO E ( >.&[, S_ pO g = x + y − &# .$−−*!' −→ x) = λ( , ) R# 0&0 *−(y,−−→

⎧ ⎨ y=λ x=λ ⎩ x+y =

⎧ ⎨ x=λ y=λ ⇒ ⎩ λ =

⇒ X =

,

( U- 4 +#cM& N# Y\ R# *f (X ) = / M ?za M .& c . +M . \ f 5F m @ .$ #E ( =&M !B$ R# &#F *f (, /) = pO 0 . f = x − y + z e0& &

"M A, .8 *0&0 x + y + z = ( >.&[, S_ pO g =−x−−−+−y−−→+ z −−−−&# .$ *!' −−−−−→ M ( RO. 5 Q R# 0&0 *( , −, ) = λ(x, y, z) E Z#. $ λ = #&0 ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

λx = λy = − λz =

x + y + z =

√

⎧ x = / λ ⎪ ⎪ ⎨ y = − /λ ⇒ z = /λ ⎪ ⎪ ⎩ /λ = ⎧ √ −−−−−−−→ ⎨ X = ( , −, ) ⇒ √ −−−−−−→ ⎩ X = − − ( , −, )

√

]3 *f (X√) = − / f (X) = / & ( − / 0 0 g = pO 0 f U- 4 +

√ −−−−−−−→ − / ( , −, ) Y\ .$ M 4 +#cM& $ O

.& √ .$ M ( / 0 0 c g√ = pO 0 f U-

*$ O .& / −(−−, −−−−,−→) Y\

}

X A% *S = 4 "M AGB 9;*H !VO * " g = pO 0 f p6 4 "M p&\

X

0 f e0& S = & +

"M

0 $

90&0 . D , +

) f = x + y + xy − x − y + , D : ≤ x ≤ , ≤ y ≤

) f = x + xy − y,

D : x + y ≤ y

) f = sin x + sin(x + y),

D : ≤ x ≤ π, ≤ y ≤ π

) f = sin x sin y sin(x + y), ) f = sin x cos y,

D : ≤ x ≤ π, ≤ y ≤ π

D : ≤ x ≤ π, ≤ y ≤ π − x

) f = x − xy + y + x − , D : ≤ x ≤ ,x ≤ y ≤

) f = x + y − x + y,

) f = x + y + z , ) f = x + y + z,

D : x + y ≤

D : x + y + z ≤

D : x + y ≤ z ≤

) f = x + y + z + x + y − z, D : x + y + z ≤

[ P' AG' N + N# 4 "M B&" .$ Z##$ I*H uL0 .$ ?z[1 M . Y&+ $ + oL6 5 . f $ E +"1 x e0& & #$ R# *$M X@ 5F 0 . e0& &

"M y N# .$ M M & O KY >_$&U N+M 0 R# S R =&# E >.&[, B&" ( =S . 1 #& S ' $ .$ . p6 4 "M *( 5F Y\ R#$ *Z$ ^-

$ O AC A;C AC >_$&U E &F λ = S #&0 ]3 *$. $ $&q AC B$&U &0 M x = y = z = &# M $ O AC B$&U E >. T R# .$ & *λ = λ = pO E & *λ = >. T R# b .$ &# x = R# 0&0 *y = z = $ O AC A;C >_$&U p&\ U# *x = ± &# x = Z#. $ AC B$&U E 0 *$ O !T&' X = (− , , ) X = ( , , ) X = (, − , ) X = (, , ) p&\ 0&6 >. T E *$$S !T&' X = (, , − ) X = (, , ) f (X) = f (X) = f (X) = f (X) = =@ D 0 f 4 +#cM& ` + .$ ]3 *f (X ) = f (X ) = 0 0 5F 4 + $ . X X p&\ .$ 0 0 *$ . X Y\ .$ }

5 Sq0 .$ p& aM ' Z' &0 !Y " XUV A .8 *0&0 . x /a + y/b + z/c = B$&U 0 & &q= E Z 6 N# .$ M ZM x= Z *!' =&M 5.&\ !B$ 0 9$. $ . 1 !Y " XUV E / . N# ZM x= *Z#0 h .$ Z 6 N# .$ . !VO ( . 1 Z 6 N# .$ M !Y " XUV E / . 5F >&j L

!Y " XUV Z' >. T R# .$ *( (x , y , z ) $. $ 4 +#cM& M ( =&M ]3 *( f = x y z 0 0 !T&' pO *Z0&0 g = x /a + y/b + z/c = pO 0 . f M $. $ $ ) λ $, M ( U R# 0 S_ )

−−−−−−−−−−−−−−−−−→ −−−−−−−→ (yz, xz, xy) = λ x/a , y/b , z/c

&F λ = S R# 0&0

⎧ ⎧ ⎨ yz = xλ/a ⎨ xyz = λx /a xz = yλ/b ⇒ xyz = λy /b ⇒ ⎩ ⎩ xy = zλ/c xyz = λz /c ⎧ ⎧ ⎨ λ/ = λx /a ⎨ x = a / ⇒ λ/ = λy /b ⇒ ⎩ y = b / ⎩ λ/ = λz /c z = c /

.$ p& !Y " XUV AGB 9*H !VO + y R0 T&= R# +M A% 5 Sq0 O $ $ pO 0 . f e0& 4 "M 0 $ $ 9M [&

) f = xy, x / + y / =

) f = x + y,

x + y =

) f = x + y ,

x/a + y/b =

) f = Ax + Bxy + Cy , ) f = cos x + cos y,

x + y =

x − y = π/

) f = x + y + z, x + y + z = ) f = sin x sin y sin z,

) f = x + y + z ,

x + y + z = π/

( e1 Z 6 N# .$ (x, y, z)√Z#$ 0 $M x= 5 Q M √ √ #$ 5&0 0 *z = c √y = b x = a √ R# 0&0 XUV

U# * f (X ) = abc M X = (a, b, c) √ √ √ a× b× c $&U0 0 √#&0 Z' aM ' &0 !Y "

*H !VO 0C $ 0 abc 0 0 5F Z' O&0 *A$ O ) GB D : S 0 f = x + y + z e0& 4 "M A .8 *0&0 . x + y + z ≤ 9Z0&# . Int(D) .$ f RV p&\ 0 *!' f = ⇔ x = y = z = ⇔ X = (, , )

N# M Z0&# . D [B 0 f 4 "M &' *f (X) = M U 0 B&" R# *∂D : x + y + z = 9( M pO 0 f 4 "M R =&#

x /a + y /b + z /c = ,

y z , x + y + z = a, < m, n, , a √ √ ) f = x + y + z − √ xyz, x + √ y + z = m n

>&j L [ & ax + by + c = y E Y\ R# V#$c A *0&0 .

g = x + y + z −

=

(B&' R# .$ *O&0 p6 4 "M B&" N# M ( Z#. $

⎧ ⎪ ⎪ ⎨

f = λg ⇔ g= ⎪ ⎪ ⎩

x = λx y = λy z = λz

x + y + z =

( ) () () ()

0 . f = xyz e0& &

"M A '() * $ *0&0 x + y + z = c x + y + z = V# pO pO *g = x + y + z − g = x + y + z &# .$ *!' U 0 S_ −−−−−−−−→! −−−−−→! −−−−−−−−→! yz, xz, xy = λ , , + μ x, y, z

#$ 5&0 0 ( ⎧ ⎨ yz = λ + μx xz = λ + μy ⎩ xy = λ + μz

⎧ ⎨ xyz = λx + μx ⇒ xyz = λy + μy ⎩ xyz = λz + μz

A*H C

5$M e+) E x + y + z = x + y + z = 5 Q M y x R# 0&0 *xyz = μ M Z#.F (0 &" R# *M T μx + λx = /μ 4$ ).$ B$&U .$ z 9Z#S h .$ . #E & B&' !1 _ nB xy = xz = yz = &F λ = μ = S AGB x+ y + z = #&0 5 Q T z y x $ , E & $ ( & R# M B&' .$ ( T c U0 $, RV+ (B&' R# ]3 *x + y + z = "#&0 #E *(" 0 0 x X"' 0 B$&U % ) &F λ = μ = S A% RV+ M x = y = z = Z E&0 ]3 ( x = *("

I 0 0 5&60qT&' M 0&0 . @ ([a $, $ A; *O&0 !1 ' 5&6, +

x + y = x + # $ & x + y = y T&= A *0&0 . x /a +y /b = q0 .$ ('&" aM ' &0 !Y " A *M p&

0 M .$ M 0&0 . E&0 . `&. `&UO AH *$ + 5 p& R `&UO x +y +z = M E (x, y, z) >&j L 0 Y\ S A8 R#$ R# S O&0 T = x − y + z & $ . $ *0&0 . M ^Y Y\ >&T 0 5F ) M 0&0 . Y " XUV AJ 5F / . N# $. $ . 1 Z 6 N# .$ ( &j L

* < a, b, c M $. $ . 1 x/a+ y/b+ z/c = T 0 (0&f & F ` + M 0&0 . @ ([a $, A7 *( aM ' 5&60qT&' M M ' @ . @ . !Y " XUV !VO 0 L A RV+ . \ !1 ' 5F ^Y ('&" $ 0 V 5F Z' *O&0 90&0 . O $ $ T & (x , y , z ) Y\ T&= A;I Ax + By + Cz + D =

√

RV+ M x = y = z = √/ &F μ = λ = S A: *x + y + z = >. T R# .$ #E (" A*H C >_$&U E &F λ

=

S A$ Z#S

= μ

λ(x − y) = −μ(x − y ) ,

*ZM 5&0 . p6 4 "M B&" =&# Z+U !VO 5 M * * * g f ≤ k < n M x= VD) % $ &

"M ("0 Y * " R 0 R E #n<\ 6 U0 g , + 0 f e0& n

S : g (X) = · · · = gk (X) =

λ(x − z) = −μ(x − z ) , λ(y − z) = −μ(y − z )

Z#S h .$ . #E & B&' 5 M &F O&0 >& z y x S (a x+z =x+y =y+z =−

λ μ

M $ O !-& &0 ]3 x−y = y−z = z−x=

*( 1& M

k

mg(X) = l@ ) f l YE . B&" % ) *Z & g (X) = ___ k

% ) N0 X S NGA C / & $ g (X) = *** g (X) = y# O 0 f e0& 4 "M B&"

g (X) *** g (X) &. $0 X ∈ S E 0 O&0 (=&# . @ λ *** λ $ , &F O&0 Y !\ "

M O

k

k

k

f (X ) = λ g (X ) + · · · + λk g (X )

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ () ⇒ ⎪ ⎪ ⎪ ⎪ ⎩

A*H C

.$ (B&' $ *( O $& 4.& Q 4$ &" E ( ) .$ 9Z#S h x = y + β = y α = x >. T R# .$ 9λ = GB *$. \\' 6#. M x + x = , + .$ 4 &" E >. T R# .$ 9λ = % y = / R# 0&0 *x = / Z#. $ A*H C & B = *α = β + β = / + λ/ α = /(λ + ) .$ *β = −/ α = / λ = / ]3 !1 ' V# f (x,√y, α, β) = / (B&' R# + M ( / = / 0 0 y & + T&= y = x + E ( /, /) Y\ T&= E ( >.&[, !VO 0C x = y + y E ( / , −/ ) Y\ & *A$ O ) % *H y# O &0 . f e0& 4 "M $. .$

0 $

90&0 O $ $

) f = xy + yz , y + z = , x + y = , x> , y> , z>

) f = x + y + z

, x + y + z = , x + y + z =

) f = x yz + ) f = xyz

,

&F z = y = x S (b x + y + z = pO E 5 M *z = −x − λ/μ &# pO E x = λ/μ M $$S x + z = y + z = −λ/μ

α = ( + λ)x β = y − λ/ α = x + λ/ y = x α=β+

x − y = y + z =

x + y + z =

R# 0&0 *(/)(λ√ /μ) = M $ O .$ *λ/μ = ±√/ z − =x=y= √ z uO 5.&\ !B$ 0 M − = x = y = − &# 9$ O !T&' Y\ √

X = √ X =

, y=x ,

) f = x + y + z + w

,

x − y + z − w −

√

, −,

− , ,

!

X = −

!

! , , − − , ,

!

√

f (X) = f (X ) = f (X) =√− / M *f (X ) = f (X ) = f (X) = / Rl+ x + y + z = pO &F x = y = z = S (c *( yb M = $ √ / 0 0 g = g = pO 0 f Z+#cM& ` + .$ ]3 0 0 5F 4 + $ . X X X p&\ .$ M√( *$ . X X X p&\ .$ M ( − / x = y + y y = x + R0 T&= R# & M A, .8 *M 3 . y = x + 0 LB$ Y\ (x, y) ZM x= *!' e0 e0& 5 M *O&0 x = y + y 0 LB$ Y\ (α, β) Z#S h .$ . (α, β) & (x, y) T&= f = d = (x − α) + (y − β)

X 0 . B&" y# O

x + y + z = x+y−z+w =

, , − X =

√

√ √ ! X = − , −, X = −

, x + y + z = , x + y = z

) f = xy + z

!

g = y − x = , g = α − β − =

U 0 S_ pO 5 M *Z#S

,

=

*#.F (0 . x = y + y = x + $ T&= AJ O $ M 0&0 @O 0 . f = αx + βy e0& 4 +#cM& A7 α + β = x + y = ZO&0 M 0&0 @O 0 . f = αx + βy + γz e0& 4 +#cM& A *α + β + γ = x + y + z =

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→! (x − α), (y − β), −(x − α), −(y − β) = −−−−−−−−−→! −−−−−−−−−−−→! λ − x, , , + μ , , , − ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

R# 0&0 *(

⎧ α = ( + λ)x (x − α) = −λx ⎪ ⎪ ⎪ ⎪ (x − β) = λ β = y − λ/ ⎪ ⎪ ⎨ −(x − α) = μ α = x + μ/ ⇒ −(y − β) = −μx β = y − μ/ ⎪ ⎪ ⎪ ⎪ y = x y = x ⎪ ⎪ ⎩ α=β+ α=β+

=

)

x y

! ! a dx + b dy ∧ a sin y dx − b dy = = −ab( + sin y) dx ∧ dy

4= α η θ M x= & $ &F O&0 #n3t 6 U0& z = f (x, y) ) (f θ) ∧ η = θ ∧ (f η) = f θ ∧ η

) θ ∧ η = −η ∧ θ

!

) θ ∧ θ =

0 * $

M $ 5&6 A

P P dx + Q dy ∧ R dx + S dy = R + θ = sin(xy) dx + x + dy

Q dx ∧ dy S

!

M M x= A; α = x dx − y dy η = y/x dx + x/y dy . J*;*H E ; y0 . *f = arcsin (x/y) *M t\ 1 η = R dx + S dy θ = P dx + Q dy M (0&f A .$ θ ∧ η M " Y !\ " X Y\ .$ 1 y\= *O&0 T GB&L X Y\ >. T R# .$ θ = P dx + Q dy S

$

dθ := dP ∧ dx + dQ ∧ dy

e1 .$ ZM G#U θ !" #$ . ∂P ∂P dx + dy ∧ dx ∂x ∂y

∂Q ∂Q + dx + dy ∧ dy ∂x ∂y

∂Q ∂P − dx ∧ dy ∂x ∂y

dθ

:=

=

Z#. $ _&0 G#U 0 ) &0

'() $

x ) d x dx − dy = − dx ∧ dy y y

) d dx − dy = x y

)

+

− (x + y)

−

+ (x + y)

, dz = Δz = z − z

*O&0 (0&f ]3 R# E LB$ $ , z y x M .&[, R .$ r4= s E . h $ !VO 0 ( θ = P (x, y) dx + Q(x, y) dy

R# $ w O *#n3 t 6 U0 Q(x, y) P (x, y) M &0 . R .$ & = + , + *Z & θ $ . e0 *Z$ 5&6 6 R $&+ θ = d(x + y ) = x dx + y dy A& '() $ *( R .$ 4= N# D : x = $ &0 4= N# θ = dx/x + x dy A, .8 *( .&[, R .$ r4= ;s E . h $ #n3t 6 U0& f (x, y) M ω = f (x, y) dx ∧ dy !VO 0 ( ; + , + *Z & ω $ . f (x, y) $ *( *Z$ 5&6 6 R $&+ &0 . R .$ & = R0 A%- ∧C S %- $ $ !VO 0 . & = dy ∧ dy = dx ∧ dx = , dx ∧ dy = −dy ∧ dx

*Z$ Z+U & = !M 0 Y >. T 0 $ + G#U η = R dx + S dy Rl+ θ = P dx + Q dy S R# 0&0 &F θ∧η

=

! ! P dx + Q dy ∧ R dx + S dy

= (P S − QR) dx ∧ dy

! d arctan(x + y) dx + arcsin(x + y) dy " # =

.$ z U0 M

( r" #$s 4= uL0 R# ` -

.&V0 4$ ` ^Y c 4$ ` y (+"1 .$ 9Z$ 4& ' $ .$ . Ǳ&O R# UB&Y *$.

*&q= T dx = Δx = x − x uL0 R# .$ $ dy = Δy = y − y

) θ ∧ (η + α) = θ ∧ η + θ ∧ α

!

F A +

− xy sin y dx ∧ dy

Z#. $ _&0 G#U 0 ) &0

dx ∧ dy

'() % $

! y ) x dx + y dy ∧ x sin y dx + dy = x

0 θ = y dx + (xy + y) dy 4= A, .8 z = ]3 *O&0 ! &M nB Adθ = U#C ( t1$ U 5 0 R# & *θ = df Z#. $ D 0 M (" f (x, y) .$ *∂f /∂y = xy + y ∂f /∂x = y M ( D = R

0

! y dx = xy + A(y)

f (x, y) = 0

!

xy + y dy = xy + y + B(x)

f (x, y) =

0 _&0 &" >.&[, $ "#&\ &0 * LB$ e0 B A M C M B(x) = C A(y) = y + C M Z#S f E ( >.&[, θ >E I R# 0&0 * LB$ (0&f $ , C *( (0&f $, C M f = xy + y + C #E ( ! &M D = R −(, ) 0 θ = x+dxx++yydy A .8 , !VO .& D & *O&0 f = x + y e0& !" #$ 0 0 *$S+ . 1 D .$ &z &+ (− , )( , ) y .&3 #E (" N# & (" 4E_ .&V 3 q1 .$ D 5$ 0 .& pO ]3 *( =&M pO ! ! θ = x +xy−y dx+ x −xy−y dy 4= A .8 E θ !"& 3 R =&# 0 *( ! &M R# 0&0 t1$ D = R 0 gc) y 0 ([" dy X#- E x 0 ([" dx X#9Z#S

0 f (x, y) = x

=

0

f (x, y) =

! x + xy − y dx + x y − xy + A(y) ! x − xy − y dy

x y − xy −

=

y

+ B(x)

U0& f O&0 4= η θ S ) d(θ + η) = dθ + dη

) d(df ) =

$

&F #n3t 6

) d(f θ) = df ∧ θ + f dθ ) d(dθ) = 0 $

θ = x/y dx + xy dy f = tan(x + ,y) M x= A .$ ; y0 . *η = xe dx + x + y dy *$ 5&6 . *;*H η = dx + arcsin (x/y) dy f = xe M x= A; *;*H .$ ; y0 . θ = cos x dx−x sin y dy *$ 5&6 . y

y

x 4= Zg S . T .$ $ $ ) z = f (x, y) U0& M ( > θ = P dx + Q dy *Z & θ !"& 3 . f >. T R# .$ *θ = df M O&0 O $ *dθ = M Zg S t1$ . θ 4= . T .$ . T .$ . D ⊆ R , + #E $ 5 0 . #E (T& &0 X ∈ D Y\ M Zg S >?@ + 9(=&# D .$ &z &+ X X y.&3 X ∈ D LB$ Y\ E 0s r*$. $ . 1 LB$ X B ( $ ) X _&0 G#U .$ M $ O ) q1 E (+"1 0 ) &0 *A$ O ) *H !VO 0C ( R# ]V, & *( t1$ &F O&0 ! &M θ S *;*H 0 =&M pO N# #E q1 *( yb M (B&' .$ XY

*$ g . . ZV' R# ]V, . 10 n

0 _&0 &" >.&[, $ "#&\ &0 * LB$ e0 B A M M $. $ $ ) C & (0&f $, Z#S f f (x, y) =

x

−

y

+ x y − xy + C

& = 4 M M M oL6 0 & $ . O $ $ 4= !"& 3 5$ 0 ! &M >. T .$ &M #E 90&0 ! ! ) x − xy dx + y − xy dy

) )

y dx − x dy

x − xy + y +

! ! x sin(xy) dx + x + y + x + y + xy dy

} !VO .& , + 9*H !VO , + 0 θ 4= S 93=_ $ $ *( ! &M D 0 θ &F O&0 t1$ D !VO .& ! &M θ = y dx + dy 4= A& '() % $ #E O&0 !VO .& D = R 5F $ V# &0 (" *(" t1$ θ nB dθ = −dx ∧ dy

! ! ! dx + dy − dz ∧ dx − dy + dz ∧ − dx + dy + dz = − dx ∧ dy ∧ dz

M x= θ = x dx − y dy + z dz

0 $

α = dx + dy + dz

η = x dx + y dy + z dz ω = y dx ∧ dy + x dy ∧ dz + y dz ∧ dx

η ∧ ω θ ∧ α θ ∧ ω θ ∧ η & .&[, E N# >. T R# .$ *M [& . α ∧ ω θ ∧ η ∧ α θ = P dx + Q dy + R dz S $ . ω θ !" #$ ω = P dx ∧ dy + Q dy ∧ dz + R dz ∧ dx ZM G#U #E >. T 0 X 0 dθ

= =

dω

= =

dP ∧ dx + dQ ∧ dy + dR ∧ dz ! ! Qx − Py dx ∧ dy + Ry − Qz dy ∧ dz ! + Rx − Pz dz ∧ dx dP ∧ dx ∧ dy + dQ ∧ dy ∧ dz + dR ∧ dz ∧ dx ! Pz + Qx + Ry dx ∧ dy ∧ dz

*$ O G#U *;*H .$ &+ ! &M t1$ = (=&# 5&Q θ 4= M Zg S > . T .$ . ω 4= ; M Zg S : . T .$ . ω 4= ; *ω = dθ M $$S *O&0 T 5F !" #$ 4= N# S ;;*;*H q1 E H & +"1 0 h .$ ZV' R# ]V, *( t1$ &F O&0 ! &M 4= ; &# . 10 0 =&M pO N# #E q1 *( yb M (B&' *$ g . . ZV' R# ]V, ; N# ω 4= $ θ θ S $ &F O&0 #n3t 6 U0& f 4= ) d(f θ ) = df ∧ θ + f dθ

) d(f ω) = df ∧ ω + f dw

) d(df ) = ) d(dθ ) =

) d(θ ∧ θ ) = dθ ∧ θ − θ ∧ dθ ) d(dω) =

, + 0 θ 4= S 93=_ $ , + 0 ω 4= ; S *( ! &M O&0 t1$ D !VO .& *( ! &M O&0 t1$ D !VO .&

! ! x + xy + y dx + x − xy + y dy ) (x + y) ! ! ) ex ey (x − y + ) + y dx + ex ey (x − y) + dy

Z+U R 0 . _&0 d'&[ $ + E&bF . 4$ ' 5 M *Z$

( .&[, R .$ 4= E . h * $ v e0 R Q P M θ = P dx + Q dy + R dz !VO 0 *Z & θ $ . e0 R# w 6 $ *#n<\ 6

!VO 0 ( .&[, R .$ 4= ; E . h

ω = P dx ∧ dy + Q dy ∧ dz + R dz ∧ dx

w 6 $ *#n3 t 6 v U0 R Q P M *Z & ω $ . e0 R# !VO 0 ( .&[, R .$ 4= E . h

Ω = f dx ∧ dy ∧ dz

$ . f $ *O&0 0n3t 6 v U0& f M *Z &

. R .$ & = & = ; & = + , +

%- *Z$ 5&6 6 R 6 R 6 R $&+ &0 X 0 . dz dy dx & & = R0 A%- ∧ &#C S ZM G#U #E >. j0 Ω

dx ∧ dx = dy ∧ dy = dz ∧ dz = dx ∧ dy = −dy ∧ dx dx ∧ dz = −dz ∧ dx

dy ∧ dz = −dz ∧ dy

dx ∧ dy ∧ dz = dz ∧ dx ∧ dy = dy ∧ dz ∧ dx = −dy ∧ dx ∧ dz = −dx ∧ dz ∧ dy = −dz ∧ dy ∧ dx dx ∧ dx ∧ dy = dx ∧ dx ∧ dz = dy ∧ dy ∧ dz = dz ∧ dz ∧ dx = dz ∧ dz ∧ dy = dx ∧ dx ∧ dx = dy ∧ dy ∧ dy = dz ∧ dz ∧ dz∧ =

Z#. $ _&0 G#U 0 ) &0

'() $

! ! y dx + z dy + x dz ∧ z dx + y dy + z dz = ! ! = y − z dx ∧ dy + z − x y dy ∧ dz ! + x z − yz dz ∧ dx ! y dx + z dy + x dz ∧ x dx ∧ dy − y dy ∧ dz ! +z dz ∧ dx ! = x − y + z dx ∧ dy ∧ dz

$ $ 4= 5$ 0 ! &M t\ R+- 0 $ $ 90&0 . ! &M & = !"& 3 O ) θ = yz dx + (xz + ) dy + (xy + z) dz

4= A& '() $ M f U#C θ !"& 3 R =&# 0 *( ! &M nB t1$ 9ZM !+, #E D. 0 Adf = θ θ = z dx+ y dy+x dz

0

dx + y dy + z dz ) θ = x+

) θ = xyz dx − x dy + dz

f

x +y +z

) θ =

−

y

+

y z

dx +

x x + z y

)

ω = x dx ∧ dy + x dy ∧ dz + z dz ∧ dx

)

ω = sin(xy) dx ∧ dy + x dy ∧ dz − xy dz ∧ dx

)

ω = z dx ∧ dy + arcsin(x + y) dy ∧ dz +

)

ω=

)

x

dx ∧ dy −

z dz ∧ dx x

x + x dy ∧ dz + y dz ∧ dx z

z

x

y

k ∂/∂z R

x

=0

4= ; $ 5&6 A .$ $ O G#U D ⊆ R !VO .& , + 0 M 5&0 0 &# *P + Q + R = M ( ! &M . T $&U

ω = P dx∧dy+Q dy∧dz+R dz∧dx

z

x

y

−−−−−−−−−−→

−−−−−−→ −−−−−−→ ∂ ∂ ∂ , , div(P, Q, R) := • (P, Q, R) = ∂x ∂y ∂z

f

= 0

f

=

f

)

dx ∧ dy =

dy = gu du + gv dv

∂(x, y) du ∧ dv ∂(u, v)

z = h(u, v, w) y = g(u, v, w) x = f (u, v, w) S AK &F

y sin z dx = xy sin z + A(y, z) 0

Q dy = 0

=

x sin z dy = xy sin z + B(x, z) 0

R dz =

xy cos z dz = xy sin z + C(x, y)

A .8 *( (0&f $, D M f = xy sin z + D .$ nB ( t1$ ω = y dx∧dy +x dy ∧dz −y dz ∧dx 4= ; θ = P dx + Q dy + R dz = R# 0&0 *O&0 ! &M U# pO R# & *dθ = ω M $. $ $ ) Qx − Py = y ,

Ry − Qz = x,

Pz − Rx = −y.

.$ & F !' ( gc) >&\ 6 &0 >_$&U & $ N# M .&V 0 & O. _z +U *(" %& M R# ' "#&0 ]3 *M+ $&# V6 Q ∼= x= ?za *O&0 R = x #&0 R# 0&0 *P = −y/ #&O P = −y R# 0&0 *O&0 R = xy U# R = y y

y

x

θ=−

&F y = g(u, v) x = f (u, v) S AnH )

x dz = xz + C(x, y)

0

P dx =

# θ=

−

y

dx + xy dz

E ( >.&[, M % ) *( B&" % ) N#

fC) f % $

) dx = fu du + fv dv

0 R dz =

0

x

i j −−−−−−→ Curl(P, Q, R) := ∂/∂x ∂/∂y P Q

=

y dy = y + B(x, z)

E ( >.&[, θ !"& 3 ?U0 *( ! &M nB t1$

.& , + 0 M θ = P dx+ Q dy 4= $ 5&6 A M ( ! &M . T .$ $$S G#U D ⊆ R !VO *P = Q 0 M θ = P dx + Q dy + R dz 4= $ 5&6 A; . T .$ $$S G#U D ⊆ R !VO .& , +

0 &# *Q = R P = R P = Q M ( ! &M $&U 5&0 y

f

Q dy =

! ! ! θ = y sin z dx + x sin z dy + xy cos z dz

+ sin(xy) dz ∧ dx

z

=

LB$ v $ e0 C B A M f = xz + y + D .$ *O&0 LB$ $, D t1$ #E (" ! &M θ = y dx + y dy + z dz 4= A, .8 *dθ = −dx ∧ dy (" 4= A .8

ω = sin(xz) dx ∧ dy + sin(zy) dy ∧ dz

y

f

z dx = xz + A(y, z) 0

0

xy dy − dz z

9( ! &M O $ $ 4= ; 4 M M M t\

0 P dx =

0

) θ = (x − yz)dx + (y − xz)dx + (z − xy)dz

=

y

"

+ A(x)

dx + B(y) dy + (xy + C(z)) dz

* LB$ e0 C B A M ! &M ω = y dx ∧ dy + x dy ∧ dz + y dz ∧ dx 4= ; A .8 *(" t1$ R# 0&0 *dω = dx ∧ dy ∧ dz = 9 #E ("

=

dx ∧ dy ∧ dz

(u − v) du ∧ dv + (v − w) dv ∧ dw + (w − u) dw ∧ du

= =

v u

9Z#. $ & = $. .$ v du ∧ dv ∧ dw w

u+w v

u = ln x − ln y M . T .$ A .8 *#.F (0 du ∧ dv X"' 0 . dx ∧ dy Z#. $ J**H ;8*;*H &#&q1 0 ) &0 *!' =

=

du ∧ dv ∂(u, v) ∂(x, y) du ∧ dv /x /y x − y y − x

=

= =

dy = gu du + gv dv + gw dw

)

dz = hu du + hv dv + hw dw

)

dx ∧ dy =

∂(x, y) ∂(x, y) du ∧ dv + dv ∧ dw ∂(u, v) ∂(v, w) +

)

dy ∧ dz =

+

)

dz ∧ dx =

=

xy du ∧ dv (x + y)

∂ (u, v, w) ∂ (x, y, z) v du ∧ dv ∧ dw ÷ u u v du ∧ dv ∧ dw (v − u) (w − u) (w − v) du ∧ dv ∧ dw ÷

w w

)

x

dx

∂(x, y) dw ∧ du ∂(w, u)

∂(y, z) ∂(y, z) du ∧ dv + dv ∧ dw ∂(u, v) ∂(v, w) ∂(y, z) dw ∧ du ∂(w, u)

∂(z, x) ∂(z, x) du ∧ dv + dv ∧ dw ∂(u, v) ∂(v, w) +

v = x + y + z u = x + y + z M . T .$ A .8 du ∧ dv ∧ dw X"' 0 . dx ∧ dy ∧ dz w = x + y + z *M [&

Z#. $ AC (+"1 [O _ &0 *!' dx ∧ dy ∧ dz

)

(u − v + w) (w − u) du ∧ dv ∧ dw

v = x + y − xy

dx ∧ dy

) dx = fu du + fv dv + fw dw

dx ∧ dy ∧ dz =

∂(z, x) dw ∧ du ∂(w, u)

∂(x, y, z) du ∧ dv ∧ dw ∂(u, v, w)

M M x= A& '() & $ vdu + udv Z#. $ >. T R# .$ y = u + v .$ dy = udu + v dv

= uv =

v dx ∧ dy = u

u − u du ∧ dv du ∧ dv = v v

y = uv + vw x = u + v + w M x= A, .8 9Z#. $ & = $. .$ >. T R# .$ z = u + v + w dx

= du + dv + dw

dy

= vdu + (u + w) dv + vdw

dz

=

udu + vdv + wdw

0 * $

[& ("0 Y y = uv x = u − v M x= A *dx ∧ dy dy dx [& ("0 Y y = r sin θ x = r cos θ M x= A; *dx ∧ dy dy dx z = u − v y = u + v x = uv M x= A dz ∧ dx dy ∧ dz dx ∧ dy dz dy dx ("0 Y

*dx ∧ dy ∧ dz ("0 Y z = uv y = ve x = ue M x= A dz ∧ dx dy ∧ dz dx ∧ dy dz dy dx [&

*dx ∧ dy ∧ dz 0 . dx ∧ dy v = sin x/y u = x − y M . T .$ AH *M [& du ∧ dv X"' u

v

9Z#. $ & = ; $. .$ dx ∧ dy

=

= dy ∧ dz

= + =

v u+w + u+w

du ∧ dv dv ∧ dw + v v

dw ∧ du v

(u − v + w) du ∧ dv + (−u + v − w) dv ∧ dw v u+w u v du ∧ dv v u + w v v v w dv ∧ dw+ w u dw ∧ du −u + v − uw du ∧ dv

+(uw − v + w )dv ∧ dw+ (uv − w )dw ∧ du v w u v dv ∧ dw dz ∧ dx = du ∧ dv + w u dw ∧ du +

0 f = cos x f = tan x e0 A& '() $ #E " U0& "0 D = (, π/) df ∧ df =

0 f

dx ∧ (− sin x dx) = cos x

f = x − y e0 A, .8 #E " U0& !\ " D = R − {(x, y)|xy = } = x + y

df ∧ df

=

(x dx − y dy) ∧ (x dx + y dy)

=

xy dx ∧ dy =

e0 A .8 U0& !\ " D = R − {(x, y, z)|x = y &# x = z &# y = z} 0 #E " f = xyz f = xy+xz +yz f = x+y+z

z

M x= A8 *dz ∧ dx dy ∧ dz dx ∧ dy [& ("0 Y

dx ∧ dy & .&[, z = x + y y = vu x = uv AJ *#.F (0 . dz ∧ dx dy ∧ dz M 0&0 B&' .$ dx ∧ dy ∧ dz X"' 0 . dρ ∧ dϕ ∧ dθ A7 *z = ρ sin ϕ y = ρ cos ϕ sin θ x = ρ cos ϕ cos θ B&' .$ dx ∧ dy ∧ dz X"' 0 . da ∧ db ∧ dc A v = a + b + c u = a + b + c M 0&0 v = u = x + y + z ?U0 w = a + b + c *w = x + y + z x + y + z = v

y

= v sinh u x = v cosh u

HS' !"# $+

df ∧ df ∧ df = (dx + dy + dz) ∧ ((y + z) dx + (x + z) dy + (x + y) dz)

= =

∧ (yz dx + xz dy + xy dz) y + z x + z x + y dx ∧ dy ∧ dz yz xz xy (x − y)(y − z)(z − x) dx ∧ dy ∧ dz =

R# *r " !\ " &v y xs M # O .&"0 ?[1 0 u3 R# 0 &3 0 ( U Q 0 &\1$ +) *( E& #E G#U 0 U0 f *** f f M x= $ !J )*E e0 R# Zg S . T .$ *O&0 D , +

E 0 M $$S (=&# G 5 Q v k e0& M " M U0 *G(f (X), f(X), · · · , f (X)) = X ∈ D * O & !J >9E O&[ U0& "0 f = sin x D = R M x= A& '() $ * " U0& "0 D 0 f f >. T R# .$ *f = cos x >. T R# .$ G(u, v) = u + v − $ O x= S #E *( T D 0 G(f, f) f = x − y f = x + y M M x= A, .8 f = (x + y)/(x − y) k

$ $ e0 M M t\ $. .$ 0 $ $ 9 " U0& !\ " , + 4 M 0 O

k

) f = x + y, f = x − y

) f = xy,

f = x/y, f = x − y

) f = x + y + z, ) f = xy z ,

f = x + y + z

f = yz x , f = zx y

) f = x + y + z + u,

f = x + y + z + u , f = x + y + z + u

) f = x + y + z + u ,

f = y + z + u + x , f = z + u + x + y

' & D = (x, y) x = y , x = −y

*H N (+

* " U0& "0 D 0 f f f e0 >. T R# .$ .$ G(u, v, w) = (u + v )/(uv) − w $ O x= S #E *( T D 0 G(f , f, f) >. T R#

$ + G#U 5 [) >_$&U N+M 0 . Ǳ&O + x +y = B$&U 0 T .$ ( VO # $ ?za

*( &q= .$ z = x − y B$&U 0 VO B Bn 5 S *O&0 Ǳ&O S R# UB&Y uL0 R# ` -

U0 f *** f f M x= $ 4E_ pO . T R# .$ *O&0 D , + 0 #n<" #$ ( R# O&0 U0& "0 D 0 e0 R# V# 0 =&M *df ∧ df ∧ · · · ∧ df = X ∈ D E 0 M k

X

X

k

X

.$ >.&[, R# 5$ $ . 1 &0 *y = r sin θ x = r cos θ (=S 9R# 0&0 * ≤ sin θ r = sin θ cos θ M $ O C −−−−−−−−−−−−−−−−−−−→ C : r (t) = sin θ cos θ, sin θ cos θ ;

≤θ≤π

*( . M V C *$ O ) % H*H !VO 0 ( 5 M0 V N# C : x = y A .8 .$ & f f (x, y) = −(−−x,−−−−−→ y) f (x, y) = x − y #E y = x y $ `&+ ) C ?U0 *$ O T (, ) ∈ C *AGB 8*H !VOC O&0 y = −x }

t 6 U0& z = f (x, y) M x= $ $% !( . C . T .$ *C : f (x, y) = c ( #n3 . S . T .$ *f (X) = 0 X ∈ C E 0 M Zg S & $ U E h mT C M Zg S )? $% !( . T .$ . C A VC *O&0 [) Y\ *O&0 . M R E , + #E 5 , 0 M Zg S N# C : x + y = x # $ A& '() $ nB f (x, y) = x + y − x (B&' R# .$ #E (

U 0 f (x, y) = −(−,−→) pO *f (x, y) = −(−−x−−−−−,−−→ y) N# C *$. tU C 0 (, ) ?U0 ( x = y = *$ O ) GB H*H !VO 0 *( . M [)

}

[) N# AGB 98*H !VO V [) N# A% V N# C

g F A .8 f (x, y) = x + y (B&' R# .$ #E *( . M −−−−−−−−−−−−−−−−−−−−→ VF 0 p6 f (x, y) = (/)x , (/)y C E (, ± ) (± , ) Y\ .& Q ]3 *y = x = R# *A% 8*H !VOC M+ T f (X) = 0 pO .$ !VO 0 [Y1 >&j L 0 5$0 &0 .

/

: x/ + y / = /

− /

− /

− −−−−−−−−−→ C : r (t) = cos t, sin t ;

≤ t ≤ π

*$ + .&3 5

}

AGB 9H*H !VO (x + y ) = xy A% ( [) N# C : x + y = y A, .8 5F t 6 .$ f (x, y) = x + y (B&' R# .$−−− #E → *(" . M C ( T GB&L

f (x, y) = (, ) *#0 h .$ . C : x + y = xy , + A .8 f (x, y) = x + y − xy (B&' R# .$ x + y = x

f (x, y) =

−−−−−−−−−−−−−−−−−−− → −−−−−−−−−−− x + xy − y , y x + y − xy

(x + y) = xy U 0 f (X) = 0 X ∈ C pO ]3 *( x + xy − y = y(x + y − xy) = x = M $$S B$&U E &F y = S M $$S 4$ B$&U E y = S & *(, ) ∈ C Z$ . 1 x + y &0 B$&U .$ S *x + y = xy x = &# ]3 *x = xy nB xy = xy Z#.F (0 R# .$ *y = R# 0&0 x = V# &# y = R# 0&0 &# x = x Z#S B$&U .$ y = x= &0 (B&' *M+ T 4 B$&U .$ x = y = & *x = . 10 O = (, ) ∈ C .$ & f (X) = 0 pO ]3 Z 0 *( V [) N# C R# 0&0 *(" N+M 5 A$ O U) J**8 0C [Y1 >&j L E C

[) N# 0 $ +, AGB 9J*H !VO [) $ R0 # E A% N# C : f (x, y) = a M x= $ M X ∈ C E 0 >. T R# .$ *( V

C : xa + yb = q0 $ $. 0 # E A .8 !VO 0 *M [& a < b M . C : xb + ya = *$ O ) % J*H x y x y *f (x, y) = b + a f(x, y) = a + b M x= *!' 9>. T R# .$ C ∩ C

: f (x, y) = f (x, y) =

− y = x : − a b

f(x, y) = B$&U .$,#$&\ R# 5$ $ . 1 &0 *y = ±x ]3 . $ M y = ±x = /a + /b = α Z#S 0 *O&0 (−α, −α) (α, −α) (−α, α) (α, α) % ) .& Q 5&+ *ZM .0 . X = (α, α) Y\ ( =&M 5.&\ !B$ C R0 θ # E ( RO. c % J*H !VO E M . @ 9U# O&0 0 0 f(X ) f(X ) R0 # E &0 X .$ C

θ

= =

=

=

∠ f (X ), f (X ) ⎞ ⎛ − −

−−−−−−→

−−−−−−→ x y x y ⎠ , ∠⎝ , , X a b b a X ⎛ ⎞ −−−−−−→ −−−−−−→

∠ ⎝α , , α , ⎠ a b b a # " # " /a b a b − − cos = cos /a + /b a + b

0 $ $

*0&0 (− , ) Y\ .$ . C : x + xy + y = Ǳ& A N# C M $ 5&6 *C : y = x + M x= A; Y\ .$ C 0 /&+ ]< *( V $ 5 M0

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)

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http://webpages.iust.ac.ir/m_nadjafikhah/r2.htm

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*O&0

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D = [− ; ] × [− ; ] , D = [; ] × [− ; ] D = [− ; ] × [; ] ,

D = [; ] × [; ]

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ij

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00

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D

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+

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m = m = m = m m = m = m = f (, ) = √ m = m = m = f (, ) =

= m

=

E AJC A;C (+"1 0 ) &0 ` + .$ R# 0&0 Z#. $ **8 q1 00

00 f (x, y) dA = D

≤

00 √

00

= ≈

+···+

+ D

D

dA +

D

=

00

00 √

00

dA +

D

√

D

D :

≤ x ≤ , ≤ y ≤

0 5 . D >. T R# .$ f (x, y) = [x + y] ?U0 9A$ O ) GB *8 !VO 0C $ + Z"\ #E >. T D = D ∪ D ∪ D , D :

≤y≤

,

D :

≤y≤

,

D :

≤y≤

,

≤x≤

− y,

− y ≤ x ≤ − y,

− y ≤ x ≤ .

T 0 0 5F c0 D # cB Zg&1 da !M 0 f (x, y) e0& M Z **8 q1 E A8C (+"1 0 h R# 0&0 ( !M 0 f (x, y) e0& *( 0 0 D !M 0 M ZM x= ( 0 0 u . (+ e- c0 D `?-_ E

e0& *O&0 0 0 D !M 0 M $ + x= 5 R# 0&0 0 0 (, ) Y\ c0 D # cB Zg&1 da !M 0 f (x, y) $ 0 0 D !M 0 M $ + x= 5 R# 0&0 ( $ **8 q1 E () () (+"1 0 ) &0 ]3 *O&0

Z#. $}

dA

D

√

Area(D ) + Area(D ) + Area(D ) √ √ √ √ π π + − + + + −

/

Z#. $ 0&6 >. T 0 00

00 f (x, y) dA = D

≥

00

D

=

* ≤

√

00 +

D

00 +···+

D

dA +

00

D

D

√

× Area(D ) = (

00 , D

00 √

00

D

D

dA +

− x − y dA ≤

dA

H**8 &a E ; & +"1 9*8 !VO 00

[x + y] dA =

00

[x + y] dA + 8 9: ; D

*( 5F U1 . \ M Z $ [B

D

00

)≈

` + .$ ]3

[x + y] dA 8 9: ;

+ =

+

=

×

D

[x + y] dA 8 9: ;

× Area(D ) + × Area(D ) +×

=

. T .$ . R E D , + #E % >. j0 F M Zg S `g−x D : a ≤ x ≤ b , h(x) ≤ y ≤ (x)

RQ *$ O ) GB *8 !VO 0 *(O 5 0 E y $ 0 |Q ( . m@ $ E #&, +

$ x X"' 0 e0& $ . $ + 0 Rg&3 _&0 E . −y E&0 .$ x M $. $ tU D 0 . T .$ (x, y) Y\ * O

*[h(x); (x)] E&0 .$ y $S . 1 [a; b] }

.

00 f (x, y) dA D

. \ $. .$

) f (x, y) = [x − y]

)

D :

0 % %

9M [&

≤x≤

, ≤ y ≤

D : x + y ≤ , ≤ x

f (x, y) =

f (x, y) = [x ] ) f (x, y) = −

)

D :

≤ x ≤ , ≤ y ≤

|x| + |y| ≤ < |x| + |y| ≤

S S

D : |x| + |y| ≤

0 Rg&3 ' N# _&0 ' N# #E $. E 0N# 0 .$ 9#.F (0 f (x, y) dA D

) f = x y

, ) f = x + y

Zh −x , + AGB 9*8 !VO Zh −x , + N# $' RU A% ( Zh −x &# . M "0 , + % ( R O !0&1 Zh −x &, + E ,&+ ) !VO 0 &# 0 0 &, + R# E & $ w O ('&" M S 0 9( T < D= D ∀ i = j : Area(D ∩ D ) = i

i

j

i

$ O (0&f &F .$ $$S E&0 W B 3 0 ZV' R# >&[f & aa E ,&+ ) >. T 0 . . M "0 , + M >&[f 0 M cQ & *(O 5 yLB `?- &0 *O&0 LB$ da N# 0 q1 ZV' 5$ $ 5&6 &

6=M)

<

+ J

U8

D :

,− ≤ y ≤

D : x + y ≤

)

f = sin(x + y)

D :

≤ x ≤ π/, ≤ y ≤ π/

)

f = x − y

D :

≤x≤

, ≤ y ≤

−x

( (i − ) ≤ x + y ≤ i \' D M x= A O G#U D : x + y ≤ R , + 0 z = f (x, y) !1 ' 0 0 X 0 M m S M $ 5&6 *( >. T R# .$ O&0 D 0 f (x, y) . \ aM ' i

i

[R]+

/ i=

00y D

i

i

00

(i− )mi ≤

f (x, y) dA ≤ D

[R]+

/ i=

(i− )Mi

p Y 0 $ da Z"\ &0 AI [#\ . \ (+"1 .& Q 0 x + y = *M [& . , + R# 0

= x = √ x + y dA

\O*' YG>X' 0

K+ %

*( O $ $ D Zh −x , + M x= /lU)−x x -= Y\ .$ . . R# M . −x 0 $ +, Y 0 (+ 0 |Q (+ , E =S h .$ M eY1 , + &0 y R# $. 0 RB .$ *Z$ (M' ( . !T&' x = b Y\ D$. 0 R#F .$ x = a Y\ D w y 5 M *A$ O ) % *8 !VO 0C *$$S

ZM x= U# Z$ . 1 & (B&' N# .$ . . Mn

0 y R# Rg&3 , E . (x, y) Y\ a ≤ x ≤ b D , + &0 Y\ R# $. 0 RB *Z$ (M' _&0

≤x≤

[& M Z. R# 0 ;**8 &a 0 ) &0 4E_ ]3 ( . O$ .&M G#U E $& &0 &S$ .&h R# *$ O ` 0 Z\ " b O. $. R# .$ ( &) 0 M Z#.F $&# 0 ) E M $S > 1 1 M G#U . 1 f (x) dx M l3 ` + ' [&

R q1 0&0C F = f M Z =&# . F (x) e0& *( &0 .&h R# *Z#. F (b) − F (a) . \ 0 Ac [#_ *$ O KY X& G#U N# 0 .$ M ( 4E_ & b a

w y %&L &0 =@ E *b = a = R# 0&0 $. 0 Y\ R#Rg&3 M ZM &6 x Y\ E .nS 5F R#_&0+ ( (x, − x) E >.&[, D+, + !&0 y R# (x) = − x R# 0&0 *O&0 x, − x E >.&[, B$&U !' E y =+ − x M ( M{ 0 4E_ *h(x) = − x B$&U !' E y = − x ( O !T&' x+y = y R# 0&0 *( F (0 ≤ y 0 ) &0 x + y = D :

+

≤ x ≤ , − x ≤ y ≤ − x

}

Y\ 5F $. 0 Y\ R#F (x, h(x)) E >.&[, h(x) e0& $. 0 RB o L R $ R# 0&0 *$ 0 *$E& oL6 . (x) e0& $. 0 R#F o L R $ 0 $ , + D ZM x= A& '() % >. T R# .$ *O&0 y = x y = x = x = p Y H*8 !VO 0C ( Zh −x , + N# D M $ 5&6 *A$ O ) GB M $ O h'? >. T R# .$ *!' (x, (x))

D :

≤ x ≤ ,

≤y≤x

M $$S &6 *$ O ) x E OnS -= y 0 *O&0 (x, x) _&0 Y\ (x, ) R3&3 Y\ >&j L

} *;*8 &a E & +"1 98*8 !VO y = x p Y 0 $ , + D M x= A .8 D M $ 5&6 >. T R# .$ *O&0 y = y = −x *( Zh −x , +

( , ) Y\ 0 y = y = x y $ $. 0 E *!' y = −x y $ $. 0 E b = R# 0&0 Z.

0 *a = − R# 0&0 Z. (− , ) Y\ 0 y = .nS w y N# S ?U0 *$ O ) % 8*8 !VO &# ≤ x ≤ V# 0 "0 &F Z#0 h .$ x Y\ E (B&' .$ * F u3 (U- $ − ≤ x ≤ (B&' .$ (x, ) (x, x) E .&[, X 0 _&0 Rg&3 p&\ (x) = ` + .$ ]3 *(x, ) (x, −x) E .&[, 4$ − ≤ x ≤ S .$C = !#_$ 0 * h(x) = −x x ≤ x ≤ S ZM Z"\ uL0 $ 0 . D ( 0 A&S$ [&

9O&0 O $ Y0&- N# & 4 M .$ h(x) M 0

*;*8 &a E ; & +"1 9H*8 !VO y = x + 0 $ , + D M x= A, .8 , + D M $ 5&6 >. T R# .$ O&0 y = y *A$ O ) % H*8 !VO 0C ( Zh −x &3 0 y = y = x E !T&' & $ !' &0 *!' a = − Z#. $ .$ *Z. ry = x = ±s _&0 0 Rg&3 E (M' -= Y N# %&L &0 *( b = (x, x ) E >.&[, D , + Y\ R#Rg&3 M ZM &6

h(x) = x R# 0&0 *(x, ) E (".&[, $. 0 Y\ R#_&0 Z#. $ ` + .$ ]3 *(x) =

D = D ∪ D

D : − ≤ x ≤ , x ≤ y ≤

D : − ≤ x ≤ , −x ≤ y ≤ D :

≤x≤

,x≤y≤

4 ).$ e0& 0 $ , + D M x= A .8 , + D >. T R# .$ *O&0 . −x y = x − x & $ !' &0 &# .$ *$ O ) J*8 !VO 0 *( Zh −x x 0 − #$&\ 0 O $ $ B$&U $ E !T&' .$ M . Y&+ *O&0 b = a = − R# 0&0 Z.

R#Rg&3 &F − ≤ x ≤ S $$S &6 J*8 !VO

O&0 T e0. E +"1 D , + M x= A .8 *( O $ x + y = y x + y = # $ 0 M 0C ( Zh −x , + D M $ 5&6 >. T R# .$ *A$ O ) GB 8*8 !VO 9Z#. $ O $ $ B$&U $ E !T&' & $ !' &0 *!'

x + y = ⇒ x+y =

x − x = y =−x

⇒

x = , y =−x

*x + = y y x = y + 0 $ , + AI x+y =

y x + y = # $ 0 $ , + A *$. $ 0 .$ . (, ) Y\ M

E . −x y = | sin x| e0& . $ + 0 $ , + A; *x = π & x = −π x = y + $ 0 $ , + A * e0. .$ e1

x = y +

x + y = x

x + y = # $ 0 $ , + A *4.& Q e0. .$ e1

p Y 0 $ , + AH & B&' 0 *a < b α < β & F .$ M x + y = b *$ O ) a α /& 0 B&" G L

∗

x + y = a y = βx y = αx

) x/ + y/ ≤

)

X 0 D , + &0 w $. 0 ! 0 Y\ R#_&0 p&\ R# ≤ x ≤ S (x, x − x) (x, ) E .&[, .$ *$ 0 (x, ) (x, x − x) E >.&[, − ≤ x ≤ S h(x) = x − x ≤ x ≤ S x − x − ≤ x ≤ S (x) = ≤ x ≤ S . @ D D uL0 $ 0 . D ( 0 = !B$ 0 Z E&0 9O&0 Y0&- N & F .$ h e0 M ZM Z"\ D = D ∪ D D : − ≤ x ≤ , D :

≤ y ≤ x − x

≤ x ≤ , x − x ≤ y ≤

}

|x| + |y| ≤

) ≤ x + y ≤ ) x ≤ x + y ≤ x , ≤ y ) (x + y) + (x − y) ≤

) (x + y ) ≤ x − y

O&0 T D D w O ('&" S % % *ZM $& D + D $&+ E D ∪ D &0 /lU)

−x

@? 2 6=M) /D1 K+ & %

Z L0 O&0 .& .$ D . M "0 , + ZM x= '& ('&" M ZM Z"\ . @ Zh −x >&UY1 0 F . , + 0 *O&0 T & F E & $ R0 w O $.

. 5F M . −x 0 $ +, y N# ]< $ + Z , E . y R# *ZM Z M eY1 x .$ . 7*8 !VO 0 Z$ (M' ( . (+ 0 |Q (+ &0 w y R# w O ! UB&Y m *$ O ) GB $ U x %&L E RV+ (U- .$ *O&0 D , +

*Z#.&+O . D , + &0 y $. 0 E !T&' & Y .&3 . −x 0 . M v &B $ U & F .$ M #& &V

. −y > E 0 @ Y !T&' p&\ E *ZM oL6

E N# X R# 0 *[0 . D , + & ZM Z y &0 $. 0 .$ U# *$ 0 Zh −x !T&' >&UY1 *$ y .&3 N# & w

*;*8 &a E H (+"1 9J*8 !VO #E &, + E N# M $ 5&6 0 $ % $ O U *M oL6 . & F $' ]< " Zh −x *O&0 Y0&- N h e0 *x = −y x = y x = − p Y 0 $ , + A &. x + y = y 0 $ , + A; *&j L

*x + y = # $ 0 $ , + A *x = y y = x + $ 0 $ , + A *y = x + y = x − y 0 $ , + AH x + y = y # $ x + y = y 0 $ , + A8 *( (, ) Y\ ! &O M # $ y = −x y = x p Y 0 $ , + AJ *( (, ) Y\ ! &O M x + y = *x = y x = y + 0 $ , + A7 y = + x x + y = p Y 0 $ , + A *. −x

}

D = D + D + D √ √ D : ≤ x ≤ , − x/ ≤ y ≤ x/ + √ D : ≤ x ≤ , x − ≤ y ≤ x/ + √ D : ≤ x ≤ , − x/ ≤ y ≤ − x −

x + y = x # $ 0 $ '& D ZM x= A, .8 − R0 -= y &3 S >. T R# .$ *( x +y = D , + y y E O ) &y .&3 $ U O&0 R# 0&0 *( 0 0 $ U R# O&0 R0 S N# 0 0 R# 0 *Z$ D0 x = y y . D , + & ( 4E_ ) % *8 !VO 0C $$S Z"\ uL0 0 D X (O 5 ` + .$ A$ O D = D + D + D

+ + D : − ≤ x ≤ , − − x ≤ y ≤ − x + + D : ≤ x ≤ , x − x ≤ y ≤ − x + + D : ≤ x ≤ , − − x ≤ y ≤ − x − x

+ x = − y 0 $ , + D ZM x= A .8 R# .$ *( N# `&UO Ǳ [ cM 0 # $ x = − y 0 |Q (+ , E . x #&3 &0 w y S >. T $ U ( − R0 x M $& *Z$ (M' ( . (+ $ U R# O&0 R0 x S *( $ O ) & Y .&3 ( N# 0 0 $ U R# O&0 R0 x S ( N# 0 0 x = − p Y y . D , + M ( 4E_ .$ 9$$S Z"\ Zh −x UY1 .& Q 0 D & Z$ D0 x = D = D + D + D + D + + D : − ≤ x ≤ − , − − x ≤ y ≤ − x + + D : − ≤ x ≤ , − x ≤ y ≤ − x + + − x D : − ≤ x ≤ , − − x ≤ y ≤ + + D : ≤ x ≤, − x ≤ y ≤ − x

Zh −x >&UY1 # , + Z"\ 97*8 !VO .&3 $ U b & a Y\ E GB 7*8 !VO .$ &a 5 , 0 N# 0 0 w y y D , + E O ) & Y $ U R# d & c Y\ 0 5.& $.n0 b E x S & *( *( N# 0 0 f & e E 0 0 e & d Y\ E *( N# 0 0 .nS . −y E p Y Z E ]3 X R# 0 Z. VQ M , + (6 0 f e d c ba p&\ E >&UY1 $ U 5O +M !B$ 0 *$ O ) % 7*8 !VO 0 , + N# 0 $ + t 5 . &, +

*$ $ !\ ( 0 . >&UY1 $ U . ' Zh −x + $ 0 $ D , + A& '() * % > E 0 + $ R# *#0 h . x = y x = y + #E *M eY1 (, − ) (, ) .$ . #V# . −x

x = y x = y +

⇒

y = x = y +

⇒

y=± x=

ZM ) . _&0 . S *A$ O ) GB *8 !VO 0C O&0 R0 w y &3 S M $M Z &6

N# 0 0 D , + y 5F E O ) & Y .&3 $ U 4E_ R# 0&0 *( $ 0 0 $ U R# O&0 .& Q R0 xS 0 D X R# 0 *$ $ D0 x = y y . !VO M ( 9$$S Z"\ Zh −x UY1 }

$ !-& !VO 0 . $ R# A$ O ) GB I*8 !VO 0C U0 &$& .$ .&M R# *(O 5 Zh −x , +

9$ 0 A&S$ 0 p 0 C D = Da − Db + + Da : − ≤ x ≤ , − − x ≤ y ≤ − x + + − x ≤ y ≤ − x Db : − ≤ x ≤ , −

7*;*8 &a E ; & +"1 9*8 !VO

}

y !0&\ .$ x %&L / + % Zh −y , + E 5 ' . 0 R# 0&0 $. #c

9(S RL D : a ≤ y ≤ b , h(y) ≤ x ≤ (y)

*O&0 ^T c $. R# .$ ;*;*8 q1 .$ Zh −u , + N# E 5 M >. T 0 9(S RL T−uv E (+"1 E D , + A& '() % #E ( Zh −y , + R# *#0 h .$ . *;*8 &a

(O 5

D :

≤y≤

, −y ≤ x ≤ y

0 , + R# M $ O ) A*$ O ) % I*8 !VO 0C *$ 0 O O Zh −x , + $ E ,&+ ) >. T 7*;*8 &a E AC (+"1 .$ , + D M x= A, .8 5 #E *( Zh −y D , + >. T R# .$ *O&0 9(O D : − ≤y≤

, y ≤ x ≤ y +

Zh −x , + E ,&+ ) >. T 0 D ?z[1 M $ O ) *A$ O ) GB *8 !VO 0C $ 0 O O }

A% 7*;*8 &a E (+"1 AGB 9I*8 !VO Zh −y , + N# $' RU >&UY1 0 . D , + $. .$ 0 % N & F (x) h(x) e0 M M Z"\ . @ Zh −x 9O&0 Y0& y = x y = y = − p Y 0 $ , + A *y = −x *x + y = x + y = x # $ 0 $ , + A; !Y " x / + y/ = q0 0 $ , + A *q0 5F 0 y

x

+

y

=

q0 $ 0 $ , + A *x + y =

M x = y y = x + $ 0 $ , + AH * ≤ x ≤ M y = − x y = x + $ 0 $ , + A8 * ≤ x ≤ *;*8 &a E ; & +"1 9*8 !VO y = x + $ 0 $ '& D M x= A .8 Z ( Zh −x Z D >. T R# .$ *( x = y .&\ !B$ 0 e1 .$ *$ O ) % *8 !VO 0 Zh −y 9Z#. $ $. $ $ ) y x R0 M D

:

D

:

≤x≤ ≤y≤

√ , x ≤ y ≤ x √ , y ≤ x ≤ y

B$&U 0 # $ 0 $ '& D ZM x= A .8 >. T R# .$ *O&0 x + y / = q0 x + y = 9(O Zh D , + $ E ,&+ ) >. T 0 5 . D A$ O ) GB ;*8 !VO 0C

*(, ) (, ) (

(, −)

, ) (, )

(, ) (−, ) g. &0 U- .& Q A7 *(, )

) x ≤ x + y ≤ x ) ) x + y ≤

y=

xy

/ g. &0 U- .& Q AJ

)

≤ |x| + |y| ≤ ≤ x + y ≤ x

0 $ , + A *. −x ≤ t ≤ π M

− cos t x = t − sin t

=

xy

=

&B Bn 0 $ , + A *y = x y = x p Y

]' !F $0 B &" !' 0 0 5& q1 M ( . 5F 5& E 0 = $ S #&B& # 5 -&#. . Z R# *ZM KY &S$ *( &. 4& 0 9( Zh −x , + D M x= %

D = D + D

,

,

D : − ≤ y ≤ , − − y / , , D : − ≤ y ≤ , − y / ≤ x ≤ − y

− y ≤ x ≤ −

}

D : a ≤ x ≤ b , h(x) ≤ y ≤ (x)

>. T R# .$ *( 3 D 0 z = f (x, y) e0& 0

00

b

(0

f (x, y)dA =

f (x, y)dy dx a

D

)

(x)

h(x)

$ 0 G+ OVBC EJ B=B] % q1 N+M 0 ]< $ + Z"\ Zh −x ' 0 . Z#S & F E N# 0 O $ $ e0& E 0 = X R# 0 *ZE e+) . !T&' $ , 4& * & B +U () Q &# N# [& 0 h $.

0 $ , + D ZM x= A& '() % e0& . \ *O&0 x = y = −x y = x p Y *M [& D 0 . f (x, y) = xy 0C $ + oL6 . D , + $' #&0 0 *!' M $$S h'? *A$ O ) % ;*8 !VO D :

≤ x ≤ , −x ≤ y ≤ x

xy dA =

0 0

x

xy dy dx =

0 ( )y=x xy

−x

D

0 =

(

x dx =

x

) =

>. T 0 . #E &, + E N# 0 % 9"# 0 Zh −y &, +

*x = y x = y + 0 $ &, + A E B x + y = q0 0 $ , + A; *|x| + |y| = *( , ) (, ) (, ) / g. &0 da A (, ) (, ) ( , ) (, ) / g. &0 U-.& Q A E B x + y = g F 0 $ , + AH *|x| + |y| = Y0 . .$ $&T (x, y) = x+yi ∈ C / g. &0 U- uO A8 *(x, y) = /

Z#. $ 0 = q1 /& 0 R# 0&0 00

*;*8 &a E (+"1 AGB 9;*8 !VO **8 &a E (+"1 A%

dx

y=−x

}

/

)

|x| + |y| ≤

)

|x + y| + |x − y| ≤

)

x/ + y / ≤

)

≤ x + y ≤ x

) (x + y + ) + (y + x + ) ≤

e0& . $ + 0 $ , + A; *. −x − ≤ x ≤ M (x + y) = a(x − y) 5 3&3 0 $ , + A * < a E&0 0 / "M / e0 . $ + 0 $ , + A *[; π] E&0 0 y

= arcsin x

∗

**8 &a E ; & +"1 9*8 !VO

9$ O !' #&0 $ ]3 (

00

00

x dA + x −

f dA = D

D

(0

0

x − x

)

00

D

−x

dA x − 0 (0

−x dy dx + = − x −x x − x − x 0 0 xy dy −xy dy = dx + dx − x − − x − x −x ( ) 0 0 0 x = x dx + x dx = x dx = = x dy x −

−

−

)

−

dx

e0& ( x + y ≤ x i1 D ZM x= A, .8 *M [& D 0 . f (x, y) = π+sin(πx/) x − x X"' 0 . x + y = x B$&U + 0 . h R# 0 *!' #E 5$ 0 ([a pO M y = ± x − x 9ZM !' y R# 0&0 *O ≤ x ≤ U 0 &V#$ . D :

00 D

⎡ √ ⎤ 0 0 x−x π sin(πx/ ) dy ⎣ √ ⎦ dx + f dA = − x−x x − x √ )y= x−x 0 ( πy sin(πx/) + = dx √ x − x =

+

+

D :

≤ x ≤ , x − x ≤ y ≤ − x

D :

≤ x ≤ , − − x ≤ y ≤ − x − x

+

00

00 f dA = D

+

>. T R# .$

00 −xy dA

xy dA + D

D

⎡ √ ⎤ 0 0 −x ⎣ √ = xy dy ⎦ dx x−x

⎡ ⎤ √ 0 0 − x−x ⎣ √ + −xy dy ⎦ dx

0 ( y x =

=

0

√ ) −x √

−

−x

√

0 ( )− x−x y dx + dx x √

x−x

! x − x dx =

−

(

x x −

)

−x

=

(, ) (, ) .$ / g. &0 aa D M x= A5 .8 [& D 0 f (x, y) = x yy−+ e0& *( ( , )

+

Z#. $ ` + .$ ]3 *$ O ) GB *8 !VO 0

}

**8 &a E J 8 & +"1 9*8 !VO x + y = x # $ 0 $ '& D M x= A4 .8 0 f *f (x, y) = x|y| O&0 . −y x + y = *M [& . D (" Zh −x , + D *ZM UB&Y . D 0 *!' GB *8 !VO 0 ( Zh −x uL0 $ 0 !#[ !0&1 & M D = D ∪ D *$ O )

+

≤ x ≤ , − x − x ≤ y ≤ x − x

π

0

y=−

π sin x dx = −

x−x

π cos x =

y = x + $ 0 $ , + D ZM x= A .8 *M [& . D 0 f *f (x, y) = xy ( x = y Z#. $ *;*8 &a E AC (+"1 0 ) &0 0 *!' D :

0 (0

00 f dA = D

≤x≤ √

x

x

√ , x ≤ y ≤ x

R# 0&0

)

xy dy

dx =

0 $ %y=√x xy dx y=x

( ) 0 x x x − x dx = − = =

M O&0 T−xy e0. E +"1 D M x= A .8 *f (x, y) = x ( O $ y = − x + y *M [& . D 0 f D $ 0 D 0 f (x, y) [& 0 *!' R# .$ A$ O ) % *8 !VO 0C *ZM #&&O . R# 0&0 D : ≤ x ≤ , ≤ y ≤ − x >. T 0 (0 −x

00 f dA =

0 $ %−x xy x dy dx = dx

D

0 =

x − x

)

(

dx =

x −

x

)

=

E AHC (+"1 .$ O G#U '& D M x= A .8 . D 0 f *f (x, y) = x|x|− O&0 *;*8 &a

*M [&

O Z"\ Zh −x , + $ 0 D 5 Q >. T R# .$ *!'

}

*M *Z0&# . da e- B$&U 0 . h R# 0 *!' e0. E&"+ E ( >.&[, (, ) (, ) p&\ E .nS y ( , ) (, ) p&\ E .nS y *y = x U# 4 ( , ) (, ) p&\ E .nS y *y = E ( >.&[, R# .$ *y = x − R# 0&0 x −− = y −− E (".&[, ) % *8 !VO 0C ( Zh −y , + D >. T (O 5 #E A$ O

**8 &a E 7 & +"1 9H*8 !VO 0 (0 /−y

00 f dA = D

/

0 ( =

/

/y

xy dx y − y −

x y y − y −

0 $ =

/

dy = −

y ≤ x ≤ y/ +

R# 0&0

dy 0 (0

00

dy

%

y − y +

≤y≤;

)

)x=/−y x=/y

D :

y/+

f dA =

y

D

y = ln x . $ + 0 $ , + D M x= A2 .8 *f (x, y) = xy /(e − e) O&0 x = e Y . −x *M [& . D 0 f }

=

0 ( x

y + y−

y

=

(y + )

)

y + dx x y−

dy

)x=y/+

0 dy =

x=y

(y + ) dy

=

B Bn 0 $ , + D M . T .$ A0 .8 . y −xyy − dA O&0 x + y = / y *M [&

$ $ y B Bn D , + $' 5O RO. 0 *!' 9Z$ $. 0 . O xy = 00

D

**8 &a E I & +"1 98*8 !VO Z ( Zh −x Z D , + >. T R# .$ _> e1 .$ *$ O ) % H*8 !VO 0 Zh −y D :

≤x≤e,

≤ y ≤ ln x

⎧ 0 (0 e ⎪ ⎪ ⎪ ⎪ 00 ⎨ f dA = 0 (0 ⎪ ⎪ ⎪ D ⎪ ⎩

:

≤y≤

, ey ≤ x ≤ e

(O 5 R# 0&0 ln x

xy ey − e

e

ey

)

dy dx )

xy dx dy ey − e

( ) ()

U# $. $ !' 0 A;C AC ( !' !0&1 b AC M 0 (

=

x y ! ey − e

⎧ ⎪ ⎨

⎧ ⎨ xy = ⎩

x − x = ⇔ ⎪ x+y = ⎩ y = −x √ ⎧ ± − ⎪ ⎨ x= ⇔ ⎪ ⎩ y = −x

⇔

⎧ ⎪ ⎨ x = , ⎪ ⎩ y=

−x

(, /) ( /, ) E .&[, $ 5F $. 0 p&\ R# 0&0 ( Zh −x , + R# *$ O ) GB H*8 !VO 0 9(O 5 #E D :

/ ≤ y ≤ , /y ≤ x ≤ / − y

)x=e

dy x=ey

.$

}

0 =

−

(

y

dy = −

y

)

=−

y = x p Y 0 $ , + D M x= A&^ .8 . D 0 f *f (x, y) = e ( x = y = *M [&

>. T R# .$ *!' x

**8 &a E ; & +"1 9J*8 !VO E ]3 .

) 0 (0 −x xey dy dx −y

A&, .8 *M !' $' X # U @ Y y = − x + 0 D &# .$ M ( RO. *!' 9( O $ x = x = y = D :

+

≤y ≤, ≤x≤ −y

>. T R# .$ *$ O ) GB J*8 !VO 0

) 0 (0 −x xey dy dx = −y ) 00 0 (0 √−y xey xey = dA = dx dy −y −y D √ ) −y 0 ( y 0 x e dx = dy = ey dy ( − y) = ey = e −

, + 0 . f (x, y) = [y − x ] e0& A& .8 *M [& x ≤ y ≤ 9$M UB&Y #&0 . ^T Ǳc) ! $ >.&[, *!' [y − x ] = n

⇔ n ≤ y − x < n + ⇔ x + n ≤ y < x + n +

0 ) &0 *O&0 LB$ ^T $, n M Z"\ (+"1 .& Q 0 . D , + (x, y) ∈ D (#$

9A$ O ) % J*8 !VO 0C ZM

D = D ∪ D ∪ D ∪ D D : x ≤ y < x + D : x +

,y≤

≤ y < x + , y ≤

D :

≤x≤

≤y≤x

:

≤y≤

, y≤x≤

) GB 8*8 !VO 0 *Zh −y Z ( Zh −x Z 9(O 5 t#@ $ 0 R# 0&0 *$ O ⎧ 0 0 x ⎪ x ⎪ ⎪ e dy dx ⎪ ⎪ 00 ⎨ f dA = ) 0 (0 ⎪ ⎪ ⎪ x D ⎪ ⎪ e dx dy ⎩

( ) ()

y

!' $ O . AC M B&' .$ AC ( !' !0&1 b A;C & 9$ + 0 $ 0 %y=x x ye dx = xex dx

=

= 0

(0

ex

√ π

y=

=

(e − )

) sin(x ) dx dy x

O $ $ A&& .8 . \ $' X # U &0 (" !' !0&1 & ( *#.F (0 . !VO 0 F $.F (0 . D $ ZM U ]3 *!' p Y x = √y + 0 D M ( RO. Z"# 0 Zh √−x % 8*8 !VO 0 ( O $ y = π y = x = π *ZM "# E&0 Zh x >. T 0 . , + R# *$ O ) (+"1 ]3 M+ T " y# O .$ (, π) Y\ 5 Q √ *D : ≤ x ≤ π , ≤ y ≤ x 9O&0 [1 $. g&3 Z#. $ .$ π

√ y

0

) sin(x ) dx dy = √ x y 00 0 √π (0 sin(x ) = dA = x

π

(0

√ π

D

0

√

π

=

D : x + ≤ y < x + , y ≤ D : x + ≤ y < x + , y ≤

,

=

−

(

sin(x ) y x

x

)y=x

0 dx =

y=

√ π cos(x ) =

) sin(x ) dy dx x √ π

x sin(x ) dx

=

R

0 πR

( − cos t) dx

}

00

>. T R# .$ y − x ] dA =

D

00 D

R

=

0 π

( − cos t) R( − cos t)dt

0 R π

=

= −

R

cos t + cos t 0 π

=

R# 0 *( cB **8 q1 .$ e0& 3 pO 1& 0 ( RV+ O&[ . 10 pO R# l&Q M U

*M ) #E &a 0 *Z0 M x= '() % ⎧ ⎪ /y < x < y < S ⎪ ⎨ − /x . T R# b .$ ⎪ ⎩ /y <x
0 (0

) f (x, y) dy dx =

$ 5&6 )

0 (0

f (x, y) dx dy

Z#. $ f Y0&- 0 ) &0 *!' 0

0 f (x, y) dy = 0

0

x

f (x, y) dy + 0

f (x, y) dy

&0 ( 0 0

√

= −

)

(0

√

x +

0

= −

√

− x

= =

x −

0

− √

dx + −

√

0 $ +

(

√

x

)√

−

( +

−√ √ √ + +

x −

x

R# 0&0

)

(0

0 (0

√

0

−

[y − x ] dA

D

√

dy dx + +

x +

dy dx )

dy dx

x +

− x

dx

% − x dx )√ −

(

√

+ x−

x

)

! &M UY1 N# 0 $ '& D M . T .$ A& .8 0C C : x = R(t − sin t) , y = R( − cos t) g V E . f (x, y) f (x, y) = y O&0 A$ O ) 7*8 !VO *M [& D 0 ( >.&[, . −x &0 C $. 0 U# y = pO *!' M ( U R# 0 C E ! &M UY1 N# ]3 *cos t = E >. T 0 . D , + * ≤ t ≤ π D :

≤ x ≤ πR , ≤ y ≤ R(

− cos t)

R# 0&0 *$ $ ^- 5

x

− dy + dy x x y y=x y= −y − = + =− y y=x x y= ) 0 0 (0 f (x, y) dy dx = −dx = − =

x

D

, x + ≤ y ≤ 00

D : − ≤ x ≤

0

dA

√ √ D ∪ D ∪ D : − ≤ x ≤ , x + ≤ y ≤ √ √ D ∪ D : − ≤ x ≤ , x + ≤ y ≤

dt + cos t

D

00

dA +

D ∪D

dA

+ D D ∪ D D ∪ D ∪ D &, + & #E " Zh −x

+ dt π + R t + sin(t) = πR +

D ∪D ∪D

00

dA +

D

00

dA +

− cos t + cos t − cos t dt

0 R π

00 dA +

D

00

=

**8 &a E (+"1 97*8 !VO ]3 ZM $& x = R(t − sin t) v v E 5 M π R# 0&0 xt πR dx = R( − cos t)dt

00

dA +

=

0 πR (0

00 ydA = D

=

R(−cos t)

) ydy dx

0 πR ( )y=R(−cos t) y

dx

y = a p Y 0 $ \Y D M . T .$ AJ e0& O&0 y = x y = x + a y = a *M [& D 0 . f (x, y) = x + y

M B&' .$ f (x, y) dx =

y = x

0 aa D M M [& . T .$ . xy dA A *O&0 (, ) ( , ) (, ) / g. D

5F .$ M M [& . T .$ .

00

xy dA

D

A;

− x , y ≥ x , x ≥

O&0 ( , ) ( , ) (, ) +/ g. &0 aa D S A;; *M [& D 0 . f (x, y) = xy − y e0& 00

#_&0 + D M M [& @60 . xy dA . \ A; *O&0 (x − ) + y = # $ 0 $ '& &. 0 /&+ a `&UO 0 # $ D M . T .$ A; &0 f (x, y) e0& O&0 e0. .$+e1 &j L

*M [& D 0 . / a − x Y0&D

0 $ !Y " 0 . f (x, y) = x y e0& A;H M [& y = b y = x = a x = p Y * LB$ \\' $ , a b U[@ $ , n m M n m

. !T&' B X # U E ]3 $. .$ 9M [&

) ) ) ) )

x

0

π

(0

0 (0

y dy dx

sin x

√ y

y

0 (0 √

0 (0 y

⎡ ⎤ 0 0 √−y ⎣ ) x dx⎦ dy

)

0 (0 x

y /

) xy dy dx )

x dx y ln x

) dy

) sin(x ) dx dy

√

)

sin(πy ) dy dx

x

0 (0 e

y

x

) dx dy

y

0 (0

)

√ y

f (x, y) dx y

y

+"1 D f (x, y) = y/(x + y) M . T .$ A;I y = p Y R0 M O&0 x + y ≤ i1 E [& D 0 . f ( O $ y = / *M D : y≤

f (x, y) dx +

=

00

0

y

0 − dx + dx y x x x=y x= x = + = x x=y y x= ) 0 0 (0 f (x, y) dx dy = dx = 0

+ $ 0 $ D M . T .$ A7 *M [& D 0 . f (x, y) = x y e0& O&0

x = y

0

0

) x exy dx dy

[& . #E .V & B E N# ) 0 (0 x dy ) dx y + 0 0

)

dy ) dx (x + y) ⎡ √ ⎤ 0 0 −x ) ⎣ xy dy ⎦dx x

⎡ ⎤ 0 0 √ −y ⎣ √ ) x dx⎦dy −

)

−

0 (0

−y

)

0 $ %

0 π 0

9M

a

rdr dθ sin θ

0 (0

)

[x + y] dy dx x

) 0 (0 −x ) (x + y) dy dx ⎡ ⎤ 0 0 √−y

) ⎣ |x − | dx⎦dy

) (x + y) dx dy

⎡ √ ⎤ 0 0 −x , ⎣ ) − x − y dy ⎦ dx 00 f (x, y) dA

9"# 0 . $' $. .$ (, ) (, ) (, ) (, ) / g. 0 !Y " D A *( x + y = y − x = B Bn 0 $ '& D A; *( Ǳ [ ! &O ( (, ) (, ) ( , ) / g. 0 `?-_ E D A *( ( , ) x + y = x y + x = # $ 0 $ '& D A *( 00 '& D M M [& . T .$ . xy dA AH ( x = p/ y y = px + 0 $

*p > x + y ≤ i1 0 . f (x, y) = |xy| e0& A8 *M [&

D

D

$ 6 $ $ Z N# E u0 Y\ N# 0 A; *O&0 \Y 4&h N# . $ & %&L A U0& A , + 0 >&j L N# E . h -&#. 5&0 0 Z R hC oL6 , + 0 A E S = N[V# ( *A*** R !#{ G#U ( T .$ >&j L K@ m M 5 M 9Z$ g . . T .$ >&j L v N# E . h % !VO 0 F : R → R #n3/ VU U0& ! F (u, v) = x(u, v), y(u, v)

(u, v) ∈ D E 0 M ( V $0 D $ &0 ∂(x, y) = J = ∂(u, v)

*Z & F >&j L v R0 M W . J $, e0& >&j L v R# E $& !B$ 0 78 % ('&" &0 , + #E 0 pO R# &S$ [& .$ *O&[ . 1 0 D E T D = V = R Z#S A& '() % F >. T R# .$ *F (u, v) = ((u + v)/, (u − v)/) ( #n3/ VU F _z #E *( >&j L v N# &f .$ F (x, y) = (x + y, x − y) −

* ∂(x, y) * =* J = ∂(u, v) *

/ / / − /

* * − *= *

=

=

} ZM x=

A, .8 >&j L v N# F >. T R# .$ F (u, v) = (uv, u/v) √ F (x, y) = xy, +x/y ( #n3/ VU F #E ( D = V = {(u, v) | u >

, v

>

v /v

* * u * = u −u/v * v

−

* ∂(x, y) * =* J = ∂(u, v) *

*$ O T D E $ O+ G#U v = y Z 0 & M >&j L v N# F E&+ ('&" &y Z R# 5 Q *M G#U F (u, v) = (x, y) $ 5&6 $. .$ 0 $ % oL6 . V D , + M G#U >&j L v N# 9M ) x = u, y = uv

)

x = u /v, y = v /u

) ) ) ) )

0 (0

y/

0 (0

ln x

0 (0

y (x − y)e(x−y) dy

) dx

) + y (x − ) + e dy dx )

π/

ln(cos y) dy dx arctan x

0 (0 x

0 (0

)

xy exp(yx ) dy +

y

x y + x y

dx

) dx dy

) dx dy ) y ln y + xy + f (x, y) = x y − x − y 0

e

(0

ln y

, + 0 . x + y

e0& AI = 0 $ e0. .$ e1 *M [&

0 $ , + 0 f (x, y) = xy e0& A 9M [& . #E &j L &.

*C : x = R cos t , y = R sin t , ≤ t ≤ π/ $ &j L &. 0 $ e0 D M x= A; Rl+ O&0 y = x = B$&U 0 y Zh −x !VO 0 . D *f (x, y) = (x − y)/(x + y) N# 0 ([" . f ]< O Zh −y c *M [&

- ^_' S& (0 O !' L 0 &# !#&" $. E .&"0 .$ v D. E $. RQ .$ * " !' !0&1 ?zT V# &# &#&O N+M e1 E .&"0 .$ M $ + $& 5 v

E .&[, &S $ .$ $. & &v v *M

&S ) . & F M Y >&j L v [Y1 >&j L v >&j L v $ R# e $0.&M !B$ 0 *$M Z .0 v M >. T 5&0 0 5&#&3 .$ ZM ^#6 . & F 0 *( $3 Z v

^- 0 U S M . Y0 @PC\) % *Zg S ( $3 RU . p&\ , + . nS Z 0 &# (#&,. "#&0 . nS 4& 0 T& T M ( RO. 9$$S O $ $ ([" Z p&\ + 0 A

}

)

)

u = xy, v = x + y

u = /x, v = /y

)

x = u + v, y = u − v , ) u = x + y , v = arctan (y/x)

.$

G+ 'BC @PC\) f % %

0 M W V $0 U $ &0 &j L v F M . T c O&0 #n3 D ⊂ V 0 z = f (x, y) e0& O&0 e0& >. T R# .$ F (D ) = D M O&0 . @ D ⊆ U S ! ?U0 ( #n3 D 0 c z = f x(u, v), y(u, v) J

0 [Y1 >&j L E , + !#[ 9;I*8 !VO ]VUB&0 .&M$ #E D S >.? @PC\) fm * % T−θr .$ 5F # j M O&0 T−xy E , +

>. T R# .$ ( D , +

00 00

f (x, y) dxdy =

0 $ '& D M x= A& '() % e0& >. T R# .$ *( x + y =+ # $ *M [& . D 0 f (x, y) = x + y 0 D 5 Q *ZM $& [Y1 >&j L v E *!' D # j U#C D ]3 $$S 5&0 x + y ≤ >. T . + r & *D : r ≤ E ( >.&[, AT−θr .$ $ ) θ 0 @O 5 Q *D : ≤ r ≤ R# 0&0 ( & U# V0 . $ > v + θ ]3 $.

D :

≤ θ ≤ π , ≤ r ≤

00 + 00 , x + y dA = r r dA D

0 π (0 = =

D

√ 0 π

)

r/ dr dθ = dθ = π

Z#. $ ` + .$ ]3

0 π

r /

dθ

*$ O ) ;*8 !VO 0 D 0 D !#[ Q Z" 0 }

J**8 E (+"1 9;*8 !VO

00

00

*

f dA = D

F − (D)

(f ◦ F ) dA

.&j 0 &#

M x= .? @PC\) & % r &0 . O & M T&= *( T−xy .$ Ǳ [ c0 Y\ *$ O ) ;I*8 !VO 0 Z$ 5&6 θ &0 . xOM # E >. T R# .$ M ( RO. M = (x, y)

x = r cos θ y = r sin θ

+ x + y = r arctan (y/x) = θ

R# 0 *r = θ = ZM G#U O = (, ) Ǳ [ $. .$ *$ O G#U F (θ, r) = (x, y) e0& X α ≤ θ < π+α ZM x= $$S N[V# F V# 0 $ ]3 * ≤ r M ( RO. c LB$ ( $, α M >. T 0 . F U : α ≤ θ < π + α ,

√

! f x(u, v), y(u, v) J dudv

D

D

! f r cos θ, r sin θ r dA

D

D

00

00

f (x, y) dA =

≤r

!M E ( >.&[, F $0 M ( ^- ?U0 *ZM G#U c T−xy ∂(x, y) −r sin θ = J = ∂(θ, r) r cos θ

cos θ = | − r| = r sin θ

R# 0&0 *O&0 T GB&L r = Y\ c0 &) + .$ M !$": . >&j L v R# *( >&j L v N# F θ [& `O [ _z +U M ( M{ 0 4E_ *Z &

N# F $ >. T .$ ?U0 *( −π &# 0 0 α U# ( RV+ M #$ V *( T−θr .$ . Z E `O &0 ( Y Z θ = θ M ( 5F O&0 # $ r = r $E& θ # E . −x &0 M Ǳ [

*$ O ) ;I*8 !VO 0 *( r `&UO Ǳ [ cM 0

=

0

π/ −π/

( =

u−

() cos θdθ =

u

)

0 −

− u du

=

du = cos θdθ R# 0&0 u = sin θ ( O x= AC .$ M π/ * uθ −π/ − }

0 M O&0 e0. E +"1 D M . T .$ A, .8 e0& . \ ( $ x + y+ = # $ *M [& . D , + 0 f (x, y) = − x − y E ( >.&[, D M ZM ) . h R# 0 *!' D : x + y ≤ ,

≤ x, ≤ y

$& r sin θ E y &0 r cos θ E x &0 D R =&# 0 9ZM

D : r ≤ ,

D :

≤ r cos θ , ≤ r sin θ ]3 ≤ r . + &

≤ r ≤ , ≤ cos θ , ≤ sin θ

≤ U 0 ≤ θ ≤ π V# 0 ) &0 F pO $ M D : ≤ θ ≤ π/ , ≤ r ≤ R# 0&0 * " θ ≤ π/ 00 ,

− x − y dA =

J**8 E (+"1 9;;*8 !VO

D

B$&U 0 "0 0 $ '& D V# x= &0 A .8 ≤ x, < a M ( (x + y) = a+(x − y) *M [& D 0 . f = x − y }

Z"# 0 Z

00 +

− r r dA

D

0

) ( π/ 0 + r − r dr dθ

0

π/

=

=

−

( − r )/

0

π/

dθ =

dθ =

π

# $ 0 $

'& D M . T .$ A .8 + f f (x, y) = x + y c O&0 x + y = x *M [& D 0 . M ZM ) . h R# 0 *!' x + y = x

D : x ≤ x + y ≤ x

9E ( >.&[, 5F D [Y1 # j

J**8 E (+"1 9;*8 !VO

D : r cos θ ≤ r ≤ r cos θ

Z#. $ & #&" & R=@ E r mn' &0

M ZM ) . h R# 0 _> D : (x + y ) ≤ a (x − y ), ≤ x

9Z#.F (0 F [1 # j D

:

(r ) ≤ a (r cos θ − r sin θ) ,

:

r ≤ a cos(θ) ,

≤ r cos θ

D : cos θ ≤ r ≤ cos θ

R# 0&0 *O&0 & c cos θ &# cos θ #&0 ]3 ≤ r 5 Q nB −π/ ≤ θ ≤ π/ D : −π/ ≤ θ ≤ π/ , cos θ ≤ r ≤ cos θ

≤ cos θ

≤ cos(θ) "#&0 ]3 A$ O ) ;*8 !VO 0C $&U 5&0 0 &# −π/ ≤ θ ≤ π/ &z cB U# ≤ cos θ R# 0&0 *−π/ ≤ θ ≤ π/ D : −π/ ≤ θ ≤ π/ ,

≤r≤a

√ cos θ

00

00 f dA = D

D

=

0

π/

−π/

Z#. $ ` + .$ *$ O ) ;;*8 !VO 0 ,

(0

x + y dA = cos θ

cos θ

)

00

r . r dA

D

r dr dθ =

0

π/

−π/

(

r

) cos θ dθ cos θ

00 I

ln x + y dA =

= D

=

00

r ln r dA =

D

()

=

=

=

0

00

00 , 00 , x − y dA = r cos(θ)r dA

ln(r )r dA

D π/

(0

)

D

( ln − )

0

r ln rdr dθ

=

)

π/

dθ =

π

=

0

a sin β

0 ⎣

a −y

ln( + x + y ) dx⎦ dy

y cot β

9M [& . O $ $ & B

⎡ √ ⎤ 0 a 0 ax−x ⎣ ) (x + y ) dy ⎦ dx

(

π/

, r

−π/

0 a

a

π/

−π/

θ

cos(θ)

) , cos(θ)dr dθ

)r=a√cos(θ) r=

cos (θ)dθ sin(θ)

+

r

π/ −π/

=

a

π

T−xy .$ O $ $ "0 . $ + &6 . h 0 9ZM Z 0&#Y\ D. 0 . 5F +[Y1 4= + x = a cos θ cos(θ) x = a cos θ cos(θ) θ

−

π

−

+√

x

a

y

−a

.

−a +√

r

π

a

π

π

+√

+

a

+√

√

+

+

a

+√ a +

+

a

a

+

√ a

√

−

⎡ ⎤ 0 0 √−y ⎣ √ I = ln(x + y ) dx⎦ dy

A .8 *M [&

>.&[, D $ M ZM ) . h R# 0 *!' E ( y

D

x +y

−x −y +x +y

−π/

= =

(0 √ a cos(θ)

π/

0

( ln − )

. f e0& I & >&#+ .$ 0 % 9M [& D , + 0 + *( x + y ≤ i1 D f = − x − y A M ( x + y ≤ i1 E +"1 D f = x + y A; *$. $ . 1 . −x #E # $ 0 $ '& D f = /( + x + y) A *( x + y = x + y = y _&0 .$ e1 '& D f = / arctan (y/x) A *( x + y = # $ ! $ y = x + x + y = # $ 0 $ '& D f = x x + y AH *( y x +y = x x +y = 0 $ '& D f = x A8 *( (x + y ) = !VO 3 0 $ '& D AJ , *f = / + x + y O&0 x − y *( x + y = # $ ! $ D f = e A . 1 4 e0. .$ M O&0 x + y ≤ , i1 E +"1 D AI *f = $. $ O&0 ([a $, a ≤ β ≤ π/ M . T .$ A 9M [& . #E ⎡ √ ⎤

D

0

⎫ 0 ⎬ r dr dθ ln r − ⎩ r⎭ ⎧ ( ) ⎫ 0 π/ ⎨ r ⎬ ln − dθ ⎩ ⎭ ⎧ 0 π/ ⎨( r

?U0

r

y ≤ x ≤ √ : x + y ≤ , y ≤ x ,

:

≤y≤

√

,

,

− y

≤y

X"' 0 y x &0 D [Y1 # j 5$.F (0 . h 0 U# Z$ . 1 θ D

: r ≤ , :

√

r sin θ ≤ r cos θ , ≤ r sin θ √ ≤ r ≤ , tan θ ≤ / , ≤ sin θ

) &0 4$ &" & ( ≤ θ ≤ π U 0 4 &" & R# 0&0 *O&0 ≤ θ ≤ π/ U 0 (#$ R# 0 ]3 *( !Y " N# M D : ≤ r ≤ , ≤ θ ≤ π/ Z#. $ ` + .$

0C 4 U . $ .$ oL6 E &0 ]& A$ M (x, y) → (x, y) h 9A$ O ) $ ;*8 !VO *O&0 . −y $ .$ E &0 "& 0C 4 U Y\ 0 ([" oL6 E &0 ]& A Y0&- &0 (O& h 9A$ O ) ;*8 !VO − / E &0 "& M (x, y) → (−x/, −y/) *O&0 Ǳ [ 0 ([" 9E .&[, _&0 d0 k#& R# +

Q &# C !VO aa '& N# # j RU 0 A *ZM # j F `?- ( =&M AU&# B Bn q0 # $C @L eY\ N# # j A; *( $ ` E @L eY\ N# A+ >&j L v N# $ Y >&j L v N# ]V, A *( Y 0 $ '& D M x= A& '() % y = x − y = x x + y = x + y = p Y [& D 0 . F Z L0 f = (x + y) ( *M M ZM ) 0 *!' D :

)

0

a

−

⎡ √ ⎤ 0 a −y ⎣ √ (x + y )/ dx⎦ dy a −y

⎡ √ ⎤ 0 0 x−x x+y ⎣ ) dy ⎦ dx x + y

)

0

⎡ a

⎣

0

⎤

√

a −x

v N# M 4=

xy

+ dy ⎦ dx x + y ? @PC\) f %

E ( >.&[, Y >&j L

x = au + bv + c , y = αu + βv + γ

* ∂(x, y) * a =* J = ∂(u, v) * α

5F .$ M

* b * * = |aβ − αb| = β *

M ( 5F A!"# 4&0C >&j L v R# . nSZ !B$ * # j &f .$ ( Y v u X"' 0 y x M !VO _z v _z T *( y N# >&j L v R# y y 9$ 4& . #E KO 0 .&M Q &# N# Y >&j L

}

≤ x + y ≤ , − ≤ y − x ≤

>. T R# .$ *v = y − x u = x + y ZM x= &' y = u/ + v/ x = u/ − v/ * ∂(x, y) * =* J = ∂(u, v) *

* / −/ * *= / / *

=

.$ D M oL6 y .& Q # j M ( RO. ?U0 *v = v = − u = u = E .&[, T −uv ) ;H*8 !VO 0 D : ≤ u ≤ , − ≤ v ≤ R# 0&0 .$ *$ O

00

(x + y) dA =

D

0 (0 =

−

00 D

u

dv

u )

dA

du =

0

u du =

&. 0 $ '& D M . T .$ A, .8 &0 e0& . \ O&0 x + y = π B$&U 0 y &j L

x−y *M [& . D 0 f (x, y) = cos x + y Y0&-

Y >&j L v 9;*8 !O GB ;*8 !VO 0C x . $0 N# E &0 &\ AGB B&\ M (x, y) → (x + a, y + b) h 9A$ O ) (a, b) E &0 *( −−→ x Y\ ' oL6 # E E &0 5 .$ A% Y0&- &0 (O& h 9A$ O ) % ;*8 !VO 0C (x, y) → (x sin θ + y cos θ, −x cos θ + y sin θ)

*( Ǳ [ ' θ E &0 .$ M ;*8 !VO 0C x y N# 0 ([" /&VU A: &VU M (x, y) → (−x, y + b) h 9A$ O ) : *( . −y 0 (["

}

*#$S !#[ v = u >. T 0 + u = v >. T (B&' R# .$ &

u = x + y − v = x − y + * * / − / J =* * / /

0 *D 00

:

x = (u − v + )/ ⇒ y = (u + v + )/ * * * = / *

E ( >.&[, D >. T R# .$ *$ O ) ;J*8 !VO , u ≤ v ≤ u

≤ u ≤

(x − x + y − y + ) dA =

D

0 0

(u + v ) dA

D

u

00

(u + v ) dv du )v=u 0 ( v = du u v+

=

=

u

0

v=u

u − u −

u

;**8 &a E (+"1 9;H*8 !VO *v = x + y u = x − y ZM x= . h R# 0 _> >. T R# .$ x = u/ + v/ y = −u/ + v/

* / * *= / *

* * / J =* * − /

=

# j . D da e- D # j D 5$.F (0 0 S *v = −u &# u/ + v/ = &F x = S 9ZM

x + y = π S *v = u &# −u/ + v/ = &F y = v = −u v = u y 0 $ '& D ]3 *v = π &F >. T 0 . D *A$ O ) ;8*8 !VO 0C ( v = π 9Z#S O Zh −u

du =

}

00 cos D

;**8 &a E (+"1 9;J*8 !VO O&0 x /a + y/b ≤00 ,q0 D M . T .$ A .8 − x /a − y /b dA *M [& . b x :L E . a ZM U 0 . h R# 0 *!' u = x/a x= &0 ` - R# *ZM mn' y :L E . y = bv x = au (B&' R# .$ *$#n3 *>. T v*= y/b >.&[, T−uv .$ D # j D J = *** a b *** = ab [Y1 v v E $ .$ *D : u + v ≤ # $ E ( u = r cos θ 9ZM $& !Y " N# 0 D !#[ 0 # j D X R# 0 *$ O ) ;7*8 !VO 0 *v = r sin θ D : ≤ θ ≤ π , ≤ r ≤ E >.&[, T −θr .$ D .$ $ 0 D

= = =

00 u x−y cos dA dA = x+y v D 0 π 0 v u du dv cos v −v 0 π$ u %u=v dv v sin v u=−v ( )π 0 π v π v sin( ) dv = sin( ) = sin( )

}

00 D

-

x y − − dA = a b 00 + 00 + = − u ab dA = − r abr dA D

D

) 0 π (0 + = ab r − r dr dθ ) 0 π ( − ( − r )/ = ab dθ /

;**8 &a E ; (+"1 9;8*8 !VO . f = (x − x + y − y + ) e0& A .8 B$&U 0 + x + y = ( . y 0 $ '& 0 *M [& x + y + = y + xy >. T R# .$ *ZM ! &M e0 . f Y0&- 0 *!' h'? ?U0 *f = ((x + y − ) + (x − y + ) ) x + y − = x − y + !VO 0 . y B$&U M $$S

x + y − = >. T 0 c . + B$&U (O 5

u = x − y + $ O x= l&Q ]3 *(x − y + ) 0 y *( f = u + v >. T 0 f &F v = x + y −

}

ab

=

0 π

πab

dθ =

# $ 0 $ '& D M . T .$ A .8 x + y + y = x *M [& D 0 . f = x − y e0& O&0 }

x + y + y = x

;**8 &a E H (+"1 9;*8 !VO &v N+M 0 D '& 0 f e0& E 0 % 9#0 v u #) y = x + y = x − p Y 0 D f = ln(x + y) A . v x − y . u *( $ x + y = x + y = *#0 x + y ( , ) (, ) / g. 0 aa D f = x + xy A; *#0 x + y . v x . u *( (, ) x+ y = x +y + xy + $ 0 D f = x+ y A M x= *( $ x + y = x + y + xy *v = x + y u = x + y x= *( $ |x|+|y| = E B 0 D f = |xy| A *v = x + y u = x − y M x + y = q0 $ 0 D f = x + y AH u = x M x= *( $ x + y = *v = y (x + y + ) + (x − y) ≤ q0 0 D f = x A8 *v = x+y+ u = x−y M M x= *( $

AJ x + y = y .& Q 0 D f = (x − y)e M x= *( $ x = + y x = y x + y = *v = x − y u = x + y y = x + x = y + y .& Q 0 D f = y − x A7 M x= *( $ y + x = y + x = *v = y + x u = y − x x= *( ≤ x, y ≤ π e0 D f = | cos(x + y)| A *v = x − y u = x + y M

;**8 &a E (+"1 9;7*8 !VO >. T R# .$ v = x − y u = x + y ZM x= _>

u=x+y ⇒ v = x−y * * / / J =* * / − /

u X"' 0 y x &0 S M $$S &6 ?U0 u + v = v > .&[, 0 Z$ . 1 O $ $ >_$&U .$ '& E ( >.&[, D U# *. Z u + v = v U# *u + u = v u + v = v # $ 0 $

v ≤ u + v ≤ v

D :

>&j L 0 . D '& v = r sin θ u = r cos θ x= &0 9Z#0 [Y1

D

r sin θ ≤ r ≤ r sin θ sin θ ≤ r ≤ sin θ , ≤ sin θ ≤ θ ≤ π , sin θ ≤ r ≤ sin θ

: : :

Z#. $ X R# 0 *$ O ) ;*8 !VO 0 00

00 (x − y) dA = D

0 =

=

π

0

=

.$ . $ U $ $. & b >&j L v #E & B&a * O %&L 5F &0 X& !VO 0 B&"

*O&0 & F ` E #& +

x = u/ + v/ y = u/ − v/ * * − *= * =

v

(x +y )/

∗

π

(0 (

v D

π

sin θ

sin θ

r

0

dA =

00

r sin θr dA

D

)

r sin θdr dθ )r= sin θ

sin θ

dθ = r= sin θ

− cos(θ)

0

dθ = · · · =

π

sin θdθ

π

*M [& D 0 . f *f .$ *v = y/x u = x + y ZM x= . h R# 0 *!' >. T R# ∂(x, y) = ÷ ∂(x, y) = ∂(u, v) ∂(u, v) * * * * *= x * = ÷* −y/x /x * x + y

J

5 *D

.$ *D

00

: 00

x D

+

dA =

y

dA =

v

D

=

ln

du = ln

R# .$ y /

≤ t ≤ π

M [& . O&0 . −x x = u( − cos t) ZM x= . h R# 0 *!' ≤ u ≤ a S M ( RO. >. T R# .$ *y = u(t−sin t) &O 3 & R# y D '& !M &F ≤ t ≤ π # j D R# 0&0 *$ O ) % I*8 !VO 0 $ O

E ( >.&[, T−uv .$ D '&

00

00 x dA =

≤ t ≤ π , ≤ u ≤ a

E ( >.&[, c !#[ R0 M W

* t − sin t * * = u | − cos t − t sin t| − cos t *

.$

u( − cos t)u − cos t sin t dA

D

D

0 π 0

a

= =

a

0 π

u ( − cos t) − cos −t sin t du dt

( − cos t) − cos t − t sin tdt

f (t) = − cos t−t sin t e0& ( ?, #&0 [& $ 0 $ O h'? >. T R# .$ *$ + oL6 [, π] 0 . M α & Y\ .$ f & *f (t) = sin t − t cos t M Z#. $ ?U0 *$$S T π ≤ α ≤ π/ t π α π f (t) f (t) f (t)

+ +

π

− +

/

/

α cos α

− −

−α cos α

− cos α

−π

i& >&j L v 9I*8 !VO = a v x = a u $ O x= S ]3 ?U0 y = av x = au >. T /

* * J =* *

D

* * u( − cos t) J =* * u sin t

/

x dA x+y

C : x = a(t − sin t) , y = a( − cos t) , 00 ( < a) x dA

D :

/

/

0 $ '& D M . T .$ A .8 *

/

y ) dv du v

0

/

(0

0

=

x

D

00

+

/

/

, ≤ y/x ≤ 5 Q ?U0 ≤ u ≤ , ≤ v ≤ (O

≤ x+y ≤

:

$ '& D M . T .$ A& '() % O&0 x + y = a A!VO .& C g F 0 + *M [& D 0 . f = / |xy| e0& 5 R# 0&0 D : x +y ≤ a M Z#. $ ) *!' I*8 !VO 0 *D : (x ) + (y ) ≤ (a ) (O *$ O ) GB }

/

/

au av

* * * = a u v *

# $ E ( >.&[, T−uv .$ D E !T&' D , +

Zh −u >. j0 . , + R# *D : u + v ≤ 3

D : − ≤ u ≤

,−

+

− u ≤ v ≤

+

− u

R# 0&0 *(O 5

00

00 D

+ .a u v dA av | |au D ⎡ √ ⎤ 0 0 −u ⎣ √ a/ |uv| dv ⎦ du

+ dA = |xy| =

−

−

−u

=

⎡ √ ⎤ 0 0 −u ⎣

a/ |u|v dv ⎦ du

=

√ )v= −u 0 ( v

a/ du |u|.

()

= ()

=

−

a /

0

−

−

a/

v=

|u|( − u ) du

0

u( − u ) du =

a/

:E B $. e0& A;C .$ AC .$ V# ^- . $M Gj . $ ]3 ( 5.&\ $ *ZM 0 0 $ x + y = y .& Q 0 $ '& D M x= A, .8 *f = /x+ /y Rl+ ( y = x y = x x+y =

}

(O 5 ( ([a [; π] !M 0 f R# 0&0 00 x dA = D

a

=

i& >&j L v 9*8 !VO y = x p Y 0 $ '& D M . T .$ A .8 x + y = x y x + y = x y $ x = y [& D , + 0 . f (x, y) = (xy) e0& O&0 *M .$ *v = y/x u = x/y ZM x= . h R# 0 *!' y = u v x = u v >. T R# −

− /

− /

− /

− /

* * − u−/ v −/ * * * *= * −/ −/ * − u v *

* * −/ −/ * − u v * * J =* * * * − u−/ v −/

y

u − v −

v

0 x = B$&U u = >. T 0 y = x B$&U

0 x + y = x y B$&U *$$S !#[ v = u >. T >. T 0 x + y = x y B$&U u + v = >. T −uv .$ D # j D R# 0&0 *#$S !#[ u +v = D : / ≤ v/u ≤ , ≤ u + v ≤ E >.&[, T D &0 O&0 5.&\ $ ( :E f e0& 5 Q *( D : u/ ≤ v ≤ u, ≤ u + v ≤ u + E 5

*$ + 0 0 $ . ( e1 e0. .$ M $M $& # j α = arctan l&Q [Y1 >&j L v N+M 0 E >.&[, T−rθ .$ D

−

(xy) D

00 dA =

00 =

=

=

(uv) dA =

D

00

≤ u + v ≤ , u ≤ v ≤ u * * * au * * * J = * bv * = abuv t = v/u s = u + v

D :

v E 5 M * ?, 0 #) v

R# .$ *ZM $& E >.&[, T−st .$ D # j D >. T

D :

00

α

(0

J

D

xy =

dθ

∂(u, v) = ÷ ∂(s, t) ∂(s, t) ∂(u, v) * * * * * *= u ÷* * −v/u /u u+v

00

00 dA =

D

π/−α 0 α sin θ dθ = (sin α − cos α) = π/−α

≤t≤

Z#. $ ` + .$ *$ O ) *8 !VO 0 *O&0

) r sin(θ) dr

= =

dA

(uv) dA

≤ s ≤ ,

0 0 c 5F R0 M W (

≤r≤

(r cos θ/r sin θ) r dA

D

0

=

D

u − v −

! dt

Z#. $ ` + .$ ]3 *$ 0

(uv)+

− cos t − t sin t

>. T R# .$ * [a $, $ b a M x= A .8 p Y 0 $

D '& 0 f (x, y) = /xy e0& + + x/a + y/b = & x/a+= y/b x/a = y/b + *M [& . x/a + y/b = + + *v = y/b u = x/a ZM x= . h R# 0 *!' y B$&U >. T R# .$ * ≤ v ≤ u i jL0 ]3 u = v U 0 [a v, u 5 Q M ( u = v U 0 u = v &# u = v B$&U 0 4$ y X R+ 0 *( + + x/a+ y/b = u+v = U 0 u = v B$&U *( .$ D # j D ` + .$ ]3 *O&0 u + v = U 0 E ( >.&[, T −uv

00

( − cos t)

t − sin t + t cos t π t dt = a π + sin t cos t − cos t

D : π/ − α ≤ θ ≤ α,

0 π

a

= =

00

D

au bv u

abuv dA =

00

00 D

uv

dA

dA = dA uv u + v st D D s= 0 (0 ) 0 ds dt dt = ln s t st s= 0 $ % ln dt = (ln )(ln t) = (ln ) t

x= c α b a X& %&L &0 #E $. E N# .$ . O $ $ '& ('&" y = br sin θ x = ar cos θ 9M [&

α

α

x y x y x y x y ∗ + + + ≤ ) ≤ h a b a b c . . x y ) + ≤ , ≤ x, ≤ y a b

)

. O $ $ & 0 $ '& ('&" $. .$ 9M [&

)

x + y = a, x + y = b, y = αx, y = βx ,

< a < b, < α < β )

xy = a , xy = a , y = x, y = x,

)

y = x, y = x, x = y, x = y

)

x = ay, x = by, x = cy , x = dy ,

≤ x, ≤ y

< a < b, < c < d )

(x/a)/ + (y/b)/ = , /

(x/a)

/

+ (y/b)

= ,

≤ x, ≤ y

$ '& D ([a $ , d c b a M x= A; x − y = c x + y = b x + y = a # $ 0 0 f = xy E c > d a > b x= &0 *( x − y = d *#0 D

F B !F N +0 -&#. & +"1 #& .$ = &$0.&M &S$ : L "= *$. $ W B 0 ' +O N#c= &+B D. 4&0 Z" O. $0.&M 0 p 0 = *Z#E $3 5F ^#6 0 M ( $, +M q M x= BV K+ $ % !0&1 & (x, y) ∈ D E 0 M ( y x 3 &v &0 $M Z"\ NQ M & Y " 0 . D l&Q *O&0 G#U 5$M %- &0 ( (0&f & F E N# 0 q M ZM x= O $ $ (+M . \ .$ NQ M !VO !Y " '& ('&"

[& !Y " 5F (= !Y " R# p&\ E V# .$ &0 *$ 0 q(x , y )Δx Δy !VO 0 !T&' >.&[, *$$S 9$$S [& D !M (= [#\ #$&\ R# + 5$E e+) (= . \ 0 j' 0 *Q ≈ 7 7 q(x , y )Δx Δy X R# 0 $ O =S ' _&0 ` + E ( =&M D !M i

j

i

j

i

j

i

j

i

j

0 $ %

# $ 0 $ D / 1 0 . f (x, y) = x + y e0& A *M [& x + y = x + y (0&f e0& E y = u sin v x = u cos v x= &0 A; &j L &. 0 $ D '& 0 f (x, y) = $, a M #0 √x + √y = √a *( ([a

e0& .$ y = uv x + y = u >&j L v &0 A &j L &. 0 $ da 0 f (x, y) = xy *M [& F x + y = y x − y = y $ 0 $ '& D M . T .$ A 0 O&0 xy = xy = B Bn $ y − x = e0& v = x − y u = xy >&j L v N+M *M [& D 0 . f (x, y) = /x + /y y $ 0 $ '& 0 f (x, y) = x + y e0& AH y = x y = x + $ x+y = x+y = *M [& . 0 $ , + 0 f (x, y) = x + y e0& A8 *M [& . x + y =

0 $ , + 0 f = exp e0& AJ x = y/ x = y y = x y = x &

*M [& . 0 $ , + 0 f = √ye + +ye e0& A7 . y = e y = e y = e y = e

*M [&

$ , + 0 f = exp +y/x+√xy e0& A . x = y x = y xy = xy = & 0 *M [&

& 0 $ , + 0 f = xy e0& AI [& . x = y x = y y = x y = x *M 0 $ , + 0 f = y ln( − x − y) e0& A *M [& . x + y = y = x = p Y + 0 $ , + 0 f = x + y e0& A; [& . x + y = x + y = p Y y = x *M ∗

x +y xy

−x

x

−x

−x

x

x

Area(D)

=

dA = 0 a #

y

"

a − y − a

=

j

dx dy

y /a

D

T&= 5& c+ 0&# u# c= y >&+"\ x >&+"\ $ U M 90&# u&M B & y T&= c B & x

)

0 a (0 a−y

00

a

dy =

Q=

B$&U 0 g F 0 $ '& ('&"

A, .8 *M [& . (x/a) + (y/b) = c u = (x/a) ZM x= . h R# 0 *!' y = bv x = au R# 0&0 *v = (y/b) /

lim N, M → ∞ Δxi , Δyj →

00

/

∂(x, y) = J = ∂(u, v)

au bv

= abu v

.$ D : (x/a) + (y/b) ≤ '& # j D . D 5 M *( D : u + v ≤ E >.&[, T−uv J = r R# 0&0 ZM !#[ [Y1 (B&' 0 /

/

≤ θ ≤ π , ≤ r ≤

D : 00

00

Area(D) =

dA =

=

ab

=

ab

abu v dA

r cos θr sin θ dA

D

00

r sin (θ) dA

D

ab

=

#0 π

sin (θ)dθ

ab θ − sin(θ)

=

! x + y ≤

.

π

" #0 ( ×

r

" r dr

)

Q=

q(x, y) dA D

9$ + T? 5 #E >. T 0 . &+B D. R# 0&0 ) )*"9 ) D J )* [ > 7<p )*$+( q B m7+ q(x, y) !F !H/ L B (x, y) MUV _`3! ) 3 ./ D ZM x= H# J= 7JD) . !) $ % Δy Δx $&U0 0 !Y " * ( T E , + #E *( Δx Δy 000 0 !Y " R# ('&" *Z#S h .$ . Area(D) = dA 0 0 D !M ('&" *H*8 0&0 ]3 *Q = Area(D) q(x, y) = &# .$ #E *( x + y = a y 0 $ ('&" A& '() $ % *M [& . ≤ y T Z .$ e1 y = ax + $. 0 Z &0 . O $ $ y + . h R# 0 *!' 9Z$

D

j

i

i

j

D

D

D

00

.$

q(xi , yj )Δxi Δyj

i= j=

M $S &S$ G#U 0 U) &0

/

/

N / M /

= πab

, + E +"1 ('&" A .8 *( e1 e0. .$ M M [&

[Y1 >&j L 0 . O $ $ D '& . h R# 0 *!' 9Z#0

a xy

y = ax x + y = a

⇔

y = −ax ± a x = a − y

'& R# 0&0 *$$S !T&' (a, a) (a, −a) p&\ M *;*8 !VO .$ O oL6 \Y E ( >.&[, h $.

#E ( Zh −y '& R# D :

≤ y ≤ a , y /a ≤ x ≤ a − y

&0 ( 0 0 5F ('&" R# 0&0

D : (r ) ≤ a r cos θr sin θ : r ≤ a sin(θ)

≤θ

*D

:

]3 ( e1 e0. + .$ D x= 0&0 5 Q ≤ θ ≤ π/ , ≤ r ≤ a sin(θ) .$ Z#. $ ` + .$ R# 0&0 ≤ π/

00

00

Area(D) =

dA = D

0

π/

(0

a

= =

√

D

sin(θ)

r dA )

r dr dθ =

a

0

π/

sin(θ)dθ = a

0

π/

(

r

i

)r=a√sin(θ) r=

}

*H*8 &a E (+"1 9;*8 !VO

} 0 $ %

. (x − y) + x = q0 0 $ '& ('&" A *M [&

x = y x = y p Y 0 $ ('&" ("0 Y A; *( ([a $, a M x + y = a x + y = a N+M 0 Z' [& AGB 9*8 !VO 8*H*8 &a E ; (+"1 A% &S$ *M [& . R `&UO 0 M Z' A& '() % $ % cM 0 Ω : x + y + z ≤ R M . h R# 0 *!' ^Y mU B$&U S *Z#S h .$ . R `&UO Ǳ [ .$ R# .$ ZM !' z X"' 0 . Ax + + y + z = R U#C 5F . $ + 0 $ Z") Ω ]3 z = ± +R − x − y >. T $ G#U $ M ( z = ± R − x − y e0& $ U# *O&0 D : x + y ≤ R & F , , Ω : (x, y) ∈ D, − R − x − y ≤ z ≤ R − x − y

Z#. $ H*H*8 0&0 ]3

V =

, 00 , R − x − y − − R − x − y dA

$ 0 $ '& ('&" O&0 ([a $ , b a S A y = −bx + b y = ax + a >_$&U 0 + *M [& . ae = bd

&0 LB$ $ , f e d c b a M . T .$ A B$&U 0 q0 ('&" O&0

(ax + by + c) + (dx + ey + f ) ≤

*M [& .

x = ay + .& Q 0 $ ('&" < a < b S AH *M [& . y = bx y = ax x = by . x + y = x + y 0 $ '& ('&" A8 *M [&

. (x/a)

0 $ '& ('&" AJ *( $= $, n 5F .$ M M [&

/n

/n

+ (y/b)

D

00 , 00 + R − x − y dA = R − r r dA = D

= =

#0

" #0

π

[θ]π

dθ

D R

r

+

"

R − r

(R − r )/

−

R =

πR

9O&0 D [Y1 # j D &# .$ M

D :

≤ θ ≤ π , ≤ r ≤ R

0 q0 5 S+ $ 0 $ Z") Z' A, .8 [& . z = − x − y z = x + y >_$&U

*M $. $ . 1 Ǳ [ .$ u . ( _&0 0 . 5 S+ *!' *$. $ Rg&3 0 . ( (, , ) .$ / . &0 4$ 5 S+ *O&0 g $ f B ]3 AT−xy 0 & F w O # j U#C D R =&# 0 *ZM !' . B$&U $ E XM & $

z = x + y z = − x − y

⇒

x + y =

⇒ x + y =

=

9M [& . & E N# 0 $ '& ('&"

)

(x + y ) = x − xy ,

)

(x + y ) = xy,

)

) (x + y) = x − y ,

. T .$

(x/a + y/b) = x y ,

≤ x, ≤ y.

2 QDR /MJ . !) $ $ %

z = f (x, y) e0 . $ + 0 $ Z' Ω M ( D , + &0 T−xy 0 5F # j M . @ O&0 0 0 Ω Z' >. T R# .$ *A$ O ) GB *8 !VO 0C

z = g(x, y)

00 Vol(Ω) =

! f (x, y) − g(x, y) dA

D

.$ . D E Δy Δx $&U0 0 Y " S #E ( . 1 !Y " R# 0 M Ω E +"1 Z' &F Z#0 h T $ R0 M ( +g&1 . 6 Z' 0 0 &z[#\ $. $ 0 5F # j $. $ . 1 z = f (x , y ) z = g(x , y ) 5F Z' R# 0&0 *( & Δx Δy !Y " 0 0 T−xy ' x= &0 *H*8 E ( f (x , y ) − g(x , y ) Δx Δy 0 0 *$$S !T&' . Mn q = f − g j

i

i

i

i

j

i

j

j

i

j

i

j

j

}

.$ *$ O ) % *8 !VO 0 D V

=

00

#0

−(x +y )

/

D

dθ (

r −

( − r )r dA

r

"

(r − r ) dr

)

= π

. a `&UO Ǳ [ cM 0 3 M E +"1 Z' A .8 *$S . 1 x + y = ax ! $ .$ M 0&0 . Mn M B$&U M ZM ) . h R# 0 *!' $ O !' z X"' 0 l&Q M ( x++ y + z = a R# .$ M ( RO. *Z. z = ± a − x − y 0 ?U0 *( g&3 + 0 0 g #_&0 + 0 0 f (B&' D : x + y ≤ ax 0 0 T−xy 0 h $. Z' # j h $. Z' ]3 *A$ O ) GB *8 !VO 0C ( &0 ( 0 0

,

00 ,

a − x − y − −

V = D

00 ,

=

0

=

0

= =

=

a − x − y

dA

00 +

a − x − y dA (=)

a − r r dA

D

D

(x/a)/ + (y/b)/ + (z/c)/ =

*M [& . . x = z x = z + y &#. 0 $ Z' A *M [&

(x + y + z ) = x + y − z #. 0 $ Z' AI *M [& . B$&U 0 #. x + y = T 0 $ Z") Z' A *M [& . z = x + y 0 #. 0 $ Z") Z' < a < b S A; [& . z = xy x + y = b x + y = a >_$&U

*M

" #0

π

$ %π = θ

T &j L >&T 0 M 0&0 . Z' A *( $ x + y + z = T z = x + y q0 5 S+ R0 e1 Z' A; *0&0 . z = M x + z = y pL E +"1 Z' a < b S A . $$S $ b a `&UO Ǳ [ cM 0 > M R0 *M [&

y = x + z q0 5 S+ $ 0 . j Z' A *M [& . y = − x − z z = x + y = &#. 0 $ Z' AH *M [& z = e x + x + y = a $ 0 $ Z") Z' A8 *M ) . z = a Z' [" Q 0 x + y = a + z q0 5 S+ AJ M ) . x + y + z = a M 0 $

#&q= g F 0 $ Z' A7

/

=

00

( − x − y ) dA =

D

0 & $ %

R# 0&0

00 ( − x − y ) − (x + y ) dA = D

8*H*8 &a E (+"1 AGB 9*8 !VO ^Y N# ('&" A%

: x + y ≤

π/ −π/

) (0 a cos θ + r a − r dr dθ

/ r=a cos θ (a − r ) dθ −π/ r= 0 π/ 0 () a cos θ dθ = a ( − s )ds −π/ − π/

a

−

(

s−

s

)

= −

a

Z# $M $& [Y1 >&j L v E AC .$ V# ^- 9( D # j D

D

: r ≤ ar cos θ : r ≤ a cos θ : :

≤ r ≤ a cos θ , ≤ cos θ −π/ ≤ θ ≤ π/ , ≤ r ≤ a cos θ

R# 0&0 *( O $& s = sin t >&j L v E A;C .$ π/ ds = cos θdθ * θ −π/ −

5 / −x −y = + + + + dA R − x − y R − x − y D 00 00 dA rdA () + + = R = R R − x − y R − r D D #0 " #0 " π R rdr + =R dθ R − r %R $ %π $ + =R θ × − R − r = πR

5 Sq0 Z' A

00 4

>&j L v E AC .$ D

:

≤ θ ≤ π , ≤ r ≤ R M

*( O $& [Y1 + M 0&0 . z = x + y pL E +"1 ('&" A, .8 *( O ) x + y = x y + f (x, y) = x + y e0& . $ + E +"1 h $. #. *!' D : x +y ≤ x i1 0 0 T−xy 0 5F # j M ( R# 0&0 *( N# `&UO 0 T1 D 5 Q *O&0

00 Area(S) = D

00 -

i=

(ai x + ai y + ai z + ai ) = a a a

a a a

M M [& . T .$ . a a = a

\V U!U) g=? 7JD) . !) * $ %

M O&0 z = f (x, y) e0& . $ + E +"1 S ZM x= (B&' R# .$ *( D , + T−xy 0 5F # j [& 0 *S : z = f (x, y) , (x, y) ∈ D 9Z"#

M Z#S h .$ . 5F E VQ M (+"1 S ^Y ('&"

[x ; x ] × [y ; y ] !Y " 0 0 T−xy 0 D# j −−−−−−−−−−−→ Y\ .$ u = −(−−, −−, (∂f /∂x)(x , y ))Δx . $0 *( 0 5F # j ( /&+ S 0 X = x , y , f (x , y )! >. T 0 *O&0 [x ; x ] × {y } e- 0 0 T−xy −−−−−−−−−−−−−−−−→ X Y\ .$ v = (, , (∂f /∂y)(x , y ))Δy . $0 0&6

e- &0 0 0 T−xy 0 5F # j ( /&+ S 0 h $. (+"1 ('&" R# 0&0 *O&0 {x } × [y ; y ] 0 . !Y " ('&" &0 0 0 &z[#\ 5 . S ^Y E ]3 *A$ O ) GB H*8 !VO 0C (" $ v u &. $0 ).& %- E 0 0 !Y " R# ('&" M R# 0 ) &0 9Z#. $ ( v .$ u i−

i

j

j−

i

i

x

i

i

i−

i

i

i

i

j

i−

00

√ x + y + dA = × dA x + y D D √ √ √ = Area(D) = × π × = π =

# " $ 0 $ Z") Ω M x= A .8 5F ).& ^Y S ( x + z = a x + y = a *M [& . S ('&" *O&0

}

i

j−

i

y + + + + dA x + y x + y

/

∗

j−

j

Area(Sij ) ≈ ≈

=

* * * *i j k * * * u × v = * (∂f /∂x)(x , y )Δx i i i* * * (∂f /∂y)(xi , yi )Δyj *

∂f ∂f (xi , yi ) + (xi , yi ) Δxi Δyj + ∂x ∂y

M Z#S *H*8 0 ) &0 R# 0&0 00

-

Area(S) =

+

∂f ∂x

+

∂f ∂y

dA

D

. R `&UO 0 M ^Y ('&" A& N+M 0 ^Y N# ('&" [& 9H*8 !VO &S$ # U &0 &# −y y # U &0 &# −x x # U &0 _> R# 0&0 *$$S+ &6 B&" >. T .$ v −z (=S h .$ 5 . $. $ . 1 Z 6 N# .$ M +"1 $M 0 0 (6 . !T&' ]< $ + [& . 5F ('&"

S . #. E uL0 R# S *$ O ) % H*8 !VO 0 z

/

'() $ %

*M [&

$ + [& . M #_&0 + ('&" . h R# 0 *!' ZM x= R# 0&0 *ZM 0 0 $ . !T&' S/ : z =

, R − x − y , (x, y) ∈ D

( ?, #E >.&[, 5$ + x= ([a E M R# 0&0 *D : x + y ≤ R 9$ O

Area(S) = Area(S/ )

. \ >. T R# .$ *( f (x, y) (+M B&Q . $ D 0 0 &z[#\ [x ; x ] × [y ; y ] !Y " .$ -&' (+M (= Q *H*8 0&0 .$ 00( f (x , y )Δx Δy 0 0 D 0 f y . \ ]3 *( f (x, y)dA 0 0 D !M 9O&0 D ('&" 0 Z"\ . \ R# i

i−

j−

j

total

i

j

i

j

D

Qmean :=

Qtotal Area(D)

⎛ ⎞ ⎛ ⎞ 00 00 = ⎝ f (x, y) dA⎠ ÷ ⎝ dA⎠ D

=

=

⎛ ⎞ 00 , 00 =⎝ x + y dA⎠ ÷ ⎝ dA⎠

⎞

D

π/

R

dθ

πR

r dr

=

R

i

Pmean

= ()

=

=

j

i−

j

i

i

j−

j

j

⎛ ⎞ # " ⎞ ⎛00 00 x y =⎝ P h − − dA⎠ ÷ ⎝ dA⎠ a b D D # " 00 hP x y − − dA πab a b D 00 hP ( − r )abr dA πab D #0 " #0 " π hP hP dθ r( − r )dr = π

c x = ar cos θ ( O x= ( ) .$ V# ^- T−rθ .$ D # j D J = abr R# 0&0 *y = br sin θ

≤ x ≤ a, ≤ y ≤

= =

=

a − x

Z#. $ ` + .$ R# 0&0

Area(S/ )

= # " 00 > > −x ? + + + () dA a − x D 00 a + dA a −x D ⎡ √ ⎤ 0 a 0 a −x dy ⎦ dx + a ⎣ a − x √ ) a −x 0 a( y + a dx = a a − x 0 $ %

*O&0 T−rθ .$ D [Y1 # j D M 0 ( O & &Q ]) N# E < A, .8 ( x /a + y/b = q0 !VO 0 5F ,&1 M z=h − x /a − y /b 0 0 D E (x, y) Y\ .$ `&. . 5F ,&1 0 < E XUV & N# E $. .&6= S *O&0

*M [& . < ,&1 0 $. y .&6= Z &0 P NQ M !Y " 0 M $ O ) ( =&M . h R# 0 *!' O&0 Δx Δy ('&" 0 M [x ; x ] × [y ; y ] ' f = P z(x , y ) E 0 M $. $ . 1 z(x , y ) `&. 0 y .&6= *H*8 0&0 ]3 *M $. .&6= < ,&1 0 E ( >.&[, i

D :

=

D

00 , 00 + y dA = x r dA πR πR D D " #0 " #0

/

Area(S) =

, + p&\ y T&= A &a '() $ % # $ 0 $

*#.F (0 . Ǳ [ & e0. .$ e1 x + y = R &0 0 0 Ǳ [ & D E (x, y) LB$ Y\ T&= 5 Q *!' + 9( RQ O y ( f (x, y) = x + y dmean

/

D

D

⎛

*M+ v z y # U &0 M ZM h'? Z &0 . 5F ('&" (=S h .$ 5 . #_&0 + R# 0&0 >. T R# .$ Z &0 S . uL0+R# S *$ + 0 0 $ 5F .$ M S : z = +a − x , (x, y) ∈ D

y M 0&0 . x + z = T E +"1 ('&" A *$ O ) x + y = a M 0&0 . x + y + z = R M E +"1 ('&" A; *$. $ . 1 z = R/ T _&0 + . z = R − x − y V+ E +"1 ('&" A *( O #0 x + y = Rx y M 0&0 z = Zg&1 pL 0 $ Z") Z' Ω M . T .$ A O&0 Ω ).& ^Y S z = T x + y *M [& . S ('&"

y M x = (z + y) pL E +"1 ('&" AH *0&0 . ( O ) x = z + y q0 5 S+ *M [& . |x| + |y| + |z| = #. ('&" A8 $ R0 M x + y + z = M E +"1 ('&" AJ . $. $ . 1 x + y = z x + y = z pL

*M [&

.$ V3 Y ) < =C) . !) $ % 0 * & . $0 5 . v Q e0& N#c= K?YT $ $ ([" A$,C B&V &q= E Y\ 0 M U R# .&0 B&Q &# 4) B&Q & $ B&V 5 h $ O

U#C ( U0 $ $& Z") N# D M x= *V# VB E (x, y) Y\ A(" ) $. B$ 0 5F 4 U0

.$ ]3 *( 00 δ(x , y )Δx Δy NQ M UY1 R# 4) R# 0&0 *( m = δ(x, y)dA 0 0 D !M 4) *H*8 0&0 ` +

$ VQ M '& D M x= A& '() $ $ % (x, y) Y\ ( x +y = q0 x = y + 0 *M [& . D 4) *( δ = x 4) B&Q . $ $ $ q0 + $. 0 ! 0 . h R# 0 *!' 9ZM [& . O i

j

i

j

D

x + y = ⇒ y + y − = x = y x = y √ ⇒ y = − ± / ⇒ y = ± / x = y x=

D : −

√

, ≤y≤ , y ≤ x ≤ − y

m

= D

=

x dA = 0

√

−

0

√

−

/

√

/

⎡ √ ⎤ 0 −y ⎣ xdx⎦ dy

√

/

− y − y

#0

√ dy =

#0

⎞

D

x

⎛

(x + y) dA⎠ ÷ ⎝

( 0

= =

y

/

Tmean = ⎝

00

=

&0 ( 0 0 h $. Z") 4) R# 0&0

00

⎛

( Zh −y , + R# *$ O ) GB 8*8 !VO 0 9(O 5 M Q √

?U0 *( D : ≤ r ≤ , ≤ θ ≤ π E >.&[, &Y\+ %qT&' .$ %- π 0 0 D ('&" M Z $

*O&0 5F e0& . $ + . −y 0 $ , + D M x= A

& $ . $ (x, y) ∈ D Y\ M ( y = y y = x y N# .$ . Z") R# l&Q *( T (x, y) = x + y & $ >. T R# .$ 0 . #&3 & $ 0 & Z$ . 1 Ǳ? *M [& . & $ R# *AO (0&f 5F p&\ + m D : ≤ x ≤ , x ≤ y ≤ M Z#. $ ) *!' R# 0&0 ( D 0 T y [&

)

⎞

00

dA⎠

D

"

(x + y)dy dx

#0 ÷ "

( x − x + − x )dx

÷ =

}

≈

( 0

#0

÷

x

)

"

dy dx "

( − x )dx

0 $ %

i1 0 z

+ a − x − y

V+ y `&. A *M [& . x + y ≤ a Y\ O&0 |x| + |y| ≤ E B D M . T .$ A; O&0 |x| + |y| V# VB .&0 B&Q . $ (x, y) ∈ D *M [& . D 0 $ ) .&0 y

&j L &. 0 $ da p&\ T&= y A *M [& . Ǳ [ & x + y = y &$ &0 z = x / + y − $ S y t+, A *M [& . D : x / + y ≤ & D : y ≤ a − x , y ≥ 0 p&\ T&= e0 y AH *M [& . : y = , −a ≤ x ≤ a y .&3 =

∗

H*H*8 &a E & +"1 98*8 !VO x = y + $ 0 $ '& D M x= A, .8 (x, y) Y\ O&0 xy = xy = B Bn $ x = y *M [& . D 4) *( y/x 4) B&Q . $ v = y /x u = xy >&j L v E . h R# 0 *!' >. T R# .$ *ZM $& J

∂(x, y) = ÷ ∂(u, v) = ∂(u, v) ∂(x, y) * * * y x * * * ÷* = y/x * = −y /x

÷

y x

=

v

E >.&[, T−uv .$ D # j D ?U0

D : ≤ u ≤ ,

≤v≤

∗

x= 2 + /DR < QR . !) $ % 0 5F 4 U0 U#C ( U0 $ $& Z") N# D ZM 5F ( &L- ( RV+ ?za *(" Z & ! .$ B$ *( δ(x, y) 4) B&Q . $ (x, y) ∈ D Y\ A*O&0 (0&f NQ M !Y " p&\ 4&+ M $ + x= 5 >. T R# .$ " δ(x , y ) 4) B&Q . $ [x ; x ] × [y ; y ] i

j

i−

i

j−

j

&0 ( 0 0 D 4) R# 0&0 *(

2 + /DR < E1n 4L) . !) & $ %

0 Y\ ( U0 $ $& Z") N# D M x= R# .$ *( δ(x, y) B&Q . $ (x, y) ∈ D >&j L

ZM Z"\ NQ M & Y " 0 . D S >. T

00

Dij := [xi− ; xi ] × [yj− ; yj ]

5F p&\ + B&Q M ZM x= Z O&0 & F E V# D !Y " &0 M ZM x= &' *( δ(x , y ) 0 0 (x , y ) >&j L &0 δ(x , y )Δx Δy 4) 0 $& Y\ N# *Z#. $ $& Y\ mn aM ' D &0 X R# 0 *Z#. $ 0 0 p&\ R# !}\f cM E x o L N#c= /.$ 0&0 ij

i

i

j

x ¯=

i

/

j

i

j

Z#. $ y = J

t1$ . \ Z#0 0' _&0 >.&[, E l&Q ]3 *( 0 m M ( x ¯ = xδ dA 0 0 D ! } \f cM E x o L

0 0 D !}\f cM E y o L 0&6 >. T 000 *( D 4) *y¯ = yδ dA &0 (

D :

D

= y¯ =

=

=

0 0 D 4) X R# 0 *( 00 m

=

πδ R

y dA = x

D

= =

ut

D

ut ( + t ) dA

+t ut

+t

#0 " " #0 t dt a t dA = du + t + t

00 D

00

a ln

*$ O ) % 8*8 !VO 0 *( 0 % $ %

π δ × × πR = δ R 00 00 yδ dA = yδ dA m πδ R D D 00 r sin θr dA πR D " #0

0 π

R

sin θdθ

r dR

* * * ut *= * ( + t ) *

≤ u ≤ a

,

dA = δ Area(D) D

≤t≤

D

00 δ dA = δ

=

t +t t +t

00

+t

E >.&[, T−tu .$ D # j D

m

∂(x, y) = ∂(t, u) * * u(−t ) * (+t ) = * * ut(−t ) * (+t )

D

`&UO 0 # $ Z N# !\f cM A& .8 '() * $ % *0&0 . ( O & R+ ]) N# E M R >.&[, . Mn # $ Z ZM x= d0 .$ (B 0 *!' *δ(x, y) = δ D : x + y ≤ R , ≤ y E ( &F O&0 [Y1 T .$ D # j D S >. T R# .$ R# 0&0 D : ≤ r ≤ R , ≤ θ ≤ π

at , ≤ t ≤

,y= + t y/x

B&Q . $ (x, y) ∈ D Y\ ( *M [&

ut ut x = + t x= &0 >. T R# .$ *!' + t

. D 4) *(

j

m

at

C : x=

i,j

m

=

.&M$ 0 0 $ '& D M x= A .8

/ xi δ(xi , yj )Δxi Δyj ÷ δ(xi , yj )Δxi Δyj

i,j

00 x dA dA = y v v D D " #0 #0 " dv

= du = v

m

=

R π

.$ *x¯ = Z#. $ . −y 0 ([" D 5.&\ !B$ 0 *C = (, R/π) e1 x /a + y/b = 0 $ q0 e0. !\f cM A, *0&0 . ( xy 0 0 (x, y) ∈ D B&Q M e0. .$ D : ≤ x , ≤ y , x /a + y/b ≤ &# .$ *!'

. $ x + y ≤ x i1 E (x,+y) Y\ M . T .$ A *M [& . D 4) O&0 x x + y 4) B&Q y = x = p Y 0 $ '& D M . T .$ A; E (x, y) >&j L 0 Y\ B&Q $ 0 x + y = *M [& . D 4) O&0 (x + y) 0 0 5F (, ) ( , ) (, ) / g. &0 da D M . T .$ A 4) O&0 xy 4) B&Q . $ (x, y) ∈ D Y\ O&0 *M [& . D "0 0 $ '& D M x= A

C:

x=a y=a

!

cos t − cos(t)! sin t − sin(t)

;

≤ t ≤ π

. D 4) *( |y| B&Q . $ (x, y) ∈ D ( *M [&

D + $ 0 . −y N+M 0 . D *$ O ) J*8 !VO 9ZM Z"\ D D D

: :

≤x≤ −

a

a

,

b x ≤ y ≤ h x a

≤ x ≤ , −

=&# Z+U [Y1 >&j L v E *δ(x, y) = xy .$ D # j D >. T R# .$ *ZM $& y = br sin θ E ( >.&[, T−rθ

x = ar cos θ

a

D :

h x ≤ y ≤ − b x a

Z#. $ R# 0&0 *( J = abr c !#[ R0 M W

a

?U0 x¯ = M ( RO. >. T R# .$ m

= =

m(D) = m(D ) = Area(D ) × δ

a a aδ δ × ×h− × ×b = (h − b)

00 m

=

00 δ dA =

D

00

>. T & *( C = (, (h + b)/) 0 0 D !\f cM U# M "#&0 R# 0&0 C = (, b) M ( & E B&"

. ! $ da `&. U# *b = h/ &# (h + b)/ = b N# E . D !VO S *(=S O $ $ da `&. / #&0 0 ZM . E w 0 . C / . Z#E&"0 X' UY1 *. Z C .$ .@ b 6+ Z") N# S *0&0 . D !\f cM $. .$ 0 $ % &+O *( (0&f B&Q M ( 5F 0 x= $ 6 M{ B&Q *M x= N# . B&Q . \ Z

*δ = x ( x + y ≤ x i1 D A x + y = x + y = ax # $ 0 $ , + D A; *δ = y ( ay ?U0 ( (, ) ( , ) (, ) / g. &0 da D A *δ = x + y *( e0. .$ e1 . g F 0 $ , + D A $ y = −x + y = x + + $ 0 D AH *( >_$&U 0 E / 1 N# 0! $ , + D! A8 *( . −x y = a − cos t x = a t − sin t ay = x + x+y = a y 0 $ , + D AJ *( (x + y ) = a (x − y ) 0 $ , + D A7 * ≤ x ( 0 # $ 0 $ &Q 1. N# D M x= A M ( x + y = bx x + y = ax >_$&U

0 0 D !\f cM M M RU . @ . a * < a < b *O&0 (a, )

≤ θ ≤ π/ , ≤ r ≤

=

abr sin θ cos θ abr dA

D

=

a b

#0

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sin θ cos θdθ (

=

xy dA D

sin θ a b

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)π/ ×

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" r dr

a b

&F O&0 D !\f cM C = (¯x, y¯) S .$ 00

x ¯ =

=

=

00

xδ dA = x y dA m a b D D

00 a b r cos θ sin θ dA a b D #0 " #0

a

π/

cos θ sin θdθ

" r dr

=

a

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}

7*H*8 &a E (+"1 9J*8 !VO h $. D da ,&1 ZM x= . h R# 0 _> 0 *O&0 . −x c D 5.&\ . $. $ . 1 . −x 0

0 $ da D M x= A& '() $ % (x, y) ∈ D M ( x + y = y &j L &.

(0 . y = y ' D .& 6S *( x B&Q . $ *#.F D : ≤ x ≤ , ≤ y ≤ − x V# 0 ) &0 *!' *( d = |y − | 0 0 : y = y & (x, y) T&= 00

|y − | .x dA =

I = D

=

0 $ (

=

x

x(y − ) x

−

%−x

)

dx =

0

)

x(y − ) dx x − x dx

D : −

00 m

≤θ≤

00

xδ dA = δ

=

0

= δ

= δ

*( R

. =

>. T R# .$

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)

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0

r cos θr dA

−π/

cos θ

dθ

M ( U0 $ '& D M x= QR +CcG ZM x= *( δ(x, y) 4) B&Q . $ (x, y) ∈ D Y\ Y\ (x , y ) ( D E Δy Δx $&U0 0 UY1 D (x , y ) Y\ .$ D 4) 4&+ ZM x= *( 5F E LB$ R# .$ *δ(x , y )Δx Δy 5 c 0 U# ( O e+) ( >.&[, x

y ' Y\ R# ! & .& 6S >. T ! T&= d (x , y ), M d (x , y ), δ(x , y )Δx Δy E ' D .& 6S ` + .$ ]3 *O&0 y & (x , y ) Y\ E >.&[, y j

j

j

i

ij

ij

i

i

j

i

j

j

i

j

i

j

i

i

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j

j

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>. T R# .$ Z#0 . −y &# . −x . S *( E $ 0 >.&[, X 0 ' .& 6S 00 00 I = y δ(x, y) dA I = x δ(x, y) dA y

D

* & ( . # fJ@ . R := +I /m $, ω (0&f # E (, &0 . D O&0 &0 S N#c= h E 0 0 .&M R# 0 4E_ W . \ Z$ 5 .$ . ' 0 . m E &0 Y\ 4) S ?U0 *( K = /I ω Z E& W . \ R+ 0 Z$ . 1 . & R T&= *(O $ 0 M Y\ ' D Z") & .& 6S 0&6 >. T 0 . M ' D00 uQ `&UO Rl+ (x , y ) >&j L

I = d (x, y), M !δ(x, y) dA >. T 0 X 0 + ! d (x, y), M 5F .$ M *$ + G#U 5 R := I /m Z#. $ AǱ [ U#C M = O 0 *( M & (x, y) T&=

πδ

πδ

=

+ (x − ) + y

0 0 (

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0 0 uQ `&UO R# 0&0

&

, ) (x, y)

00 00 (x − ) + y δ dA = |x| dA D

. !) $ %

D

r cos θdr dθ

D : − ≤ x ≤ , x ≤ y ≤ x +

IM =

+CcG

x

D

) cos θ

r

π/

Iy = m

πδ

y y = x + 0 $ '& D M x= A .8 |x| 0 0 (x, y) ∈ D Y\ B&Q ( y = x+ (x − ) + y *M [& . M = ( , ) Y\ ' D .& 6S *( (, ) p&\ .$ O $ $ y + M Z#. $ ) *!' R# 0&0 . $ $. 0 (− , ) Z#. $ (

, )

(

−π/

δ

y

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00

( π/ 0 cos θ −π/

0

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,

Z8

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D

Iy

π

]

i

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9ZM $& [Y1 >&j L E . h R# 0 *!' π

∗

i

x 0 $ &Q '& D M x= A, .8

x + y =

∗

+

dx =

0 (0 −x

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D

T&= 5 Q

M

D

M

00

IO =

M

! x + y δ(x, y) dA

D

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00 !0&\ .$ . I .& 6S *( M = d (x, y), !δ(x, y) dA E *M + r& s (T &0 Xb M

D

x= &0 . . −y . −x ' .& 6S $. .$ 9M [& δ = )

(x − a) + (y − a) ≥ a ,

=

−

0 =

≤x≤a )

0 (0

,

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x = u(t − sin t) , y = u( − cos t) ,

≤u≤a

,

)

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)

(x + y ) ≤ a xy ,

)

x/ + y / ≤

−

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x

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−

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− x ) dx +

a

a→a

a

Da

00 D = lim Da ⇒

00 f dA := lim

a→a

f dA

a→a

D

Da

9$ + $& 5 $. $ .$ & R# E M ( M{ 0 4E_ $. $ $ ) ( . (+ ' M $ O (0&f S AGB >. T .$C |Q (+ & M (=S Z #n3[& U# *O&0 h $. ' 0 0 A$ ) (" $ ) 0 B$

x

0 x.(x +

− x ) dx

=

+

) B&Q &0 '& D M . T .$ A .8 O&0 x + y = a (x + y) 0 $ (xx ++yy) *M [& . Ǳ [ ' D & .& 6S . D : x + y ≤ a(x + y) '& . h R# 0 *!' 9Z#0 [Y1 >&j L 0 : r cos θ + r sin θ ≤ a r a : r ≤ cos θ + sin θ

R# 0&0 $ O+ T &l cos θ + sin θ 5 Q M D :

≤ θ ≤ π , ≤ r ≤ +

00 IO = D

00

=

(x + y )δ dA =

00

&0 ( 0 0 h $. .& 6S x + y dA (x + y )

D

! cos θ + sin θ .r dA

D

⎡ 0 π 0 ⎣ = 0 π ( = =

a

a

cos θ + sin θ

√

a/

cos θ+sin θ

⎤ ! cos θ + sin θ .r dr⎦ dθ

D

&+ 9 OVBC EJ NC % % E ' 0 . & B R# ( =&M B +U & *ZM !#[ $&, & B E $ & {D } ZM x= . h R# 0 & B E N# lim D = D M O&0 &, +

00 9ZM G#U >. T R# .$ *O&0 $&, f dA

−

− x ) dx

D

O&[ . M &# "0 D , + S % % &# M ! (#& 0 0 D E Y\ .$ z = f (x, y) V# &# Y\ Q00E u0 D 0 f & 3& V# &# $$ G#U + . f dA >. T R# .$ O&0 Q &# *Z &

0 $ %x+ |x|y dx

δ =

/ F DF B !F 00 f dA

|x| dy dx =

|x|.(x +

D

00

)

0

=

≤ t ≤ π

x+

! r cos θ + sin θ

0 π

√ )a/ cos θ+sin θ dθ

dθ = a π 0 $ %

0 . −y 0 ([" uQ `&UO & .& 6S A x = p Y . −x y $ wE& T δ(x, y) = + x/ pO &0 . y = x + x = − *M [&

e0& . $ + . −x 0 $ &Q '& & .& 6S A; . . −y ' π ≤ x ≤ π &0 f (x) = (sin x/x) *M [&

$ y = y = −x y = x p Y 0 D M x= A uQ `&UO & .& 6S *δ(x, y) = y+ ( O *M [& . Ǳ [ ' D

0

()

∞

=

00

e−(x

= ≤x,y ()

=

0 e−x dx +y )

00

lim

R→∞ DR

e

#0

=

π

R→∞

e−r

00

$. $ ) ( . (+ ' M $ O (0&f S A% *$. $ ) |Q (+ & M (=S Z *( R T !M D M x= A& '() % % *M [& . D 0 f (x, 00 y) = /(x + y + ) . M D #E ( & f dA >. T R# .$ *!' 9( R `&UO Ǳ [ cM 0 # $ D ZM x= *(" 5 M *D = lim D M ( RO. *D : x + y ≤ R 9(O 5 [Y1 >&j L v N+V0

e−r dA

dA = lim

R→∞ DR

R

dθ

lim

()

" #0

π/

R→∞

=

e−y dy

dA

−(x +y )

lim

∞

re R =

π

−r

D

"

R

dr

R→∞

− lim e−R = R→∞

00

π

R

R# 0&0 v 5$ 0 & E AC .$ V# ^- * D M ( O x= A;C .$ Z# $M $& RU .$ e0. .$ M ( Ǳ [ .$ cM R `&UO 0 # $ e0. Z# $M $& [Y1 v v E AC .$ $. $ . 1 T *O&0 [Y1 T .$ D # j D √ I = π/

R

00 dA dA = lim R→∞ (x + y + ) (x + y + ) DR 00 rdA = lim R→∞ (r + ) DR

R

" #0

#0 π

=

lim

dθ

R→∞

R

rdr (r + ) R

"

$ %π − θ × R→∞ r + = π lim − =π R→∞ R +

R

R

R

=

lim

9( D [Y1 # j D &# .$ M R

R

DR : ≤ θ ≤ π ,

} e0& A .8 *M [&

, + E . h R# 0 *!'

Da : ≤ x ,

−(x+y)

≤ y , x + y ≤ a

∂(u, v) = ÷ ∂(x, y)

* * ÷* *

* * *= *

!V6 . D `?- y = x = x + y = a p Y 5 Q T−uv .$ D # j D 5$.F (0 0 $

9Z0&# . `?- R# #&j a

a

a

x= ⇒ v=

ε

00

dA = lim − x − y ε→ +

D

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>&j L &.

0 $ '& D M x= A, .8 00 dA ( + &#F *( x + y = y −x−y f (x, y) = /( − x − y) #E ( & R# *!' y R# 0 (x, y) S $ O+ G#U x + y = y 0 D ZM x= *$ + ! (#& 0 0 f e0& $ O N#$c x + y = − ε y >&j L &. 0 $ , +

A7*8 !VOC D = lim D >. T R# .$ *( D

;*8*8 &a E ; (+"1 97*8 !VO T e0. 0 . f (x, y) = xe

00

Dε

ε→ +

=

lim ε→

= = = =

0 −ε $

lim

ε→ +

0 −ε

lim

ε→ +

lim

ε→ +

$

ε

dA −x−y

0 −ε $ 0 −x−ε

% dx

−x−y %−x−ε − ln( − x − y) dx ln( − x) − ln ε dx

− x ln( − x) + x + ln( − x) − x ln ε

%−ε dx

lim (ε − ) ln ε = +∞

ε→ +

*( S h $. U# *M [& . I = 1 e dx A .8 M ZM ) . h R# 0 *!' ∞ −x

y= ⇒ u=v

x + y = a ⇒ u = a

≤r≤R

I

=

I ×I

R→∞ lim

=

#0

×

R

#0

r(m+n)− e−r dr

R# 0&0 D 00

"

π

B(m, n) R→∞ lim

=

"

cosm− θ sinn− θ dθ 0

0

R

√ R

r(m+n)− e−r dr m+n− −s

s e ds B(m, n) lim R→∞ 0 ∞ sm+n− e−s ds = B(m, n) Γ(m + n) B(m, n)

= =

= =

( O x= AC .$ V# ^- *( 4&+ 5&0 v v E A;C .$ D : ≤ x, ≤ y, x + y ≤ R *D : ≤ θ ≤ π, ≤ r ≤ R ( O $& [Y1 00 >. T .$ . f dA $. .$ 0 % % 9M [& $ )

=

00 a→∞ Da

R

D

! ) f = / (x + )(y + ) , D = R /(x y), D :

)

f=

)

f = exp(−x − y ) cos(x + y ), D = R

)

f = (x − y )/(x + y ), D :

≤ x, e

−x

=

=

ve−u 0

a

(

lim

a→∞

v

xe−(x+y) dA

dA = lim

a→∞

0

0

a

u

)v=u e

−u

ve−u dv du

0 du =

a

lim

a→∞

u e−u du

v= $ %a 0 a −u lim − u e + ue−u du a→∞ %a 0 a $ lim −a e−a − ue−u + e−u du

a→∞

lim

a→∞

−a e−a − ae−a − e−a +

a + a + = a→∞ ea

− lim

− lim

a→∞

=

ea

− lim

a→∞

a + ea

.&0 $ Ǳc) 0 Ǳc) D. E .&0 $ _&0 >&[& .$ V# ^- *Z# $ + $& &3 ,&1 E c 9ZM G#U [a $ , n m a M x= A .8

≤y≤

0

≤x≤y

∞

Γ(a) :=

)

f = exp(−x − y), D : ≤ x ≤ y , ) f = / − x − y , D : x + y ≤

xa− e−x dx

0 B(m, n) :=

xm− ( − x)n− dx

M M (0&f * & & 0 e0& & &S e0& X 0 . & # = R# i& (B&' *B(m, n) = Γ(m)Γ(n)/Γ(m + n) ! U[@ $ , n m M = (m + n)!/m!n! 9( RQ * " dt = x dx Z#. $0 Γ(a) .$ t = x x= &0 *!' 0 *Γ(a) = x e dx R# 0&0 t ∞ ∞ Z#. $ B e0& .$ x = cos θ x= &0 0&6 >. T

)

f = y + , D : − ≤ x ≤ , + + − / − x ≤ y ≤ / − x

) f = exp −x /a − y /b ,

m n

D : x /a + y /b ≥ ) f = √xy exp −x − y , D =

)

00

a→∞ Da

lim

R

=

≤ v ≤ u E ( >.&[, Da ]3

xe−(x+y) dA = lim

≤x,y

=

: ≤ u ≤ a,

a

∞

e0. f = sin(x + y ), D = 4$ e0.

B&Q . $ R E (x, y) LB$ Y\ M . T .$ A ' R & .& 6S O&0 δ = exp(−x − y) 4) *M [& . >&j L Ǳ [ . −y . −x , >. T R# .$ *Ω : ≤ z ≤ / x + y M x= A; ^Y ('&" B ( & Ω Z' M $ 5&6 [& R# E &#F ( & & 5F ).& (=S 5

0 $ , + 0 . f (x, y) = y e0& A 9M [& #E x = a ( cos t − cos(t)) * C : y = a ( sin t − sin(t)) , ≤ t ≤ π

∗

B(m, n) =

0

a−

π/

−x

cosm− θ sinn− θ dθ

0 ∞ 0 Γ(m)Γ(n) = xm− e−x dx 00 = xm− y n− e−(x +y ) dA x≥ y≥

( )

=

R→∞ lim

00

xm− y n− e−(x

+y )

.$

y n− e−y dy

∞

dA

DR ( )

=

R→∞ lim

00

DR

r(m+n)− e−r cosm− θ sinn− θ dA

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http://webpages.iust.ac.ir/m_nadjafikhah/r2.htm

*( O $.F MnB = d'&[ 0 p 0 e0& & B&a

\' 0 $ , + 0 f (x, y) = x e0& A 9M [& . O $ $ .$ $ )

*C : x = at/(t + ) , y = at/(t + ) y y M O&0 e0. E +"1 D S M M (0&f AH &F ( O #0 x + y =

00 exp

y−x y+x

dA =

D

e − e

y y M O&0 e0. E +"1 D S M M (0&f A8 &F ( O #0 x + y = π/ 00 sin x sin y sin(x + y) dA = D

π

y y M O&0 e0. E +"1 D S M M (0&f AJ &F ( O #0 x+y = 00 0 xm− y n− f (x, y) dA =

Γ(m)Γ(n) Γ(m + n)

tm+n− f (t) dt

D

M (0&f v = y u = x +y v) v E $& &0 A7 0 (0

−x

exp

y x+y

dy dx =

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+ , -$ P

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& ' #P := min n, m, ,

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: e = z < z < . . . < z − < z = g.

Δxi := xi − xi− ,

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F B !F U& 2 .$ M ( &S$ [O &S +. G#U *O $.F **8 }

[xi− ; xi ] ∩ [yj− ; yj ] ∩ [zk− ; zk ] ∩ Ω

.$ w O R# 0 0 . Ω O&0 ! &M !Y " XUV

&# & Ω R# 0&0 *Z#S ∅ . 5F >. T R# b GB ;*J !VO 0C " &# &M !Y " XUV

*J !VO E $ & P E = E KV N# E . h *A$ O ) . M "0 Ω ⊆ R ZM x= & M S 0 ( Ω .$ X = &(x , y , z )' p&\ $ ) . @ g e d c b a $ , R# 0&0 *O&0 XUV .$ Ω U# *Ω ⊆ [a; b] × [c; d] × [e; g] M (O $ *Ω = ∅ M k j i E 0 x , y , z ! ∈ Ω E&0 0 0 . −x 0 5F # j M ( p& !0&1 Y "

P B< Ω , + ZM x= & [e, g] E&0 0 0 . −z 0 [c; d] E&0 0 0 . −y 0 [a; b] u = f (x, y, z) O&0 ;**8 &+ X KV 0 B< N# E . h *A$ O ) GB *J !VO 0C ( >. T R# .$ *$$S G#U Ω! 0 M ( U0& 5F .$ M P = P × P × P !VO 0 ( 0qT&' Ω e+T&' 7 f x , y , z Δx Δy Δz E . h 9 " [e; g] [c; d] [a; b] 0 #&E = X 0 P P P j i E 0 M ( O $ $ !V60 & .&[, + P : a = x < x < . . . < x < x = b, e0& Zg S . T .$ *Ω = ∅ h $. k P : c = y < y < . . . < y < y = d, I $, M ( ]oH/ Ω , + 0 u = f (x, y, z) J ijk

ijk

ijk

ijk

ijk

ijk

P

ijk

ijk

ijk

i

ijk

ijk

ijk

j

ijk

ijk

k

n−

n

m−

m

( [; 000 Ω

]

' XUV Ω M x= '() & >. T R# .$ *f = xy − z

n / m / / i j k , , f dV = lim f m,n, →∞ mn n m i= j= k=

n / m / / ij k − m,n, →∞ mn mn i= j= k= ⎧ ⎫ ⎛ ⎞ # n " m ⎨ ⎬ / / / ⎝ ⎠− = lim i j k m,n, →∞ ⎩ m n ⎭ i= j= k=

m+ n+ + lim lim − lim = n→∞ m→∞ m n →∞ − = =

lim

#n. T R# .$ O&0 \\' $, a O&0 000

)

000

af dV = a

f dV

Ω

)

Ω

000

000

(f + g) dV = Ω

000 f dV +

Ω

g dV Ω

f (x, y, z) ≤ g(x, y, z) (x, y, z) ∈ Ω E 0 S A >. T R# .$ 000

000

f dV ≤ Ω

g dV Ω

&0 Ω Ω , + $ w O $. '& Z' S A >. T R# .$ O&0 T ('&"

000

000

f dV = Ω ∪Ω

000

f dV + Ω

f dV Ω

& g f & F .$ M Ω E @&\ h $. Z' S AH >. T R# .$ O&0 T 000

000

f dV = Ω

g dV Ω

.$ O&0 T Z' &0 Ω Ω , + $ m? S A8 >. T R# 000

000

f dV = Ω

f dV Ω

&F Z$ 5&6 Vol $&+ &0 . Ω Z' M . T .$ AJ 000 dV = Vol(Ω) Ω

N# δ ([a $, N# < ε E 0 M $$S (=&# 5&Q 0 P E = E 0 M O (=&# S 0 n U[@ $, O $ P E X %&L E 0! #P ≤ δ |P | ≥ n M Ω 7 Δx Δy Δz − I < ε ZO&0 R# .$ * f x ,y ,z 000 f dV $&+ &0 & Ω f )* 3)+ ./ . I >. T 9$ O O (B&' R# .$ M ( +U ?U0 *Z$ 5&6 ijk

P

ijk

ijk

i

j

k

Ω

000

f dV = lim

|P |→∞

#P →

Ω

/

! f xijk , yijk , zijk Δxi Δyj Δzk

P

?z+, $. $ h [) _&0 G#U 78 & 5F $& R#+, (" &S [& 0 *O&0 &S &$0.&M ) c & &[f &1 .$ UB&Y $. & B $ ) E M ( 4E_ d0 $ 0 *M !' . B&" &z 1 #E q1 *ZM !T&' 5&+@ $ & . M "0 , + Ω S #$S G#U Ω 0 M O&0 x U0& u = f (x, y, z) Q &# N# Y\ Q &# N# E h mT ( . M &F O&0 3 Ω !M 0 & ('&" &0 #. Q &# N# *$. $ $ ) Ω 0 f $ ) 0 =&M y# O _&0 q1 78 % & * " 4E_ ) 0 y# O R# M =U . O&0 LB$ f T Z' &0 Ω &# f = LB$ Ω S ?za

*( T 0 0 O $ $ ) Ω 0 f &F y# O .$ f e0& 0Ω00 , + l&Q & & Zg S T . f dV M T H**J q1 + ]3 R# E *Z & + . 5F >. T R# b .$ 5&0 &z ' T u=? VF $&, d0 $. & B *$$S G#U N+M 0 &S [& 0 !#{ q1 E *$ + $& 5

[a; b]×[c; 000 d]×[e; h] !Y " XUV Ω S * & 0 0 f dV >. T R# .$ O&0 #n3 Ω 0 f $ 0 Ω

Ω

lim

(b − a)(d − c)(h − e) × mn

n / m / / d−c h−e b−a ,c+ j ,e + k f a+i n m

n,m, →∞

i= j= k=

M $M x= 5 [& .$ (B ( ) 0 [B *( *$ + [& . ' N# & m = n =

# U &0 M A$ O ) % ;*J !VO 0C ZM h'?

.$ *Z. Ω , + 0 Ω .$ −x 0 x +

000 =

Ω

mVol(Ω) ≤

000 x dV +

Ω+

f dV ≤ M Vol(Ω) Ω

x dV

000 000 f dV ≤ |f | dV

*

Ω−

000

000 x dV +

=

0 f . \ R# +M M 0 0 Ω 0 f . \ R# 60 S A7 >. T R# .$ O&0 m 0 0 000

−

000 f dV

Ω

Ω+

−x dV =

Ω+

% $+ x + y + z

0 f =

Ω

Ω

e0& E A&

A

'() &

*#0 Ω : x + y + z ≤ "0 S 0 f M $M Z"\ 5 . @ . Ω >. T R# .$ *!' 9O&0 (0&f UY1

**J &a 9;*J !VO 000 f dV Ω

. \ $. .$

) f (x, y, z) = [x + y + z] , Ω :

) f (x, y, z) = ,

≤ X < X ≤

⇒ ≤ X <

000 f dV

9M [&

≤ x, y, z ≤

≤ X < ⇒ X <

Ω :

≤ X <

(+"1 0&0 ]3 *A$ O ) GB ;*J !VO 0C ( T (O 5 I**J E 000

000

f dV = Ω : |x| + |y| + |z| ≤

≤ x, ≤ y, ≤ z, x+y+z ≤

) f (x, y, z) = x + y + z , ) f (x, y, z) = z,

⇒

' & Ω − Ω ∪ Ω ∪ Ω ∪ Ω = X X =

0 &

0 Rg&3 ' N# _&0 ' N# $. .$ 90&0

) f (x, y, z) = xy,

f (X) = X ∈Ω

0 0 0 Ω : ≤ X < 0 f 0&6 >. T 0 M Z' ?U0 *( 0 0 Ω : ≤ X <

Ω : x + y ≤ z ≤ − x − y

) f (x, y, z) = xyz, Ω

0 f 0&6 >. T 0 *( T 0 0 Ω : X < 0 f ]3 9( N# 0 0 Ω : ≤ X <

}

.

f (X) = ⇒ X ∈Ω

Ω

000 = Ω ( )

=

=

Ω : |x| + |y| + |z| ≤

Ω : x + y ≤ , x + y ≤

\O*'−xy DYG>X' 2 &# .$ &S$ $. .$ O KY d'&[ 0 h [& 0 Z\ " b O. M ( &0 ?z &M c Zh −xy , + 4 d0 R# \ *$ O KY &S *( Oxyz &q= .$

=

f dV

Ω ∪Ω ∪Ω ∪Ω

dV

000

000

+

dV + Ω

Ω

dV

000 +

dV

Ω

+ Vol(Ω ) + Vol(Ω ) + Vol(Ω )

π × − π ×

+ π× − π×

+ π× − π× ! π + + = π

*f = x Ω : x + y ≤ z ≤ M x= A, .8 *M [& . Ω 0 f Ω Z"\ &0 #E ( T Ω 0 f >. T R# .$ *!' pL Z $ 0 Ω+ : x + y ≤ z ≤

,

Ω− : x + y ≤ z ≤

, x≤

≤x

}

e1 .$ * " &6 . Z# $ + !' ?z[1 M #&" $.

& F Z (+'E ZM Z" . & F & VO Z &0 *Z#+ $ 0 . 0 $ Z' Ω ZM x= A& '() & *O&0 y = x = x + y + z = &j L >&T ( Zh −xy , + Ω &#F !Y " E >.&[, T−xy 0 Ω #& _z #E 0 *!' D :

≤ x ≤ , ≤ y ≤

.$ . D ∈ (x, y) E .nS -= y S &f .$ ( D$. 0 Y\ R#Rg&3! M $M Z &6 Z#0 h *O&0 x, y, − x − y &0 0 0 R#_&0 (x, y, ) 0 0 Ω &0 ` + .$ ]3 *(x, y) = − x − y h(x, y) = R# 0&0 Z#. $ Ω : (x, y) ∈ D ,

≤z ≤−x−y

*$ O ) GB *J !VO 0 ( Zh −xy , + Ω }

Zh −xy , + 9*J !VO . T .$ . Ω ⊆ R , +

& z = h(x, y) e0 D ⊆ R , + M Zg S `g−xy M O (=&# S 0 z = (x, y) Ω : (x, y) ∈ D, h(x, y) ≤ z ≤ (x, y)

e0& . $ + 0 _&0 E M ( &q= E +"1 Ω #$ 5&0 0 m@ E z = h(x, y) e0& . $ + 0 Rg&3 E z = (x, y) 0C ( D , + E 5F ,&1 M ( $ 0 *A$ O ) GB *J !VO ( Zh −xy &# . M "0 , + & R O !0&1 Zh −xy &, + E ,&+ ) !VO 0 &# R# E & $ w O $. '& Z' M S 0 ( 9( T 0 0 &, +

< Ω= Ω ∀i = j : Vol(Ω ∩ Ω ) = i

*;*J &a E ; & +"1 9*J !VO , + Ω : x +y +z ≤ S M $ 5&6 A, .8 *( Zh −xy >.&[, 0 z X"'+0 O $ $ &" & !' &0 *!' & *Z. |z| ≤ − x − y &# z ≤ − x − y T−xy 0 Ω #& ]3 O&0 ([a &V#$ . #E ( ?, #&0 D : x + y ≤ i1 E >.&[, .&[, X 0 h ( + + ) % *J !VO 0 − x − y − − x − y E .$ *$ O Ω : (x, y) ∈ D , −

,

,

− x − y ≤ z ≤ − x − y

[B 0 M mn' E !T&' Z' Ω M x= A .8 x = ± >&T 0 $ 3 XUV E x + y + z < 5&6 *$ O ) GB H*J !VO 0 *( z = ± y = ± *(" Zh −xy , + Ω >. T R# .$ M $

i

j

i

6=M)

<

+ J

U8

K+ &

.& .$ Ω Zh −xy , + M x= /lU)−xy .&M R# _z +U *Z0&# . T−xy 0 Ω #& 0 *O&0 & F G#U $ RU c Ω $' mU >_$&U !' &0 *J !VO 0 O&0 !+, R# D ZM x= *( "

E =S h .$ D .$ (x, y) -= Y\ *$ O ) % B A ZM x= *Z#. _&0 0 Rg&3 ,

.$ *O&0 Ω , + &0 $. 0 Y\ R#F RB X 0 *B = x, y, h(x, y)! A = x, y, (x, y)! >. T R# 0 =. =. B +U !#&" E X& $ U !' &0 [B D Z !VO Z 50 M Z. y" E U- N+V0 & >_&' .&"0 .$ ?U0 *ZM oL6 . h RQ .$ *$ + oL6 . B&" $' 5 & #&" & !'

, + &#F M M oL6 $. .$ 0 $ & $' $ 0 ([a &3 S * &# ( Zh −xy O $ $ 0 . , + >. T R# b .$ M oL6 . , +

9M Z"\ Zh −xy >&UY1 *( x + y + z = x M 0 $ , + Ω A >&T x + y = 0 $ , + Ω A; *( z = − T &j L

T &j L >&T 0 $ !Y " XUV Ω A *( z = x = y = − + Rg&3 E z = x + y pL 0 $ Z' Ω A *( _&0 E z = T z = −x −y q0 5 S+ $ 0 $ Z' Ω AH *( z = x + y x + y = $ R0 $ '& Ω A8 *( z = z = T $ x + y = x +y +z = x +y +z = M $ R0 Z' Ω AJ *(

)

|x| + |y| + |z| ≤

) x ≤ x + y + z ≤ x )

x + y ≤ , x + z ≤

& ' ) max |x|, |y|, |z| ≤

)

x + y + z ≤ , z ≤

)

x + y / ≤ , x + z / ≤

E Zh −xy , + [O (.$ 78 % & *(S RL 5 Zh −yz &# Zh −xz &, +

Ouvw 5 &q= N# .$ Zh −uv , + E M . @ 0 0 [O ?z &M Zh −uv &, + M i *(S RL *( Zh −xy &, +

pL 0 $ Z' Ω M x= A& '() & & Ω >. T R# .$ *( y = T x + z = y !VO 0 ( Zh −yz Zh −xy Zh −xy , +

X 0 D D D S \1$ 5&0 0 *$ O ) 8*J T−yz T−xz T−xy 0 Ω # j E >.&[, &F O&0 yz

xz

xy

Dxy :

≤ x ≤ , −x ≤ y ≤ x

Dyz :

≤ z ≤ , −z ≤ y ≤ z

}

*;*J &a E (+"1 9H*J !VO T−xy 0 5F #& #E (" Zh −xy Ω , + _> B&' .$ ( D : − ≤ x ≤ , − ≤ y ≤ e0 0 0 , + .$ z #&0 &F Z#0 (, ) Y\ . (x, y) S M M B&' .$C (" Q.&
Ω = Ω ∪ Ω ∪ Ω

Ω : (x, y) ∈ D , − ≤ z ≤ , Ω : (x, y) ∈ D , − ≤ z ≤ − − x − y , Ω : (x, y) ∈ D , − x − y ≤ z ≤

pL 0 $ √, + Ω M x= A .8 .$ M $ 5&6 *O&0 z = x + T x + y = z *( Zh −xy , + Ω >. T R# $ $ $. 0 . T pL B$&U . h R# 0 *!' M Z#S

x + y = z √ z = x +

⇒

(x − )

+ y =

3 q0 E ( >.&[, T−xy 0 Ω # j nB D : (x − ) / + y ≤

l&Q #F 0 GB 8*J !VO E M . @ 5&+ Rl+ h .$ . . −z > E

0 (x,! y) E .nS -= y + Y\ R# _&0 x, y, x + y Y\ R# Rg&3√Z#0 R# 0&0 *O&0 x, y, (x + )/ Ω : (x, y) ∈ D ,

, √ x + y ≤ z ≤ (x + )/

*$ O ) % J*J !VO 0 x + y + z = M 0 $ Z' Ω M x= A .8 &q= Z .$ M ( x / + y/ + z/ = 5 S q0 Zh −yz , + Ω >. T R# .$ *( O e1 ≤ x E ( >.&[, T−yz 0 '& R# # j _z #E ( x = # $ .$ . #V# . Mn #. $ D : y + z ≤ R# 0&0 *M eY1 y + z = Ω : (x, y) ∈ D ,

,

Dxz : − ≤ x ≤ , −

+

+

− x ≤ z ≤ − x

9(O $ O #E !VO 0 . Ω

, , Ω : (x, y) ∈ Dxy , − y − x ≤ z ≤ y − x + Ω : (x, z) ∈ Dxz , x + z ≤ y ≤ , , Ω : (y, z) ∈ Dyz , − y − z ≤ x ≤ y − z

,

− y − z ≤ x ≤ − y − z

Zh −xz &# Zh −xy >. T 0 . , + R# l&Q *ZM Z"\ (+"1 0 . Ω M ( 4E_ Z"# 0 ( , , ) (, , ) / g. &0 ) k3 Ω M x= AH &a

!VO 0 *O&0 &q=−uvw .$ (, , ) ( , , ) (, , ) ( Zh −uv Ω >. T R# .$ *$ O ) 7*J ,&+ ) >. T 0 F & *( Zh −vw Zh −uw #&0 0 `O 0 *(O 5 Zh −uv , + E 0 T−uv 0 C D p&\ # j *$M Z . Ω E . $ + R# 0 *C = (, , ) D = ( , , ) E ( >.&[, X OC AB U- .& Q E >.&[, T−uv 0 Ω # j X *$ 0 }

}

(+"1 *;*J &a AGB 98*J !VO (+"1 J*;*J &a A% Q.&3 $ 5 S B Bn 0 $ Z") Ω M x= A, .8 Zh −yz , + Ω >. T R# .$ *( y +z = x − #. $ R# $. 0 &0 *$ O ) GB J*J !VO 0 ( # $ $ E ( >.&[, $ R# $. 0 ! M $ O h'?

0 0 T−yz 0 $ # j M y + z = , x = ± R# 0&0 *O&0 ,D : y + z ≤ , Ω : (y, z) ∈ D, −

y + z −

≤x≤

y + z −

&# Zh −xy >. T 0 . , + R# ZO&0 !#& l&Q *ZM Z"\ uL0 0 . Ω "#&0 Z"# 0 Zh −xz }

H (+"1 J*;*J &a 97*J !VO .& Q OBD da # j X 0 U U U ZM x= E Y\ *O&0 T−uv 0 OAC da ABDC Uy u = v B$&U 0 OA y $. 0 ! E ( >.&[, *E = ( , /, ) U# T−uv .$ u = B$&U 0 BD .$

U

:

U

:

U

:

≤u≤

≤ v ≤ u ≤ u ≤ , ≤ v ≤ u/ ≤ u ≤ , u/ ≤ v ≤ u ,

. 1 5F 0 OBD da M T B$&U V# 0 ) &0 E ( >.&[, $. $ u− − −

v− w− − − − −

= ⇔

u = v + w

; (+"1 J*;*J &a 9J*J !VO + & 0 $ , + Ω M x= A .8 .$ *( y + z = y = >&T x = − z x = z $ _z #E ( Zh −xz , + Ω >. T R# $ .$ & F $. 0 ! . $ . 1 . −y $ .$ O $ $ $ R# $. 0 ! &f .$ *( x = T .$ e1 y E ( >.&[, y = T &0 D : − ≤z≤

, z ≤ x ≤ − z

Ω : (x, z) ∈ D ,

(O 5 .$

≤y ≤−z

da E ( >.&[, T−uv 0 da R# # j 5 #E !VO 0 F #E Z") .$ U = OBD >. T 0 *Ω : (u, v) ∈ U , ≤ w ≤ u − v 9(O >. T 0 . T−uv _&0 ABD da #E Z") 0&6

9Z$ 5&6 Z #E

c Zh −xy Zh −xy , + 5 U0 F *"# 0 Zh

, + 5 U0 . |u| + |v| + |w| ≤ , + AH *"# 0 Zh −uw Zh −uv Zh −uv (, , ) ( , , ) ( , , ) (, , ) / g. &0 4 A8 Zh −uv , + >. T 0 . &q= −uvw .$ *"# 0 Zh −uw Zh −uw (u − v) + (u + v) = X#. 0 $ Z' AJ 5 U0 . &q=−uvw .$ w = w = − >&T *"# 0 Zh −uw , +

u = T v +w = 0 $ Z' A7 Zh −uv , + N# 5 U0 . Z 6 N# .$ e1 *"# 0 −yz

Ω : (u, v) ∈ U ,

>. T 0 . OAC da _&0 ADC da #E Z")

Ω : (u, v) ∈ U , u/ − v ≤ w ≤ − u − v

*Ω = Ω ∪ Ω ∪ Ω .$ *(O 5

5 S q0 S M x= A4 .8

(ρ + ) + (ρ + ϕ + ) + (ρ + ϕ + θ + ) =

F B ]' !F $2 5 U0 M ZM KY .

uL0 R# .$ *$S . 1 $& $. &S B&" !' 0 [&

*O&0 &S$ 0 [O ?z &M ` - R# ( Zh −xy , + Ω ZM x= & Ω : (x, y) ∈ D , h(x, y) ≤ z ≤ (x, y)

>. T R# .$ *( 3 Ω 0 u = f (x, y, z) 00 (0

000 f dV = Ω

/

D : (ρ + ) + (ρ + ϕ + ) ≤

E ( >.&[, !M .$ Ω $

f (x, y, z)dz dA D

h(x,y)

[& 0 G 'BC EJ B=B] & 0 . Ω 0 4cB >. T .$ Ω , + 0 f &S ?za C Z"# Zh −xy &, + E ,&+ ) >. T R0 w O $. '& Z' M . @ AΩ = Ω ∪ · · · ∪ Ω 0 **J q1 N+M 0 f E ]< *O&0 T & F E & $ . !T&' #$&\ F ($ Z#S & F E N# 9ZE e+) k

000

000

000 f dV + · · · +

f dV = Ω

, + Ω >. T R# .$ *O&0 S 0 $ Z' Ω #&0 T−ρϕ 0 Ω # j D R =&# 0 *( Zh −ρϕ ! ]< ZM /&+ S :.& 0 . . −θ E Y .$ N# ZM oL6 . T−ρϕ &0 D$. 0 x= *O&0 D , + 5F ! $ M $ 0 T−ρθ S *O&0 : ρ = ρ , ϕ = ϕ -= y R# ZM "#&0 &F O&0 X 5F /&+ ! O&0 /&+ S 0 .$ (ρ + ) + (ρ + ϕ + ) + (ρ + ϕ + θ + ) = *θ = − − ρ ± { − (ρ + ) − (ρ + ϕ + )} #&0 θ .$ &V#$ . &z cB ]3 O&0 Y\ N# & #&0 X & 3 q0 E ( >.&[, D .$ O&0 T

)

(x,y)

Ω

≤w ≤−u−v

f dV Ωk

0 $ Z' Ω M . T .$ A& '() & O&0 x + y/ + z/ = T >&j L >&T *M [& Ω 0 . f (x, y, z) = z Z"# Zh −xy !VO 0 . Ω 0 . h R# 0 *!' 9A$ O ) GB *J !VO 0C

Ω : (ρ, ϕ) ∈ D ,

,

− (ρ + ) − (ρ + ϕ + ) ≤ θ , − (ρ + ) − (ρ + ϕ + ) ≤− −ρ−ϕ+

− −ρ−ϕ−

0 * &

, + 5 U0 . u + v + w = vw A *"# 0 Zh −vw Zh −uv q0 5 S + $ R0 Z' Ω M x= A; >. T 0 F *( x = − y − z x = y + z *"# 0 Zh −yz , + N# (, , ) (, , ) ( , , ) (, , ) / g. &0 4 A *"# 0 Zh −yz Zh −xy , + 5 U0 . >&T 0 $ o1& pL Ω M . T .$ A O&0 y = x + z @L ^Y y = y =

}

D :

≤x≤

,

Ω : (x, y) ∈ D ,

≤ y ≤ (

− x)

( − x − y)

≤z≤

Z#. $ **J 0 &0 .$ 00 (0 (−x−y)/ ) z dV = z dz dA

000

000 f dV =

; (+"1 **J &a 9*J !VO Zg&1 0 $ Z' Ω M . T .$ A .8 O&0 x + z = y + Q.&
Ω

Ω

=

&, + $. .$ **J 0 [O ZV' 0&0 .$ Z#. $ Zh −xz 000

000

|y| dV =

f dV = Ω

⎡

00

⎣

=

0

00

Ω

=

√

y dy ⎦ dA = ()

D

=

x +z −

−

⎤

x +z −

(x + z − ) dA =

#0

⎣

" #0

π

dθ

0 (0 (−x)

=

x +z −

00

|y| dy ⎦ dA

√ x +z − [y ]y= dA

=

(r − )r dA

0

( − x) dx =

>&T 0 $ Z' Ω M . T .$ A, .8 0 . f = |x|y O&0 y = x + z = *M [& Ω Z"# Zh −xy !VO 0 . Ω 0 . h R# 0 *!' 9A$ O ) % *J !VO 0C y+z =

, x ≤ y ≤

D : − ≤x≤ Ω : (x, y) ∈ D ,

≤z≤

−y

Z#. $ **J 0 &0 .$ 00 (0 −y

000

000 f dV =

x = r cos θ ( O x= ( ) .$ V# ^- E ( >.&[, T −rθ .$ D # j D .$

z = r sin θ

Ω

≤r≤

pL 0 $ !VO M$ Z' Ω M . T .$ A .8 &q= Z .$ M O&0 x + y+ + z = M y + z = x *M [& Ω 0 . f = x y + z $. $ . 1 ≤ x Zh −yz , + !VO 0 . Ω 0 . h R# 0 *!' 9A$ O ) % I*J !VO 0C Z"#

− y − z

xy dV = Ω

=

) |x|y dz dA

D

00 $ 00 %z=−y |x|yz dA = |x|y( − y) dA z=

D : y + z ≤ , , Ω : (x, y) ∈ D, y + z ≤ x ≤

dx

y=

"

π

D : ≤ θ ≤ π ,

)

( − x − y) dy

(r − r) dr

( ) $ %π r r = θ × − =

D

)y=(−x) 0 ( −( − x − y)

=

z=

D

D

00

⎤

√

0

D

√

D

00

⎡

D

D:

+ ≤ y ≤ x + z −

D

00 ( )z=(−x−y)/ 00 z dA = ( − x − y) dA

D

D

)

0 (0

=

−

0 =

−

0 =

=

−

|x|y( − y) dy dx

x

(

|x|

y

4

|x|

0 4 x

− −

−

y

)y= dx

y=x

x

x

x

+

+

x

5 dx

5 dx =

π/

(0 cos θ

−π/

cos θ

0 =

0 =

0 =

(

π/

−π/

$

π/

−π/

)

r − r sin θ + r sin θ dr dθ

r − r sin θ + r sin θ

)r= cos θ dθ r= cos θ

cos θ − sin θ cos θ

+ sin θ cos θ

%

f dV = Ω

3 . 0 E >.&[, Ω M x= A4 .8 [& Ω 0 . f = |s| + |t| + |n| O&0 &q=−stn .$ *M 0 t −s 0 s # U &0 M ZM ) . h R# 0 *!' x= l&Q nB f M v Ω −n 0 n &# −t . 1 &q=−stn Z 6 N# .$ M O&0 Ω E +"1 Ω ZM 9A$ O ) % *J !VO 0C "#&0 &F O&0 $. $ |s| + |t| + |n| ≤

000

Ω

:

Ω

:

000 Ω

≤s≤

≤t≤ (s, t) ∈ D , ≤ n ≤

−s

,

−s−t

Z#. $ ` + .$ ]3

000

(s + t + n)dV

) 00 (0 −s−t (s + t + n)dn dA =

√ )x= −y −z 00 ( , x = y + z x=√y +z dA D 00 00 , () ( − y − z ) y + z dA = ( − r )r dA = D

#0

" #0

π

=

dθ

" (r − r ) dr

D

=

π

$& z = r sin θ y = r cos θ v v E ( ) V# ^- *Z# $ + }

(+"1 **J &a 9I*J !VO # " & 0 $ Z' Ω ZM x= A .8 u+w = w = >&T u +v = u u +v = u *M [& Ω 0 . f = w O&0 0C Z"# Zh −uv !VO 0 . Ω . h R# 0 *!' 9A$ O ) GB *J !VO

00 $ %n=−s−t (s + t + n) dA

−uv

n= D

00

! − (s + t) dA

D

)

0 (0 −s &

' − (s + t) dt ds

0

t=−s = ds t − (s + t) t= 5 0 4 s = −s+ ds =

≤w ≤−v

Ω : (u, v) ∈ D ,

D

=

−y −z

D : u ≤ u + v ≤ u

Ω

=

⎣ √

⎤ , x y + z dx⎦ dA

Ω

f dV =

=

Ω

000 f dV =

f dV

(, , ) ( , , ) (, , ) / g. &0 4 N# Ω & −st , + Ω .$ *( &q=−stn .$ (, , ) 9( Zh

D

, x y + z dV

0 √y +z

D

r cos θ ≤ r ≤ r cos θ −π/ ≤ θ ≤ π/ , cos θ ≤ r ≤ cos θ

:

⎡

00 =

dθ = π

:

000

000

$& v = r cos θ u = r sin θ >&j L v AC .$ M E ( >.&[, T−rθ .$ D # j D ( O D

&, + $. .$ **J 0&6 ZV' 0 &0 .$ Z#. $ Zh

−yz

&, + $. .$ **J 0&6 ZV' 0 &0 .$ Z#. $ Zh

f dV =

Ω

00 (0 −v

000

000

w dV = Ω

=

00 ( )w=−v w D

()

=

00 D

w=

) wdw dA

D

00 dA =

( − r sin θ) r dA

D

( − v) dA

}

) f = , Ω : x + y + z ≤ R x +y +z )

f = |s| + |t|, Ω : |s| + |t| ≤ , |s| + |n| ≤

F B !F - ^_' S& (2 !V6 .&"0 &# !' !0&1 b & B&' 0 Xb &S v 4&0 F .&M O. ` 0 0 .&Q& M $ O

e0& _z +U *Z# O >?V6 S R# e=. 0 >&j L

K?T q \ 0 . $. $ &# $.

*$ + 5

v N# E . h @PC\) f & ( F : R → R #n3/ VU U0& &q= .$>&j L

&0 F (u, v, w) = x(u, v, w), y(u, v, w), z(u, v, w) !VO 0 M . @ Γ $0 Ω $ ∂(x, y, z) = J = ∂(u, v, w)

(u, v, w) ∈ Ω E 0 A*J C

*Z & F >&j L v R0 M W . J #E N# 0 A*J C pO l&Q 78 & (.$ d0 $ Z E&0 O&[ . 10 Ω E T Z' &0 , +

R# !B$ *Z $ >&j L v . F 5&l+ & ( *O&0 &S [& .$ >&j L v R# E $& M x= A&

'() &

F (u, v, w) = u + v, v + w, w + u

u − y + z

,

y − z + n z − n + y . ,

* * ∂(x, y, z) * =* J = ∂(u, v, w) * *

Z#. $ Ω !M 0 &f .$ * * * * = = * *

F (u, v, w) = w cosh u cos v, w cosh u sin v, w sinh u

!

0

0

π

π

sin(x) cos(y + z) dz dy dx 0 π (0

)

π/

(0

)

)

ρ sin ϕ dρ dϕ dθ

) sin(z) dy dz dx −z ⎡ √ ⎤ ⎤ 0

0 (0 −x 0

)

x

⎡ 0 0 √−y ⎣ ⎣ ) 0

)

−

)

−x −y

z dz ⎦ dy ⎦ dx

⎡

⎡ ⎤ ⎤ 0 0 √y−x + ⎣ ⎣ √ x + z dz ⎦ dy ⎦ dx x

−

0 (0 (0 √ z

y−x

ln

) ) e x sin(πy ) x dy dz y 000 f dV

9M [& .

$. .$

Ω

f = z, Ω : x + y ≤ z ≤ − x − y , ∗

) f = x + y , Ω : x + y + z ≤ x

)

f = x y , Ω : |x| + |y| ≤ , |z| ≤

)

f = xyz, Ω : ≤ x ≤ y ≤ z ≤

) f = |z|, Ω : x + y ≤ , x + z ≤

)

f = x + y, Ω : ≤ y ≤ x ≤ , ≤ z ≤ y

)

f = u + v, Ω : u + v ≤ u, |w| ≤ u

f = uα v β w γ , Ω : u + v + w ≤ + ) f = y x + z , Ω : x + z ≤ y + , ≤ y ≤

) )

M x= A, .8

π

)

)

*( >&j L v N# F >. T R# .$ *Ω = Γ = R _z #E F − (x, y, z) =

0

8 H (+"1 **J &a 9*J !VO 9M [& . #E & B 0 &

∗ )

f = , Ω : y ≥ z, y ≤ − z, x ≥ z, x ≤ − z f =y− ,

(, , ) ( , , ), ( , , ), (, , )

)

/ g. &0 4 = Ω

f = uv + vw + wu, Ω : ≤ w −

+ u + v ≤

.&j 0 &#

000

000 f dV = H

F − (H)

R# .$ *Ω

=

*r = θ = z = z ZM G#U M = (, , z ) Ǳ E 0 0 *$$S G#U F (r, θ, z) = (x, y, z) e0& X R# 0 LB$ α E 0 M ( 4E_ F 5$ 0 N# 0 N# R & . h

G#U nB * ≤ r c α ≤ θ < α + π (0&f ]3 R# E B 9A$ O ) % ;*J !VO 0C ZM

≤r,

z∈R

E ( >.&[, F R0 M W Rl+

* * ∂(x, y, z) * −r sin θ = * r cos θ J = ∂(r, θ, z) * *

cos θ sin θ

* * * *=r * *

v N# F .$ *( T . −z c0 &) + .$ M 0 4E_ *Z & ) + . R# *( >&j L

*M .& −π &# . θ [& α [ Xb M ( M{ 000 , 5F .$ M x + y dV A& '() & & ( O $ z = T x + y = z pL 0 Ω *M [& . >. T 0 . Ω $ *!' Ω

Ω : x + y ≤

, , x + y ≤ z ≤

≤ w, u ∈ R

z , arctanh + x + y y , arctan , x + y − z x

Rl+ −w sinh u sin v w cosh u cos v

w cosh u

=

+ ⎧ ⎨ r = x + y θ = arctan (y/x) ⎩ z=z

π ,

∂(x, y, z) ∂(u, v, w) * * w sinh u cos v * * w sinh u sin v * * w cosh u

=

>&j L 9;*J !VO Y\ M M x= =C @PC\) % & ZM x= *( (x, y, z) >&j L &0 &q=−xyz .$ Ǳ [ c0 T&= z . H & M T&= *O&0 T−xy 0 M # j H GB ;*J !VO 0C Z & θ . HOx # E r . O & M >. T R# .$ M ( RO. *A$ O )

Γ = R #E ( >&j L v N# F >. T

≤ v ≤

F − (x, y, z) =

J

Ω : α ≤ θ < α + π ,

(f ◦ F ) J dV

KY . U0 $. & >&j L v E + $ .$ *Y M 9ZM

}

⎧ ⎨ x = r cos θ y = r sin θ ⎩ z=z

:

cosh u cos v cosh u sin v sinh u

* * * * * *

Z' . $ T R# M ( T Ω E w = T 0 & *O&0 Ω .$ T (O& M $ 5&6 $. .$ 0 & M G#U >&j L v N# F (u, v, w) = (x, y, z) 9M oL6 . Γ Ω &, +

) x = u − v + w , y = −u + v + w , z = u + v + w,

) u = x + y + z , v = y + z + x , w = z + x + y,

) u = cos x cos y , v = cos y cos z , w = cos z cos x, ) u = x + ey , v = y + ez , w = z + ex , ) x = u cos v , y = u sin v , z = w, ) x = u , y = uv , z = uvw, ) x = u − vw , y = v − uw , z = w − uv,

) x = uev , y = vew , z = weu , ) u = x , v = y , w = z , ) u =

/x , v = /y , w = /z,

Ω $ &0 >&j L v F M . T .$ $ & #n. T R# .$

000

f (x, y, z) dV = H

000 = H

! f x(u, v, w), y(u, v, w), z(u, v, w) .J dV

>.&[, &q=−rθz .$ Ω '& # j .$ *(O 5

E ( Ω

r ≤ r cos θ , : r − r cos θ + ≤ z ≤ − r + r cos θ −π/ ≤ θ ≤ π/ , ≤ r ≤ cos θ , : r − r cos θ + ≤ z ≤ − r + r cos θ

E ( >.&[, O ` + .$ ]3 000

+ dV = z dV x + y Ω 0 π/ (0 cos θ (0 −r +r cos θ

Ω

=

−π/

0 π/ (0 cos θ =

−π/

0 =

π/

−π/

r −r cos θ+

r cos θ − r

cos θ

Ω : r ≤

Ω :

Ω :

z dz

#0

dθ =

≤ x , ≤ y , (x + y ) ≤ x − y

≤ x, ≤ y, ≤ z ,

≤ x , +≤ y , (x + y ) ≤ x − y , ≤ z ≤ x + y

Ω : :

≤ r sin θ , (r ) ≤ r cos(θ) , z ≤ r

≤ r cos θ , ≤ cos θ , ≤ sin θ , r ≤ cos(θ) , z ≤ r

.$ * ≤ cos(θ) M $ O pO E &" & $ E kπ − π/ ≤ θ ≤ kπ + π/ r ≤ cos(θ)

Ω

0 π (0 (0 =

dr dθ

9ZM # j &q=−rθz 0 #E >. T 0 . , + R#

≤ θ ≤ π , ≤ r ≤

Ω

)

$M !' ≤ z pO &0 . F &" & *(O 5

M Z#S

Ω :

,r≤z≤

,r≤z≤

000 000 , x + y dV = r r dV

dθ

(x + y ) ≤ x − y , z ≤ x + y

≤r≤

0 5 M * O $ $ 5&6 Ω Ω &, + *J !VO .$ Z#. $ 8**J N+M

))

!VO 0 . Ω R# 0&0

:

R# 0&0 * ≤ θ ≤ π ]3 (" θ 0 @O 5 Q

" #0

π

=

dθ

)

r dz $

zr

) dr dθ

%z=

" dr

z=r

Zg&1 0 $ '& Ω M x= A .8 .$ *( x + y = z pL (x + y ) = x − y *M [& Ω 0 . f = z(x − y) >. T R# z &# −y # y &# −x 0 x # U &0 M $ O h'? *!' +"1 Ω 5 .$ f M v Ω −z 0 ]3 =S h .$ . $. $ . 1 &q= Z 6 N# .$ M Ω E D & *$ + 0 0 (6 . !T&' 5F 0 f [& E E ( >.&[, T−xy 0 Ω # j D :

,r≤z≤

000

z

=&M &q=−rθz .$ $ 0 Ω !#[ 0 *(O 5

. 1 . r sin θ . \ y &) 0 r cos θ . \ x &) 0 ( 9Z#. $ X R# 0 *Z$

=

π

0

( − r)r dr =

π

}

J**J &a E (+"1 9*J !VO q0 & S+ 0 $ Z' Ω M x= A, .8 *( z = − (x − ) − y z = (x+ − ) + y *M [& . Ω 0 f = z/ x + y e0& ( Rg&3 0 . $ _&0 0 . q0 5 S+ *!' 0 A$ O ) GB *J !VO 0C Ω # j [& 0 nB !' & $ N# .$ . $ 5F >_$&U ( =&M T−xy 9ZM

z = (x − ) + y z = − (x − ) − y

⇒ (x − ) + y =

T−xy 0 O $ $ '& . j .$ D : (x − ) + y ≤

!VO 0 . Ω R# 0&0 *( Ω : (x, y) ∈ D , (x − ) + y ≤ z ≤ − (x − ) − y x + y ≤ x , : x + x + + y ≤ z ≤ − x + x − y

pL 0 $ Z' Ω M . T .$ A8 < a < b M O&0 z = b+ z = a >&T *M [& Ω 0 . f = z + x + y e0& ≤ z ≤ y, z ≤ & #&" & &0 Ω M . T .$ AJ O&0 O 5&0 x + y ≤ x c x + y ≤ *M [& Ω 0 . f = x e0&

x + y = z

M x= + L @PC\) & ) GB H*J !VO 0 *O&0 &q= .$ Ǳ [ E b Y\ H = (x, y, ) Y\ $M # j T−xy 0 . M *$ O ϕ . zOM # E ρ . O & M T&= *Z#.F (0 . 0 0 O / . # E OM H da .$ *Z & θ . xOH # E .$ ( π/ − ϕ M = (x, y, z)

z

=

M H = ρ sin (π/ − ϕ) = ρ cos ϕ

OH

=

ρ cos (π/ − ϕ) = ρ sin ϕ

k = #&0 R# 0&0 * ≤ θ ≤ π/ M $$S c Z#. $ ` + .$ ]3 * ≤ θ ≤ π/ Ω : 000

≤ θ ≤ π/ , ≤ r ≤

, cos(θ) ,

≤z≤r

R# 0&0 *$ O ) % *J !VO 0 z(x − y ) dV =

Ω

=

000

=

=

=

0 0

z(x − y ) dV

Ω

zr cos(θ) dV

Ω

0

000

π/

π/

(0 √

cos(θ) 0 r

(0 √

π/

cos(θ)

zr cos(θ) dz

) dθ

) r cos(θ) dr

cos (θ) dθ =

dθ

π

}

R# 0&0 r = OH Z#. $ T−xy .$ & x =

OH cos θ = ρ sin ϕ cos θ

y

OH sin θ = ρ sin ϕ sin θ

=

Z#. $ ` + .$ ]3 ⎧ ⎨ x = ρ sin ϕ cos θ y = ρ sin ϕ sin θ ⎩ z = ρ cos ϕ

J**J &a E ; (+"1 9*J !VO

⎧ , ⎪ ⎪ ρ = x + y + z ⎪ ⎪

, ⎨ ϕ = arctan x +y ⎪ ⎪ yz ⎪ ⎪ ⎩ θ = arctan x

pO *Z. (ρ, ϕ, θ) ≤ρ,

A;*J C

U0& 0 X R# 0 M ( 5F e0& R# #n< VU

→ (x, y, z)

α ≤ θ ≤ α + π ,

≤ϕ≤π

M >&j L v 9H*J !VO

A*J C }

0 * &

, / x + y

M 0 $ S 0 . f = e0& A *M [& x + y + z = z B$&U 0 x + y = x 0 $ Z' Ω M . T .$ A; e0& O&0 z = a > z = +>&T *M [& Ω 0 . f = z x + y Q.&3 $ 5 SB Bn 0√ $ Z' Ω M . T .$ A e0& O&0 z = T x + y + = z [& Ω 0 . f = z( − x − y )/( + x + y) *M Zg&1 & 0 $ Z' Ω M . T .$ A z = b z =, a T $ x +y = b x +y = a 0 . f = / x + y e0& < a < b M O&0 *M [& Ω q0 & S+ 0 $ Z' Ω M . T .$ AH e0& O&0 y = − x − z y = x + z *M [& Ω 0 . f = |xyz|

#0

" #0

π

=

π/

dθ

" #0

sin ϕ dϕ

"

H*J 0C O&0 A−π &# 0 0 _z +U C LB$ $, α M E ( >.&[, e0& R# 0 M W *A$ O ) %

ρ dρ

( ) $ %π π/ ρ = θ ϕ − sin(ϕ) + sin(ϕ)

=

π (π − )

J

∂(x, y, z) ∂(ρ, ϕ, θ) * * sin ϕ cos θ * * sin ϕ sin θ * * cos ϕ

= =

}

ρ cos ϕ cos θ ρ cos ϕ sin θ −ρ sin ϕ

−ρ sin ϕ sin θ ρ sin ϕ cos θ

ρ sin ϕ

= A = (ρ, ϕ, θ) | ρ =

&# ϕ

, + 0 M *JC R# 0&0 *O&0 T Z' &0 &q=−ρϕθ .$ M ( T >&j L R# *M G#U A*J C $ &0 v v N# A *Z & M . R# .$ O&0 &q=−ρϕθ .$ Ω # j Ω S ` + .$ >. T =

&# ϕ

* * * * * *

= π

8*J !VO $ x + y + z = z M 0 Ω M+ . T .$ A, .8 *M [& Ω 0 . f f = x + y + z O&0 *J & B = Ω : x + y + z ≤ z 0 ) &0 0 *!' 9ZM oL6 . &q=−ρϕθ .$ Ω # j **J .$ Ω : ρ ≤ ρ cos ϕ :

≤ ρ ≤ cos ϕ

&z cB M >&j L v .$ & * ≤ cos ϕ #&0 ]3 R# 0&0 ≤ ϕ ≤ π/ .$ ≤ ϕ ≤ π Ω :

f (x, y, z) dV =

000 000 , x + y + z dV = ρ ρ sin ϕ dV π

= = π

" #0

Ω

( π/ 0 cos ϕ

dθ 0

π/

cos ϕ sin ϕ dϕ =

)

0 $ Z' Ω M x= A& '() & Z .$ e1 x + y + z = M x + y = z pL

Ω 0 . f = (x + y ) e0& *O&0 ≤ z &q= *M [&

9ZM 5&0 & #&" & N+M 0 . Ω 0 *!' /

z

y x 0 &q=−ρϕθ .$ Ω # j Ω R =&# 0 ]< 9ZM $& **J .$ A*J C & B = E X 0

Ω

"

: ρ sin ϕ ≤ ρ cos ϕ , ρ ≤ : sin ϕ ≤ cos ϕ ,

ρ sin ϕ dρ dϕ

π

ρ sin ϕ sin θ, ρ cos ϕ ρ sin ϕ dV

, Ω : x + y ≤ z , x + y + z ≤

Z#. $ X R# 0 *$ O ) J*J !VO 0 #0

f ρ sin ϕ cos θ,

Ω

Ω

≤ θ ≤ π , ≤ ϕ ≤ π/ , ≤ ρ ≤ cos ϕ

Ω

000

000

}

≤ρ≤

( U R# 0 &" & ]3 ≤ ϕ ≤ π ( 4E_ & E ( >.&[, Ω R# 0&0 ≤ ϕ ≤ π/ M

Ω :

≤ ϕ ≤ π/ , ≤ ρ ≤

Z#. $ $. $ ) θ 0 #$ 5 Q Ω :

≤ θ ≤ π , ≤ ϕ ≤ π/ , ≤ ρ ≤

.$ !Y " XUV !VO 0 '& N# Ω M $ O ) >. T R# .$ *$ O ) 8*J !VO 0 *( &q=−ρϕθ

000

M >&j L 9J*J !VO

Ω

(x + y )/ dV =

000 Ω

ρ sin ϕ ρ sin ϕ dV

*Ω : (x + y + z) ≤ x + y − z f = |z| A *Ω : (x + y + z) ≤ xyz f = AI f = (x + y + z) A

*Ω : x + y + z ≤ a , x + y ≤ az

0 $ M 3 E +"1 Ω M . T .$ A .8 N# .$ + M O&0 ; X 0 `&UO Ǳ [ cM 0 &M 0 f = + (x + y + z) e0& E $. $ . 1 4$ Z 6 *#0 Ω M $ O ) *!' /

Ω : x ≤ ,

>_$&U 0 &M 0 $ Z' A; *M [& . x + y + z = y

x + y + z = z

Y v v E . h

? @PC\) &

−

−

* * * *=|− |= * *

: :

≤ x, ≤ y, ≤ z , x + y + z ≤ ≤ x ≤ , ≤ y ≤ − x, ≤ z ≤

−x−y

Z"# w v u X"' 0 . z y x ]< Ω :

≤w≤

,w≤v≤

,v≤u≤

ρ sin ϕ cos θ ≤ , ≤ ρ sin ϕ sin θ , ≤ ρ cos ϕ , ≤ ρ ≤ cos θ ≤ , ≤ sin θ , ≤ ρ cos ϕ , ≤ ρ ≤ π/ ≤ θ ≤ π , ≤ ϕ ≤ π/ , ≤ ρ ≤

: :

Ω

0 . Ω 0 &q=−uvw .$ Ω # j Ω 5$M oL6 0 9ZM 5&0 & #&" & X"' Ω

Ω

000 ,

M O&0 Ω M . T .$ A& '() & c $ O ) Z 6 N# E x + y + z = T y *M [& Ω 0 . f O&0 f = /(x + y + z + ) w = x v = x + y u = x + y + z >&j L v E *!' z = u − v y = v − w x = w >. T R# .$ *ZM $& ?U0

:

([" ]& /&VU 5 .$ &\ X) v v RQ *$$S T &# y N# 0 ([" ]& &# Y\ N# 0 S S M ( 5F .$ &j L v RQ S# R#+, 4$ ).$ #. &# @L eY\ T y Y\ N# M U R# 0 *$ 0 RQ c S =&# !#[ &F O&0 ( 5 S q0 N# S =&# !#[ &F O&0 M S S ?za

*b_

* * ∂(x, y, z) * =* J = ∂(u, v, w) * *

E ( >.&[, &q=−ρϕθ .$ Ω # j Ω .$

!VO 0 ( v v

⎧ ⎨ x = a u + a v + a w + a y = a u + a v + a w + a ⎩ z = a u + a v + a w + a * * * a a a * * * * J =* * a a a * = * a a a *

≤ x + y + z ≤

≤ y, ≤z,

R# 0&0 + (x + y + z )/ dV =

000 ,

+ ρ .ρ sin ϕ dV

= Ω

#0 = =

"#0

π/

dθ π

×

×

"#0

π

π/

sin ϕ dϕ

ρ

,

" + ρ dρ

( − ) = π ( − √) √

9Z#0 Ω 0 f E $. .$ 0 & *( $ x + y + z = z M 0 Ω f = z A Ǳ [ cM 0 &M 0 Ω f = (x + y + z) A; * < a < b M ( $ b a `&UO + − x − y − z A M E +"1 0 Ω f = . 1 Z 6 N# .$ ( $ x + y + z = *$. $ + x + y = z pL 0 Ω f = x + y + z A *( $ z = T + x + y = z pL 0 Ω f = / x + y + z AH * < a < b M ( $ z = b z = a >&T x + y + z = R &M 0 Ω f = z A8 *( $ x + y + z = Rz + x + y + z = y M 0 Ω f = y/ x + y + z AJ *( O $

M E +"1 0 Ω f = ln(x + y + z + ) A7 *$. $ . 1 4$ Z 6 N# .$ M ( x + y + z = − /

0 × 0

π π+π/

=

π

#0

sin ϕ dϕ

" ρ dρ

000

|cos θ + sin θ| dθ

−π/ 0 π π/ (cos θ + sin θ) dθ = π √ −π/

=

Z#. $ .$

Ω

}

000 dV = × dV (x + y + z + ) (u + ) Ω ) ) 0 (0 (0 du = dv dw w v (u + ) ) 0 (0

= − dv dw (v + ) w 0

+w− dw = ln − = w+

*$ O ) 7*J !VO 0 }

3 .$ 3 >&j L v $ 9*J !VO M O&0 Z 6 N# E +"1 Ω M . T .$ A .8 9M [& . Ω Z' ( O ) S #. y S : x /a + y /b + z /c

!

= (xyz)/(abc)

v = y/b u = x/a >&j L v N+M 0 0 _> Ω X R# 0 *ZM mn' . c b a & .&3 w = z/c E $ 0 >.&[, &q=−uvw .$ Ω # j

M >&j L v 97*J !VO 5 Sq0 0 $ Z' Ω M . T .$ A, .8

Ω : ≤ u , ≤ v , ≤ w , (u + v + w ) ≤ uvw

∂(x, y, z) = abc J = ∂(u, v, w)

000

000

Vol(Ω) =

dV = Ω

Ω

abc dV = abc

000

.$ dV

Ω

Ω

Ω

: : :

≤ ρ sin ϕ cos θ , ≤ ρ sin ϕ sin θ , ≤ ρ cos ϕ , ρ ≤ sin ϕ cos ϕ sin θ cos θ ≤ cos θ , ≤ sin θ , ≤ cos ϕ ,

0

Ω

( π/ 0

0 = abc

π/

(0

Ω : −π/ ≤ θ ≤ π − π/ , 000

000 f dV =

sin ϕ cos ϕ sin θ cos θ

)

)

ρ dρ sin ϕ dϕ dθ

Ω

sin ϕ cos ϕ sin θ cos θ dϕ dθ

000 =

)

π/

Ω

Ω

ρ sin ϕ dV

= abc

∂(x, y, z) ∂(u, v, w) = ÷ J = ∂(u, v, w) ∂(x, y, z) * * * ** * * = ÷* * * = * * 000 000 000 f dV = |x + y + | dV = |u + v|

dV

R# 0&0

( π/ 0

R# .$ ZM $& Ω 0 M >&j L v E 5 M E ( >.&[, &q=−ρϕθ .$ Ω # j Ω >. T

≤ θ ≤ π/ , ≤ ϕ ≤ π/ , ≤ ρ ≤ sin ϕ cos ϕ sin θ cos θ

Vol(Ω) = abc

*M [& Ω 0 . f f = |x + y + | O&0 v = (x + y + ) u = x + ZM x= *!' .$ Ω # j Ω >. T R# .$ *w = (x + y + z + ) .$ *Ω : u + v + w ≤ S E ( >.&[, &q= −uvw

Ω

ρ ≤ sin ϕ cos ϕ sin θ cos θ

000

.$ ZM $& M >&j L v E 5 M E ( >.&[, &q=−ρϕθ .$ Ω # j

(x+ ) + (x+ y + ) + (x+ y + z + ) =

=

≤ ϕ ≤ π, ≤ ρ ≤

.$ *$ O ) *J !VO 0 |u + v| dV

Ω

ρ sin ϕ |cos θ + sin θ| dV

Ω

#0 π−π/ −π/

" |cos θ + sin θ| dθ

>_$&U 0 pL 5 Sq0 0 $ Z") Z' A x /a +y /b = z /c x /a +y /b +z /c = *M [& . |x| + |y| + |z| = #. 0 $ Z' Ω M . T .$ A; *M [& Ω 0 . f = |x| + |y| e0& O&0 B$&U 0 "0 [) #. 0 $ Z") Z' A *M [& . x /a + y/b + z/c = x/h x /a + y /b + z /c = #. 0 $ Z") Z' A *0&0 . B$&U 0 "0 [) #. 0 $ Z' 0 AH Z 6 N# .$ e1 (x/a + y/b + z/c) = x/h − y/k *#0 f = |x| e0& E b >&j L v M ( RO. 78 $ & & #E & B&a /& R+ 0 *( ` .&"0 $. & M . @ 5&+ * " & F E i jL0 #& + 0 ( r.&V 0 s $& $. . c0 & & F %&L .$ #$ v F $ ) 0 (U- 0 "0 B&" .$ M U R# *ZM ' @ . X& >&j L

uO 0 $ Z' Ω M x= A& '() % & x = z z = y z = y y = x y = x Ω 0 f E >. T R# .$ *f = /(xyz) ( x = z *#0 Z"# #E !VO 0 . Ω , + *!'

#0 =

≤ y/x ≤ ,

:

≤ z/y ≤ ,

()

=

=

∂(x, y, z) ∂(u, v, w) = ÷ J = ∂(u, v, w) ∂(x, y, z) * * −y/x /x * * = ÷* −z/y /y * /z −x/z 000

000 xyz

Ω

dV = #0

=

Ω

du

u

uvw

" #0

v

" #0

dw

w

#0

abc

(xyz)

Z#. $ R# 0&0

s ( − s ) ds

=

" sin θ cos θ dθ

" #0

" t ( − t ) dt

abc

s = sin ϕ >&j L v E AC .$ V# ^- *Z# $ + $&

pL 0 $ Z' Ω M . T .$ A .8 O&0 y + z = T x + xy + xz = yz *M [& Ω 0 . f = y + z (x + y) + (x + z) = (y + z) !VO 0 . pL B$&U *!' f e0& T B$&U 0 h !B$ R+ 0 (O 5

R# .$ *w = y + z v = x + z u = x + y ZM x= Ω : (x + y) + (x + z) ≤ (y + z) , y + z ≤ 5 Q >. T &q=−uvw .$ Ω # j Ω R# 0&0

Ω : u + v ≤ w , w ≤

:

+ u + v ≤ w ≤

5 Q ?U0 *(

J

= =

∂(x, y, z) = ÷ ∂(u, v, w) ∂(u, v, w) ∂(x, y, z) * * * ** − * *= ÷* * = * * *

Z#. $ ` + .$ 000

000 f dV = Ω

Ω

000 (y + z) dV =

w Ω

Ω

dV

.$ *ZM $& >&j L v E 5 M E ( >.&[, &q=−rθz .$ Ω # j

Ω : r ≤ z ≤

:

≤ θ ≤ π , ≤ r ≤

000

000 f dV

=

Ω

=

(ln )

w dV = Ω

#0

000

" #0

π

,r≤z≤

zr dV

Ω

( 0

dθ

= π

"

π/

Z#. $ R# 0&0

dV

dv

* * * *= * *

#0

t = sin θ

≤ x/z ≤

v = z/y u = y/x M $ O x= ( X& R# 0&0 >.&[, &q=−uvw .$ Ω # j Ω >. T R# .$ *w = x/z .$ *Ω : ≤ u ≤ , ≤ v ≤ , ≤ w ≤ E ( Z#. $

abc

" sin ϕ cos ϕ dϕ ×

: x ≤ y ≤ x , y ≤ z ≤ y , z ≤ x ≤ z

Ω

π/

0 r

)

"

zr dz dr r

( − r ) dr =

π

0 &

v = r sin θ u = r cos θ >&j L v E 5 M &q=−rθz .$ Ω # j Ω .$ ZM $& w = z E $ 0 >.&[,

Ω

/

≤ θ ≤ π ,

:

000

000 f dV = #0

Ω

" #0

π

=

dθ

000

−

/

/

" = π

|z| dz

/

M uO 0 $ , + 0 f = /xyz e0& E A .8 A = c x A = cx A = b x A = bx A = a x A = ax < b < b < a < a A = x + y + z 5F .$ M *#0 < c < c &v w = A/z v = A/y u = A/x ZM x= *!' >. T R# .$ *O&0 #)

/

v = (y/a) u = (x/a) x= &0 (O 5

Ω e1 .$ $$S !#[ M N# 0 S #. w = (z/a) *Ω : u +v +w ≤ E ( >.&[, &q=−uvw .$ Ω # j >. T R# .$ &

|z|r dV

" #0

r dr

.$

Ω

/

/ / / + (y/a) + (z/a) = (x/a)

≤ r ≤ , − ≤ z ≤

|w| dV =

/

− /

≤ r ≤ , − ≤ z ≤

:

Ω

#&q= g F 0 $ Z' Ω M x= A, .8 e0& *O&0 x + y + z = a B$&U 0 *M [& Ω 0 . f = (xyz) >. T 0 . S O $ $ #. B$&U *!'

J

* * ∂(x, y, z) * * ∂(u, v, w) = * *

=

a u v w

=

au av aw

Z#. $ .$

A = =

J

=

f dV

A u v w

A /uw A /vw A(A − w)/w

Ω

000 =

Ω

=

=

=

)

a

000

dV = aVol(Ω ) = aπ

Ω

* * * * * *

0 #. .& Q 0 $ Z' Ω M . T .$ A .8 z = x + (x − y) + (y − z) = >_$&U

O&0 z = x − (x − y) + (y − z) = *M [& . Ω 0 f = (z − x)( xyz − ) e0& >_$&U .$ z − x y − z x − y ! , V# !B$ 0 *!' R# .$ *Z#S w v u X 0 . & F O . V O $ $ E >.&[, &q=−uvw .$ Ω # j Ω >. T Ω :

A u v w dV A A A u v w Ω " #0 " #0 " #0 a b c du dv dw u v w a b c

a b c ln ln ln a b c

.$ * [a $ , c b a M x= 0 & & 9M [& . O $ $ #. 0 $ Z' $.

y

a

≤ u + v ≤ , − ≤ w ≤

?U0 (

000

=

x

(au av aw )−/ a u v w dV

Ω

dV xyz

(xyz)−/ dV

>.&[, T−uvw .$ Ω O $ $ , + # j Ω =@ E .$ Ω : a ≤ u ≤ a , b ≤ v ≤ b , c ≤ w ≤ c E ( 000

=

Ω

.$

∂(x, y, z) = ∂(u, v, w) * * A(A − u)/u A /uv * = * A(A − v)/v * A /vu * A /wu A /wv

000

000

A A A x +y +z = + + u v w

A + + u v w A = ÷ /u + /v + /w

?U0 *

* * * * * *

z

+ + = b c

#

)

x a

y

+ b

"

z

+ = c

J

= =

∂(x, y, z) = ÷ ∂(u, v, w) ∂(u, v, w) ∂(x, y, z) * * * x − ** * y − ** = ÷* * * − z * | xyz − | 000

000 f dV Ω

R# 0&0

f dV | xyz − | Ω 000 000 = |z − x| dV = |w| dV =

Ω

Ω

R O $ E& &S &$0.&M E .&"0 Z = M ( *( 4 , #& .$ ojL M x= h # QDR /MJ . !) $ & 0 !Y " XUV (= >. T R# .$ q ∼= Ω ⊆ R Z' &z\1$ M ( Δx Δy Δz 0 0 Δz Δy Δx $&U0 000 R# E #& B&a ?z[1 *Vol(Ω) = dV R# 0&0 *( 5F M{ E nB Z# $ + &6 . 5F &0 p&[. .$ >&#+ $0.&M *ZM . $$ & F i

j

k

k

j

i

Ω

9fC) V3 Y ) < =C) . !) $ &

A$, (+MC B&V 5 N# q Ω ⊆ R M x= XUV 0 q . \ >. T R# .$ *$$S G#U Ω 0 M O&0 *( Δx Δy Δz 0 0000Δz Δy Δx $&U0 0 !Y "

.$ *O&0 Vol(Ω) = dV 0 0 Ω !M 0 q . \ R# 0&0 (= (+"1 :.& E ( >.&[, Ω 0 q . \ y U# !M Z' 0 !M i

j

k

k

j

#

) #

)

x y z + + a b c x y z + + a b c

x

qmean := mean(q) = ⎝ Ω

000

⎞

q dV ⎠ ÷ ⎝

Ω

000

⎞ dV ⎠

Ω

E (x, y, z) Y\ & $ M x= A& '() $ & + .$ *( T = |z| x + y 0 0 Ω : x + y + z ≤ S & Z$ . 1 #& S Bc# y N# .$ . Z") R# M . T *M [& . #& (0&f & $ 0 . #&3 & $ 0 & $ R =&# 0 R# 0&0 #& S W y U# & $ *!' S *ZM [& . Ω 0 T y ( =&M T #& B$&U &0 ( 0 0 h $. & $ O&0 &q=−ρϕθ .$ Ω # j Ω

⎛ Tmean = ⎝

=

π ×

= =

π

− 000

000

⎞

A ≤ αx S R0 w 6 '& Ω M . T .$ A5 *A = x + y + z M O&0 A ≤ γz A ≤ βy 0 0 M $ 5&6 $M [& Ω 0 . f = xyz *( (/ )( /α + /β + /γ )

F B !F N +2

#0 π

" 0 dθ

(π)

# 0

π/

j

k

i

j

k

i,j,k

Ω

, |z| x + y dV

k

Ω

Ω

-&#. & +"1 #& .$ = &$0.&M &S T [ *$. $ &+ ' .& F .$ ' N#c= $ O & &S & F .$ M #& B = : L &S $ (B&' 0 [O ?z &M M ( / FH q *A$ O ) *H*8 0C O&0

+M q(x, y, z) M x= BV K+ $ & E 0 M ( z y x 3 !\ " &v &0 $, 0 . Ω l&Q *$$S G#U (x, y, z) ∈ Ω T−yz T−xz T−xy E >&T N+M (= [& 0 ZM Z"\ NQ M & Y " XUV 0 &S ) . & Y " XUV R# E N# (= Ω !M 9$M e+) Z &0 . !T&' $ , / ]< $ + [& Z

. \ 5$.F (0 0 *Q ≈ q(x , y , z ) Δx Δy Δz M Z#0 ' . T .$ ( . (+ >.&[, E =&M Q U1 *# 0 T 0 Δz Δy Δx (#& 0 0 >&+"\ $ U &0 ( 0 0 Q e1 0 i

dV ⎠

| cos ϕ| sin ϕρ dV

π π

⎛

T dV ⎠ ÷ ⎝

Ω

000

=

⎞

000

x y z = + − a b c

+

i

⎛

"

x y = + a b

xyz y z = + , ≤ x, ≤ y, ≤ z a b c abc m n p ) xm + yn + zp = a b c m, n, p ∈ , , · · · , ≤ x, ≤ y, ≤ z

)

Ω

⎛

"

n / m / /

lim

" #0 π | cos ϕ| sin ϕ dϕ ρ dρ "# cos ϕ sin ϕ dϕ

"

=

. . −z & Ω 3 p&\ T&= y A, .8 *Ω : x + y ≤ x , ≤ z ≤ 9M [&

j

n,m,→∞

i

! q xijk , yijk , zijk × Δxi Δyj Δzk

Δxi ,Δyj ,Δzk → i= j= k=

000

*Q = q dV &S$ G#U 0&0 .$ ) )*"9 ) Ω J )+ )* > 7<p )*$+( r U# m7+ q(x, y, z) !F !H/ L B (x, y, z) MUV &$0.&M E + Q $ .$ s_`3! ./ Ω q B #0 *ZM KY . . $

+, [) M &S Ω

0 $ $ &

9M [& Ω 0 . f y $. .$ >&j L >&T 0 $ 4 Ω f = x + y + z A *( x + y + z = *( Ω : ≤ x, y, z ≤ ' XUV Ω f = xyz A; +

Q.&&T x + y = + z

Y\ M . T .$ E1n 4L) + QR . !) % $ & m 4) &F O&0 δ(x, y, z) 4) B&Q . $ (x, y, z) ∈ Ω 5 #E D. 0 . Ω Z") C = (x , y , z ) !\f cM 9$ + [&

x =

m

zδ dV.

y =

Ω

xδ dV,

m

yδ dV, 000

000

δ dV, Ω

$ ( +") Ω M x= A& '() & $ & .$ e1 y = / T z = − x + 0 E (x, y, z) Y\ ) B&Q M . T .$ * Z 6 N# *M [& . 5F !\f cM 4) O&0 y 0 0 5F 9(O 5 #E >. T 0 . Z' R# *!' Ω : − ≤ x ≤ ,

≤y≤

,

≤z≤ : −π/ ≤ θ ≤ π/ , ≤ r ≤ cos θ , ≤ z ≤

[& . . −z & T&= f = fmean =

π × ×

= =

π

≤ z ≤ − x . )

m

=

−

=

−

0

=

−

x

= =

xδ dV =

m Ω 0 (0 / −

−

cos θ

(0

cos θ

(0

)

π )

r dV

Ω

r dz dr dθ )

r dr dθ

Ω

z =

x = ρ sin ϕ cos θ, z = ρ sin ϕ sin θ, y = ρ cos ϕ

( >.&[, !#[ R# R0 M W >. T R# .$ *ZM $& E ( >.&[, &q=−ρϕθ .$ Ω # j Ω J = ρ sin ϕ E

≤ρ≤

,

≤ ϕ ≤ π , ≤ θ ≤ π

E ( >.&[, O y .$

000 Vol(Ω)

q dV Ω

⎛ ⎞ 000 = ⎝ |ρ cos ϕ| (ρ sin ϕ) ρ sin ϕ dV ⎠ Ω

÷

.$ )

=

)

xy dz dy dx

=

)

xy( − x ) dy dx

π/

Ω

(0

000

M x= A .8 B$&U 0 5 Sq0 0 $

i T.$ . $ 5F E (x, y, z) >&j L +0 Y\ M . Ω i y *( q = |y| x + z *M oL6

. h R# 0 *ZM [& Ω 0 . f y ( =&M *!' >&j L v E

qmean =

y( − x ) dy dx

0 (0 / (0 −x

000

)

)

( − x ) dx =

−π/

0

y +

Ω :

y dz dy dx

0 (0 /

π/

000 , x + y dV =

π −π/ 0 π/ cos θ dθ = π −π/ π

&0 ( 0 0 Ω 4) X R# 0 0 (0 / (0 −x

e0& y 5 M 9ZM

Ω

0 =

+ x + y

⎛ ⎞ ⎛ ⎞ 000 000 =⎝ f dV ⎠ ÷ ⎝ dV ⎠ Ω

x +

Ω

m=

Ω

Ω : r ≤ r cos θ ,

4 !VO 0 ( cL

000

000

R# 0&0 *ZM $& >&j L v E *!' E $ 0 >.&[, &q=−rθz .$ Ω # j

∗

$ x + y + z = z M 0 Ω f = x + y + z A *(

z =

Ω

=

π

000 Ω

#0

π × × ×

ρ sin ϕ| cos ϕ| dV " 0

#0

π π dθ sin ϕ| cos ϕ| dϕ π 0 π/

× sin ϕ cos ϕ dϕ =

" ρ dρ

=&M 5 M *C = (, , z ) U# *$ $ . 1 . −z 0 5F $& M >&j L E . h R# 0 *Z0&0 . z ( >. T R# .$ O&0 &q=−ρϕθ .$ Ω # j Ω S *ZM

Ω

000 m

=

Ω

π b − π a 000

π(b − a )

=

#0

×

Vol(Ω) × δ − 000

zδ dV

=

ρ cos ϕρ sin ϕ dV

dθ " #0

π/

b

" ρ dρ

a

(b + ab + a )

#. 0 $ M Ω XT Z") 4) A .8 B&Q . $ z = y = x = +>&T *M [& . O&0 δ = x + y E >.&[, Ω $ *ZM [& . Ω 4) 0 *!'

z = xy

m

=

≤y≤

0 (0 0

000

xy

δ dV = Ω

0 (0 =

, xy

m Ω 0 (0 / −

,

≤ z ≤ xy

Z#. $ .$ *(

)

, x + y dz dy dx

m

√ −

≤z≤

!VO 0 >&j L .$ 5F # j M Ω : −π/ ≤ θ ≤ π/ ,

−

Ω

0 (0 / −

0

Ω

−

≤ r ≤ cos θ , ≤ z ≤

)

y( − x ) dy

( − x ) dx =

dx

: x + y ≤ z ≤ − x − y : x + y ≤

, x + y ≤ z ≤ − x − y

Z#[0 >&j L 0 . 5F l&Q *$ + 5&0 5

Z#. $ Ω :

≤ θ ≤ π , ≤ r ≤

, r ≤ z ≤ − r .

&0 ( 0 0 Ω 4) .$ =

δ dV = Ω

#0

0 $ !VO @ 1 Z") Ω M . T .$ A .8 Y\ O&0 z = T T−xy x + y = x 4) O&0 δ = x + y + z ) B&Q . $ (x, y, z) ∈ Ω *0&0 . 5F !\f cM

(B&' R# .$ M ( RO. *!' Ω : x + y ≤ x ,

000

0 x (x + )/ − x dx = =

0

dx

*O&0 (, /, / ) 0 0 Ω !\f cM R# 0&0 5 S+ $ 0 $ Z") Ω M . T .$ A, .8 Y\ $ 0 z =+ x + y z = − x − y q0 4) O&0 δ = x + y ) B&Q . $ (x, y, z) ∈ Ω *M [& . 5F >. T 0 . Z' R# *!'

m

dx

)

y ( − x ) dy

( − x ) dx = − ) ) 000 0 (0 / (0 −x zδ dV = yz dz dy dx

) x + y dy

−

=

"

#0 π

,

−

Ω

cos ϕ sin ϕ dϕ

≤x≤

=

Ω

(b + b a + ba + a )

Ω :

= z

z dV

Ω

π(b − a )

=

=

(x − x ) dx = ) ) 0 (0 / (0 −x 000 yδ dV = y dz dy dx

=

.$

000 zδ dV =

=

y

≤ ϕ ≤ π/ , ≤ θ ≤ π , a ≤ ρ ≤ b

:

=

=

≤ ρ cos ϕ , a ≤ ρ ≤ b

:

z

0

= =

π.

000 , 000 x + y dV = r dV Ω

Ω

" #0 ( 0 ) " π −r dθ . r dz dr 0

r

(r − r ) dr =

π

B& V+ N# !VO 0 3 Ω M x= A .8 ?U0 *( a 5F $ `&UO b 5F ).& `&UO M ( A(0&f B&Q &0 U#C &Q ]) N# E Ω M M x= *M oL6 . Ω !\f cM O&0 O & 0 0 p&\ + .$ Ω 4) B&Q M ZM x= Z *!' !VO 0 . Ω M ZM x= Z ?U0 *( δ = δ R# 0 *Z#S h .$ Ω : ≤ z , a ≤ x + y + z ≤ b !\f cM .$ ( 5.&\ . −z 0 ([" Ω X

+ 000 z =

m

000

=

π

=

π

dθ =

π

E ( >.&[, Ω 4) .$ *O&0

π/

(0

δ dV =

z(x + y + z) dV Ω

Ω

cos θ

(0

)

= )

+zr sin θ + z r dz dr dθ 0 = =

π π

π/

−π/

0

π/

−π/

cos θ

Ω

0

zr cos θ

=

"

# r cos θ + r sin θ + r

) dr dθ

cos θ + cos θ sin θ + cos θ dθ =

*( Ω !\f cM C = (/

,

/

, /) .$ 0 * $ &

pL 0 $ Z") Ω M . T .$ A B&Q . $ 5F E (x, y, z) Y\ O&0 y =+ T *0&0 . Ω !\f cM 4) O&0 δ = y x + z x + y + z = z M 0 $ Z") Ω M . T .$ A; + cM 4) O&0 δ = x + y + z B&Q e0& &0 *0&0 . Ω !\f 0 5 Sq0 y M O&0 Z 6 N# E 6L0 Ω S A ]) E O ) x /a + y/b + z/c = B$&U

oL6 . Ω !\f cM &F ( O & &Q *M z = x − y #. 0 $ Z") Ω M . T .$ A δ = z 5F B&Q e0& $ 0 y = T T−xy *0&0 . Ω !\f cM 4) O&0 *( R `&UO Ǳ [ cM 0 M Ω M x= AH Ω 4) [" Q 0 z = x + y q0 5 S+ + x= δ = x + y . B&Q e0& *M Z"\ . *#0 B$&U 0 "0 #. 0 &Q Z") N#! Ω M . T .$ A8 . 5F 4) O&0 x /a + y/b + z/c = *M [&

B$&U 0 "0 #. 0 $ &Q! Z") !\f cM AJ . Z 6 N# .$ e1 x/a + y/b + z/c = *0&0

(x + y + z) dV

000 r cos θ + r sin θ + z r dV = Ω

−π/

(0

000

000

z r cos θ + r sin θ + z r dV

Ω

0

000

zδ dV = Ω

cos θ sin θ

=

π/

(0 ( cos θ 0

−π/

0

) (0 cos θ r cos θ + r sin θ + r dr dθ

π/

−π/

0

) ) r cos θ+r sin θ+rz dz dr dθ

π/

−π/

cos θ + sin θ + cos θ

dθ = π

&0 ( 0 0 x Z &0 C = (x , y , z ) . Ω !\f cM l&Q 000 x =

000 xδ dV =

m

Ω

000

=

y = x + z

π

Ω

0 =

π

π

x(x + y + z) dV Ω

r cos θ r cos θ + r sin θ + z r dV

π/

(0

cos θ

(0

−π/

r cos θ )

)

+r cos θ sin θ + r z cos θ dz dr dθ 0 =

π

π/

(0

cos θ

r cos θ

−π/

)

+r cos θ sin θ + r cos θ dr dθ 0 =

π

π/

−π/

cos θ + cos θ sin θ +

cos θ

dθ =

?U0 000 y =

m

000 yδ dV =

Ω

000

=

π

Ω

0 =

π

π

y(x + y + z) dV Ω

r sin θ r cos θ + r sin θ + z .r dV

π/

(0

cos θ

(0

−π/

r sin θ cos θ

)

)

+r sin θ + r z sin θ dz dr dθ

∗

0 =

π

π/

−π/

(0

cos θ

r sin θ cos θ

)

+r sin θ + r sin θ dr dθ 0 =

π

π/

−π/

sin θ cos θ + cos θ sin θ

>. T R# .$ O&0 δ 0 0 (0&f Ω B&Q ( O ) T−xy . −z . −y . −x ' Ω & .& 6S *M [& . Ǳ [ T−yz T−xz B&" R# .$ M Z# $ ) *!' Ω : x /a + y /b + z /c ≤

,

≤ x, ≤ y, ≤ z

y = br sin θ x = ar cos θ >&j L v E ' Ω # j Ω J = abcr >. T R# .$ *ZM $& 9ZM [& . T−xy z = ct

Z 6 N# .$ e1 + &Q +XT Z") N# Ω M . T .$ A7 + O&0 x/a+ y/b+ z/c = #. 0 $ *M [& . 5F 4) [T Z") Ω M x= ) +CcG . !) $ & *( δ(x, y, z) 4) B&Q . $ 5F E (x, y, z) Y\ M ( *( &q=−xyz .$ T &# y Y\ N# S M x= E >.&[, S ' Ω & .& 6S G#U 0&0

000

000 Ixy

= Ω

=

z δ dV

000

δ Ω

=

= =

abc δ

z dV = δ 0

abc δ

0

abc δ

0

000

π/

(0

π/

dθ =

c t abcr dV

) ( − r )/ r dr π

dθ

Ix Iy

= Ixy + Iyz =

π

π

π

= Ixy + Ixz + Iyz =

000

Ixy (Ω) =

π

Iyz (Ω) =

−

:

≤ x, ≤ y, y ≤ x ≤ , (x + y )/ ≤ z ≤ x + y ≤ x ≤ , ≤ y ≤ x, (x + y )/ ≤ z ≤ x + y

y δ(x, y, z) dV

Ω

000

abcδ (a + b + c )

z δ(x, y, z) dV

Ω

000 Ixz (Ω) =

B&Q e0& &0 XT Z") Ω M . T .$ A, .8 x + y = ± >&T 0 $ δ = z(x + y) z = x + y z = x + y 5 S+ $ x − y = ± *M [& . . −z ' Ω & .& 6S O&0 *Ω M v δ −y 0 y &# −x 0 x # U &0 *!' . 1 Z 6 N# .$ M Ω E +"1 Ω E Ω &0 .$ 5 O&0 $ x = y = x >&T R0 O $ 9$M 0 0 (6 . % ) ]< $ + $& :

Ω

T−xz T−yz T−xy . S l&Q *O T−xy ' Ω & .& 6S >. T R# .$ Z## &0 0 0 X 0 T−xz T−yz

abcδ (a + b )

IO

Ω

000 z + y δ(x, y, z) dV = 000Ω = x + z δ(x, y, z) dV 000Ω x + y δ(x, y, z) dV =

abcδ (a + c )

= Ixz + Iyz =

Iz (Ω)

abcδ (b + c )

Iz

.$ Z#0 . −z . −y . −x . S S *O . −z . −y . −x ' Ω & .& 6S >. T R# 0 0 X 0 Iy (Ω)

abc δ

xz

= Ixy + Ixz =

*O&0 S & (x, y, z) Y\ T&= d(x, y, z) 5F .$ M ( Ǳ [ ' Ω & .& 6S &F Z#0 Ǳ [ . S l&Q

Ix (Ω)

Z [& 50 $. $ $ ) Ω δ .$ M .&\ !B$ 0 π π I = a bcδ I = ab cδ Z"# 0 Z#. $ *H*J .$ & B = N+M 0 yz

d (x, y, z)δ(x, y, z) dV

Ω

Ω

⎡ √ ⎤ ⎤ 0 0 −t ⎣ ⎣ t r dt⎦ dr⎦ dθ

π/

IS (Ω) :=

000 x + y + z δ(x, y, z) dV IO (Ω) =

Ω

⎡

x δ(x, y, z) dV

Ω

M $$S . @ R# _&0 G#.&U E *O Ix = Ixy + Ixz , Iy = Ixy + Iyz , Iz = Ixz + Iyz IO = Ixy + Ixz + Iyz =

(Ix + Iy + Iz )

G#U R := +I /m >. T 0 . S ' Ω uQ `&UO N# (, &0 M Ω Z") 6[) W >. T R# .$ *ZM

0 M m 4) 0 Y\ W &0 Q S ' &f .$ .$ *( 0 0 Q # E (, 5&+ &0 S E R T&= N# E +"1 Ω M . T .$ A& '() $ & 5 Sq0 y M O&0 Z 6 S

S

S

x /a + y /b + z /c =

= =

δ a + b + c abcδ

×

4

ac b

+

a bc

+

a b c

5

000

a b + a c + b c

Iz

=

a + b + c

000 − x + y δ dV = z x + y dV

Ω

0 (0

B$&U 0 #. 0 $ &Q Z") & .& 6S A *0&0 . . −y ' (x + y + z) = a y x + y + z = 0 $ &Q Z") & .& 6S A; *M [& . . −z ' x + y = z B$&U 0 "0 #. 0 $ &Q Z") & .& 6S A *M [& . Ǳ [ ' (x + y + z) = x + y >&j L >&T 0 $ Z") Ω M . T .$ A ' Ω & .& 6S O&0 x + y + z = a T T−xz T−xy . −z . −y . −x B&Q e0& 5$ 0 (0&f x= &0 . Ǳ [ T−yz *M [& δ = δ

h

=

= =

0 0

Ω

x +y

(x +y )/ x

dy dx =

) ) − z x + y dz dy dx

*−−−−−−−−−−−−−−−→ −−−−→* * * *(x − , y − , z − ) × (a, b, c)* = *−−−−→* * * ||v || *(a, b, c)* *−−−−−−−−−−−−−−−−−−−→* * * *(yc − zb, za − xc, xb − ya)* + a + b + z + (yc − zb) + (za − xc) + (xb − ya) + a + b + z

* *−−−→ * * *X X × v *

E ( >.&[, h $. '& ?U0 Ω :

000

≤ x ≤ a, ≤ y ≤ b, ≤ z ≤ c

&0 ( 0 0 I O .& 6S .$ h δ dV =

Ω

000

/ F DF B !F 02

=

l&Q O S J**J .$ M . @ 5&+ b .$ $&, . 5F M T H**J q1 y# O .$ 5$ $ 5&6 uL0 R# E m *Z & & . 5F >. T R# *( & B # [& Q

=

f dV

Ω

&# O&[ . M "0 Ω , + S % & G#U &# M ! (#& 0 # Ω , + p&\ E 0 0 f b Z' . $ Ω 0000f & 3& , + V# &# $$ *Z & + . f dV >. T R# .$ O&0 T

(0

>&T 0 $ !Y " XUV & .& 6S A .8 . DY1 ' z = c y = b x = a >&T &j L

*M [&

6 $.F B&Q e0& E L B&" >. T .$ 5 Q *!' Y XUV R# Y1 *( (0&f δ = δ ZM x= ( O . 5F &S[V ]3 *$.nS (a, b, c) Y\ [ E M ( Y\ T&= *$ + %&L 5 −(a,−−b,−→c) . 5F $& . $0 E ( >.&[, y R# & (x, y, z) LB$

∗

000

=

O&0 R `&UO Ǳ [ cM 0 S E +"1 Ω M x= AH (x, y, z) Y\ B&Q $. $ . 1 Z 6 N# .$ M + `&UO & .& 6S *O&0 δ = x + y + z 5F E *M [& . x = y = z y ' Ω uQ Ǳ [ cM 0 M 0 $ &Q S Ω M x= A8 X = (a, b, c) Y\ ' Ω & .& 6S *O&0 R `&UO R G L & U- 9#&+ . *M [& . *#0 h .$ . X

x

=

0 $ &

Ω

&0 ( 0 0 O .& 6S .$

=

=

(yc − zb) + (za − xc) + (xb − ya) dV a + b + c Ω 0 a (0 b 0 c δ (yc − zb) a + b + c % % +(za − xc) + (xb − ya) dz dy dx 0 a (0 b δ c +c y (b − y) b b a + b + c ) c c x + (a − x) + + c(xb − ya) dy dx a a 0 a4 δ c b c b (a − x) + a a + b + c 5 c b cb cb x + (a − x) + x + dx a a a

δ

0 =

a

0

a

a→∞

=

0

a

4

lim

a→∞ 4

=

lim

a→∞

− (v + ) (a + )

lim

w

w + + (w + ) (a + )

+

(a +

a − (a + )

5

a

dw

000

) 5

x +y +z ≤

a→a

=

a

p

a

a→ −

x +y +z ≤

= lim

a→−

Ωa

000

= lim

a→−

Ωa

#0

dV

= lim

dθ

a→−

()

=

= ()

=

= =

π π

π

lim

a→−

0

000

exp − (x + y + z )/ dV =

Ω

000 =

0

a

0

√ a

π

#0 sin ϕ

lim

a→∞

exp − (x + y + z )/ dV

Ωa

000

=

lim

a→∞

e−ρ ρ sin ϕ dV

Ωa

#0

" 0

π

lim

dθ

a→∞

lim (π)()

a→∞

−

e

π

0 sin ϕ dϕ

−a

+

=

a

ρ e−ρ dρ

π

*( &q=−ρϕθ .$ Ω # j Ω &# .$ M Z 6 N# 0 f = /(x + y + z + ) e0& E A, .8 *#0 y Z 6 N# E O ) 4 . Ω . h R# 0 *!' 9ZM G#U x + y + z = a T a

ρ sin ϕ dV ( − ρ )p " 0

'() % &

*#0 cM 0 M . Ω , + LB$ a > E 0 *!' R# .$ *Ω : x + y + z ≤ a 9Z#S h .$ a `&UO O Z#. $ M >&j L v N+M 0 >. T

=

( − x − y − z )p

= lim

a→−

/

=

π

Ωa

0 exp(−(x + y + z) ) E A&

R

a

dV ( − x − y − z )p

000

f dV

a→a

a

( − x − y − z )p

p

Ip =

f dV := lim Ω

$ ) U#C ( + I p E #$&\ 4 M Ǳ E 0 *0&0 $ ) >. T .$ . 5F . \ A$. $ VO 0 . Ω R# 0&# ( ' S [B .$ I !V6 *!' M [B V# Z O&0 + Ω 0 Z M ZM %&L ZM G#U < a < E 0 9O&[ & F .$ ' M $$S h'? >. T R# .$ *Ω : x + y + z ≤ a 9Z#. $ M >&j L v N+M 0 lim Ω = Ω 000

000

⇒

Ω = lim Ωa

dV

Ip :=

Ω

a

( ([a $, p M x= A .8 000

x= 9 OVBC . !) B=B] % & 000 . @ . Ω &, + *O&0 & f dV M $&, Ω , + 0 f a Ǳ E&0 M ZM %&L R# .$ *a → a M O&0 & Ω ' (B&' Ω &f .$ O&0 9Z"# (B&' a

a − − (a + ) (a + )

−a − (a + )

dv dw

a

ρ dρ ( − ρ )p

"

a

ρ ( − ρ )−p dρ u/ ( − u)−p du

u/ ( − u)−p du

πB , − p = π Γ (/) Γ( − p) Γ (/ − p) π (/)Γ ( /) ( − p)Γ(−p) (/ − p) (/ − p) Γ ( / − p) π/ ( − p) × Γ(−p) ( − p)( − p) Γ ( / − p)

O $& 8*8*H E () .$ u = ρ >&j L v E ( ) .$ # + 0 M $$S h'? & &S $ 0 ) &0 *(

≤ x, ≤ y, ≤ z, x + y + z ≤ a

Ωa :

$& w = x v = x + y u = x + y + z >&j L v E &q= −uvw .$ Ω# j Ω J = >. T R# .$ *ZM

*Ω : ≤ w ≤ a , w ≤ v ≤ a , v ≤ u ≤ a E ( >.&[, Z#. $ .$ a

a

000

Z 6 N# =

dV = (x + y + z + ) 000

lim

a→∞ Ωa

0

=

a

dV = lim a→∞ (x + y + z + ) 0

a

0

lim

a→∞

w

v

a

du (u + )

000 Ωa

dv dw

dV (u + )

)

000

f (x + y + z)xp− y q− z r− dV =

p

Ω

Γ(p)Γ(q)Γ(r) = Γ(p + q + r)

0

^T − p −p $ , E 4 M ( 4E_ I *O&[ .$ . #E & B E N# . \ 0 % & 9M [& $ ) >. T 000

f (u)up+q+r− du

58' 9 / 22 ;H7 T 0 !< . c= 4 E $& >& \ &6 0 *$ O U) doc.pdf !#&= &# e0& [& 0 G 'BC & & x Zh −xy , + 0 f (x, y, z) v

Ω : a ≤ x ≤ b , h(x) ≤ y ≤ l(x) , u(x, y) ≤ z ≤ v(x, y)|

. $ E int(int(int(f(x,y,z),z=u(x,y)..v(x,y)), y=h(x)..l(x)),x=a..b)

*ZM $&

RV+ G 'BC . 1 1) . !) & & !< C $ O & 6 [& [1 . $ ) &0 ( evalf . $ E >. T R# .$ AO&[ 5F !' 0 .$&1 9ZM $& 5F [#\ . \ [& 0

)

dV

Z 6 N# 000

)

000

R

)

# /.$F .$

78 & &

http://webpages.iust.ac.ir/m_nadjafikhah/r2.htm

*( O $.F MnB = d'&[ 0 p 0 e0& & B&a

(x + y + z )a

exp −(x + y + z )/ dV

000

Z 6 N# 000

)

≤x,y,z≤

zdV (x + y + z + )

dV p x yq z r

)

000

−≤x,y,z≤

dV x+y−z

O&0 5.&\ \\' × ]#& [a ] M . T .$ AJ Aa = a U#C ij

ij

a a a

a a a 000

evalf(int(int(int(f(x,y,z),z=u(x,y)..v(x,y)), y=h(x)..l(x)),x=a..b))

dV

x +y +z ≥

)

(x + y + z + )

R

a a > , a

a a

ji

a < , a

9( ⎛ + #E &F ⎞ a > exp ⎝−

/ /

aij xi xj ⎠ dV

j= i=

$ Z 6 N# .$ e1 4 Ω S M $ 5&6 A7 &F O&0 x + y + z = T 0 000

xp y q z r dV =

) Ω

Γ(p + )Γ(q + )Γ(r + ) Γ(p + q + r + )

. , -$ ( 0 0 C E r(t ) & r(t ) UY1 0 f . \ &0 &z[#\ M Q A$ O ) GB *7 !VO 0C i

ΔQi

i−

! ≈ f r(ti ) Δsi ! ≈ f r(ti ) dsi ! = f r(ti ) r (ti ) Δti

}

$ M Z# $ $ . 1 .0 $. . #& B &# & Z+U !j= R# E m *$ 0 m&T & F B *( & 0 0 A&E&0 0 v N# e0 E C RU

& $. .$ ` *( ` $ 0 y U0 N# Ǳ&O (= [& 0 5F E ( B&V . $0 & $. .$ 4$ ` *$ + $& 5

( w y O 4& .&M [& 0 ?z+, ( 0 G#U &6 0 *$. .&V0 . $0 5 f& *$ O U) !j= E ; uL0

! `GF !F 4 * / 1 @ AGB 9*7 !VO ` y A% .&3 %&L E !\ " **7 G#U * 0 . 1 f ds 5$.F (0 0 >&[& S U# *( *( V# F Z$ 4& & .&3 E %&L $ f f C C C M x= * R# .$ *O&0 LB$ $ , a a v Q e0 f >. T 0 0 0 ! * af + af ds = a f ds + a f ds A &F f(X) ≤ f(X) X 0∈ C E 0 0 S A; * f ds ≤ f ds >. T R# .$ *O&0 Y\ Q &# C ∩ C S A Z#. $ B&'0 RQ .$ *C 0+ C : = C0 ∪ C Z"#

N# C : r(t) ; a ≤ t ≤ b M x= * 0&6 >. T 0C O&0 R &q= .$ .&3 N# & &0

f : R → R M x= *AO&0 c R T .$ G#U C 0 M ( A&q= .$ B&V 5 C v U0& !VO 0 . C 0 f >. T R# .$ *$$S

C

C

f ds = C +C

C

f ds + C

b

f ds : =

! f r(t) r (t) dt

a

C

O =U .&3 N# E u0 &0 C S *ZM G#U Z &0 C *** C C . 5F & .&3 E N# $0 O&0 ZM G#U #E !VO 0 . C 0 f >. T R# .$ n

0

0

f ds : =

C

C

0

0

C

0

f ds + · · · +

f ds +

C

C

0

C

f ds Cn

Z &0 & C E & $ w O M ( 5F 4E_ pO & @ V# &# O&0 Y\ & $ U . $ V# &# $ 0 &+B D. h Y\ E *O&0 T & F E & $ w O A$ O U) N#

+, -&#. %& M E 8 !j= 0C

f ds C

i

! ΔQi = f r(ti ) r (ti ) Δti

Zg&1 pL E O ) (+"1 S M x= A .8 Z 6 N# .$ z = z = >&T y x + y = z 0 . f = x(y + z) *O&0 S #. [B C ( *M [& C C *Z# $ $ 5&6 % *7 !VO .$ . C S #. *!' # $ e0. C C *ZM Z"\ C & C uL0 .& Q 0 . 9ZM .&3 #E !VO 0 . & F * " y .&3 C C C

:

C

C

:

−π π

&aa ( ) &0 &aa ( ) &0

=

r(t) =

π

≤t≤

−−−−−−−−−−−−→ sin t, cos t, ) ;

f ds = C

0

f ds + C

π/

= 0 + 0

≤t≤

+

f ds +

π( + π )

C = C + C + C C = ( , , )(, , ) : r(t) = ( − t)i + tj ;

: r(t) = ( − t)j + tk ;

π

f ds +

: r(t) = ( − t)k + ti ;

f ds =

f ds + C

0

=

! −−−−−−−−−−−−−→! cos t, sin t, dt

sin t, cos t,

! −−−−−−−−−→! f − t, , − t − , , − dt

f 0

f ds + =

0

=

√ 0

√

,

f t, ,

! −−−−−−−→! − t, t , − , dt ! −−−−−−−→! − t , , − dt

0 √ √ ( − t) dt + (t − ) dt +

=

f ds C

! −−−−−−−−→! − t, t, − , , dt 0

0

f ds + C

f

C

≤t≤

&0 ( 0 0 0 O 0 R# 0&0

0

0

+

! −−−−−→! , , dt f , t + , t +

≤t≤

C = (, , )( , , )

0

C

≤t≤

C = (, , )(, , )

! −−−−−−−−−−−−→! f cos t, sin t, − sin t, cos t, dt

f 0

0

C

π/

+

(, , ) ( , , ) / g. &0 da C M x= A, .8 [& . C 0 f *f = x − y + z ( (, , ) *M ( y .&3 `&+ ) C M ZM ) 0 *!' 9ZM .&3 &S ) . & Y .&3 R# E N#

Z#. $ ` + .$ ]3 0

! f r(t) r (t) dt

( cos t) + (t) + ( sin t) −π , (− sin t) + () + ( cos t) dt 0 π + t dt

=

C

0

) + (y + ) + z

−π

= (, , )( , , ) −−−−−→! −−−−−→! : r(t) = ( − t) , , + t , , −−−−−−−−−−−→! = − t, , − t ; ≤ t ≤

0

π

= 0

&aa ]V, ( ) &0

: C

0

&aa ]V, ( ) &0

:

( O .&3 *M [& . C 0 Z#. $ **7 G#U N+M 0 *!'

*f = (x −

=

y .&3

:

f

−−−−−−−−−−−−−−−−−−−−→! cos t + , t − , sin t ; −π ≤ t ≤ π

C

(, , )(, , ) −−−−−→! −−−−−→! + t , , r(t) = ( − t) , , −−−−−−−−−−−→! = , t + , t + ; ≤t≤ ⎧ x + y = z ⎪ ⎪ ⎨ z= ⎪ ≤ x, y ⎪ ⎩ ⎧ ⎪ ⎪ x +y = ⎨ z= ≤ x, y ⎪ ⎪ ⎩

:

C : r(t) =

f ds

−−−−−−−−−−→! r(t) = cos t, sin t, ;

:

C

0

⎧ x + y = z ⎪ ⎪ ⎨ z= ≤ x, y ⎪ ⎪ ⎩ ⎧ x + y = ⎪ ⎪ ⎨ z= ≤ x, y ⎪ ⎪ ⎩

:

0

*( @ 0 0 ds A 3.& N# E +"1 C M x= A '() * !VO 0 M (

dt

√ ( − t) dt

M O&0 y + z = E +"1 S M . T .$ A8 C ( O ) Z 6 N# .$ x = T y [& C 0 . f = x(y + z) e0& O&0 5F [B *M M M %&"' . T .$ .

1 C

|y| ds

C : (x + y ) = x − y 0 x ds C : y = a cosh a y C

.&3 &0 0 .

e0& A

C f = xy + z −−−−−−−−−−−−−−−−−−−−−−− √−−−→ r(t) = t sin t + cos t, t cos t − sin t, t

cos t( + sin t) dt + 0

π/

+

=

−−−→! C : r(t) = t, t ;

0

0

δ ds,

y =

m

x =

0 yδ ds, C

z =

m m

xδ ds, 0

C

zδ ds C

B$&U 0 # $ Z C M x= A& '() * .$ e1 x + y = R E (x, y) Y\ M ( . −x _&0 + C !\f cM 4) *O&0 δ = y x + y B&Q . $ 5F *0&0 . 9ZM .&3 . C 0 *!' −−−−−−−−−−−→ C : r(t) = (R cos t, R sin t), ≤ t ≤ π

0

dt

≤t≤

+ (t)(t ) + t dt ( + (t ))/

0

t dt + t

= ()

=

=

0

u−

du

u

( − ln )

*u =

C

C

√

! −−−−→! f t, t , t dt

=

! `GF !F N 4

m=

=

C

*M [&

0

( − t)

.$ ZM .&3

≤t≤

0

0

/

f ds

.&0 & S 4) h V#c= (+M N# B&Q f S M . @ 0 $$S G#U C 0 M O&0 ***� V# VB 0 *$ 0 C !M (= E >.&[, f ds &F &# &S $ 0&6 &$0.&M 0 ) &0 /& R+ 9$ + KY 5 . #E KO 0 & B = &S (x, y, z) Y\ S E1n 4L) + QR . !) * >. T R# .$ O&0 δ(x, y, z) ) B&Q . $ C E 5 #E D. 0 F P = (x , y , z ) !\f cM C 4) 9$ + [&

cos t( + sin t) dt +

M ( y = x + E +"1 C M x= A .8 0 . f = xy/( + x ) * ≤ x ≤ *M [& C !VO 0 . R# *!'

* ≤ t ≤ π M M [&

0 . f = xz + yz e0& AI

dt

√ ++ + √ +

=

−− −−−−−−−→ t−−−−−−−t− r(t) = + ln t, − ln t, t ,

0

π/

=

y AJ

y A7 *M [&

5F .$ M .

0

+ t

( O x= AC .$ M 0 $ *

(

/ g. &0 da 0 f = x + y + z e0& E A *#0 (, , ) (, , ) !VO 0 C f = x + y M . T .$ A; e0& O&0 r(t) = −−−t,−−−−−t,−→t! ; − ≤ t ≤ *M [& C 0 . f 0 . f = x sin z e0& A , , )

−−−−−−−−−→! C : r(t) = cos t, sin t, t ; −π ≤ t ≤ π/

*M [&

x + y + z = M $. 0 ! C M . T .$ A C 0 . f = |z| e0& O&0 x = y T *M [&

O&0 x + y + z = z M E +"1 S M . T .$ AH [B C ( O ) z = z = >&T y M *M [& C 0 . f = z(x − y) e0& O&0 5F

0 z

= = =

#

√

m

zδ ds C

0 π

πδ √ π

√

tδ

m

dt

= =

√

√

, − , π − π π

"

* E ( >.&[, C !\f cM ]3 & . N# .$ e1 b Y\ C B A M x= A .8 *0&0 . ABC da !\f cM O&0 &q= .$ ZM x= ( & ) B&Q E L 5 Q *!' 9ZM .&3 #E !V60 . ΔABC da *δ =

= =

x

= =

AB : r(t) = ( − t)A + tB ≤ t ≤

y

=

BC : r(t) = ( − t)B + tC ≤ t ≤

=

CA : r(t) = ( − t)C + tA ≤ t ≤

=

m

=

δ ds = Δ

= x

= = =

=

.$

0 Δ

ds = Δ

−→ −→ −→ AB + AC + CA 0 xδ ds m Δ 0 0 0 x ds + x ds + x ds m AB m BC m CA 40 ! −→ ( − t)xA + txB AB dt m 0 ! −→ ( − t)xB + txC BC dt + 5 0 −→ + (( − t)xC + txA )CA dt

0 *

xδ ds 0 π R cos t . R sin t dt = R 0 yδ ds m C 0 π R sin t . R sin t dt R π R m

C

>. T R# .$ O&0 δ 0 0 C (0&f B&Q ZM x= *!' 0 m

=

δds C

0 π =

x

⎫ ⎧ − → − → − → − → → − → ⎨ −→ − A+B −→ B + C −→ C + A ⎬ + BC + CA AB m⎩ ⎭

−→ − → − → −→ − → − → −→ − → − → AB( A + B ) + BC( B + C ) + CA( C + A ) → −→ −→ (− AB + AB + CA)

R0

√ √ −−−−−→! −−−−−−−→! C : r(t) = ( cos t) , , + ( sin t) , − , −−−−→ √t− + (, , ) ; ≤ t ≤ π

#$ 5&0 0

* * *−−−−−−−−−−−−→* *(−R cos t, R sin t)* dt * *

*(, πR/) E ( >.&[, C cM .$ & &Q ]) N# E #E 3.& M . T .$ A, .8 90&0 . 5F!\f cM O&0 O

−→ xB + xC −→ xA + xB + BC AB m −→ xC + xA +CA

!\f cM .$ ( [& !0&1 z y 0&6 >. T 0 E ( >.&[, Δ aa P = (x , y , z )

δ ds 0Cπ δ(R cos t, R sin t) 0 π (R sin t)R . R dt 0 π 9: t dt; 8R sin δ ds

=

ΔABC = AB + BC + CA

0

X R# 0

0

=

√ −−−−−−−→ +( cos t)( , − , ) + 0 π δ dt

=

πδ

= = =

y

* √ −−−−−→ * δ *(− sin t)( , , )

= = =

√ −−−−−→** dt ( , , )*

0

m

xδ ds C

0 π √

πδ √ − π 0 m

√

√

cos t + sin t δ

dt

yδ ds C

πδ √ − π

0 π √

cos t + sin t δ

dt

[& . . −z ' C & .& 6S *( δ = x +y +z *M >. T 0 . C y .&3 *!' −−−−−−−→ −−−−−→ C : r(t) = ( − t)(, , − ) + t( , , ) −−−−−−−−−−−−→ = (t, − t, t − ) ; ≤ t ≤

.$ *ZM .&3

0 Iz

! x + y δ ds

= C

0 =

0 =

t + ( − t)

t + ( − t) + (t − )

*−−−−−−−→!* * * ×* , − , * dt √ (t − t + )(t − t + ) dt √

=

. T−xy .$ e1 x + y = R &Q # $ A, .8 5F & .& 6S RU R+- Z$ 5 .$ x = y = z ' *M oL6 . 5F uQ `&UO . x = y = z y & LB$ Y\ N# T&= ( 4E_ *!' 5F $& . $0 X = (, , ) y R# &SV 5 Q *Z0&0 &0 ( 0 0 . Mn T&= R# 0&0 *( v = −( −,−−,−→) −−−→ X X × v v −−−−−−−−−−−−−−→ (y − z, z − x, x − y) √ . (y − z) + (z − x) + (x − y)

h = = = 0 I

= C

= =

= =

δ

δ

&0 ( 0 0 O .& 6S R# 0&0 h δ ds 0

& ' (y − x) + (y − z) + (z − x) ds

C

0 π

(R sin t − R cos t) + (R sin t − )

*−−−−−−−−−−−−−−→* * * +( − R cos t) *(−R sin t, R cos t, )* dt 0 δ π R ( − sin t cos t) dt

πδ

R

( πR # $ / 1 @ ( (0&f δ = δ 5 Q ?U0 &0 ( 0 0 C uQ `&UO O&0 πRδ. 0 0 C. 4) *R = mI = R

−−−−−−−−→! C : r(t) = t − , t ;

B$&U 0 + 4) A ≤t≤

*M [& . ( δ = y/ 5F ) B&Q M B$&U 0 !\f cM 4) A; −−−−√ −−−−−−−−→ t ; C : r = t, t / ,

≤t≤

5&V .$ 5F ) B&Q M M [& . T .$ . *( q = /(x + ) 0 0 (x, y, z) &0 x + y = # $ Z E (x, y) Y\ M . T .$ A cM 4) O&0 δ = |x + y| 4) B&Q . $ ≤ y *M oL6 F !\f 0 $ x + y = z o1& pL S M . T .$ A Y\ ) B&Q O&0+z = − z = − >&T !\f cM O&0 δ = −z x + y 0 0 5F E (x, y, z) *0&0 . S [B x + y = # $ 0 $ i1 Z D M x= AH Y\ S *O&0 D [B C $ 0 x ≤ y T Z .$ e1 + O&0 δ = x + y ) B&Q . $ (x, y) ∈ D *M oL6 . C !\f cM

B = (, b, ) A = (a, , ) M x= A8 ]) N# E ABC da M . T .$ *C = (, , c) M M %&L . @ . c b a O&0 O & &Q *$0 . 1 ( , , ) 5&V .$ 5F !\f cM

.$ ] Z8 + ) +CcG . !) * δ(x, y, z) ) B&Q . $ C E (x, y, z) Y\ M . T .& 6S O&0 &q=−xyz 0.$ T &# y Y\ N# S O&0 T&= h M I = hδds E ( >.&[, S ' C . −x o Ǳ [ 0 0 S l&Q *( S & X = (x, y, z) &0 0 0 X 0 T−yz T−xy . −z . −y 0 ∗

S

C

(x + y + z )δ ds,

IO = 0

C

(y + z )δ ds,

Ix = 0

0 C

(x + y )δ ds,

Iz =

0 Ixy =

C

0 Ixz =

y δ ds,

C

G#U

(x + z )δ ds,

Iy =

C

+ R = I/m

0

z δ ds,

C

Iyz =

x δ ds.

C

>. T 0 . # fJ@ $. .$ *ZM

(, , − ) E y .&3 C M x= A& '() $ * ) B&Q . $ 5F E (x, y, z) Y\ ( ( , , ) &

!VO 0 . R# *!' C : r(t)

−−−−−→ −−−−−−−→ ( − t)( , , ) + t( , − , ) −−−−−−−−−→ ( , − t, t) ; ≤ t ≤

= =

*ZM [& . C 0 T y ( =&M *ZM .&3 Z#. $ ( 5F $ R0 !T E 0 0 y .&3 @ 5 Q T

= =

mean T C 0 T ds C C #0

=

*−−−−−−−→!* * * ( + t)* , −, * dt

*−−−−−−−→! −−−−−→!* * * − , , * ÷* , − ,

0 =

"

(

&

C : r(t) =

) C : x + y = , ≤ x, δ = |xy| −−−−−−−−−−−−−−−−−→! ) C : r(t) = sin t, t − , cos t , −π/ ≤ t ≤ π/, δ = |z|

) C

: |x| + |y| = , δ = |x|

) C

: max{|x| + |y|} = a, δ =

) C

: x + y + z = , z = , δ = z

C : r(t) =

g F 0 p&\ T&= y A *M [& . Ǳ [

3.& 0 f = z e0& y A;

−−−−−−−−−−−−−−−−−−−−→! cos t+ , sin t− , − t , ≤ t ≤ π

C

. $ C E (x, y, z) Y\ M x= ?1 < 2 U!U) 4) 0 Y\ P (x , y , z ) ( δ(x, y, z) ) B&Q −−−−−→ F = (α, β, γ) . $0 &0 P >. T R# .$ *O&0 m 5F .$ M $ O 6M C (+ 0

#E *

E#J

2oR

+

. !) *

0

α = β

=

γ

=

δ(x, y, z)(x − x ) ds C ((x − x ) + (y − y ) + (z − z 0 δ(x, y, z)(y − y ) ds Gm C ((x − x ) + (y − y ) + (z − z 0 δ(x, y, z)(z − z ) ds Gm C ((x − x ) + (y − y ) + (z − z Gm

) )/

*( u S (0&f G = × m /(kg.s) E ( >.&[, c C 0{&) 5 !"& 3 −

0

U= C

x +z =

C

:

Gm δ(x, y, z) ds ((x − x ) + (y − y ) + (z − z ) )/

C

x + z = x y=

(x − ) + z = y= −−−−−−−−−−→! : r(t) = cos t, , sin t :

C

= π ×

≤ t ≤ π

M $$S h'? >. T R# .$ ?U0 *( `&UO 0 # $ C

= π 0

mean f C

= =

π π

) )/ ) )/

C

e0& . \ y A& '() * * x &0 y = T $. 0 E !T&' # $ 0 *M [& . 9ZM .&3 . C 0 *!'

f = x + y + z

:

S A4 /z 5F B&Q e0& O&0 3.&

*M [& . T−xy π

M x= U!U) 2 a2 =C) . !) & * y >. j# .$ *$$S G#U C 0 f (x, y, z) e0& @ 0 C (= !M (+"1 :.& E ( >.&[, C 0 f . \

0 *mean f = f ds 9C / 1

*M [& . x + y ≤ x + y + B$&U 0 i1 D M x= A x + y 0 0 (x, y) ∈ D & $ S *( D [B C ( [& . C & $ y c D & $ y O&0 0 0 &#F M ( k j i .$ / g. &0 da C M x= A [& . C 0 f e0& y *f = x + y + z *M <

, x + y

−−−−−−−−−−−→! t cos t, t sin t, t ; π ≤ t ≤

' C 5F & .& 6S δ =

+ t) dt = 0 *

x/ + y / = a/

`&UO & .& 6S H & >&#+ .$ 0 % * 9M [& . Ǳ [ ' C uQ

=

f ds C

0 π &

' cos t + + sin t * * *−−−−−−−−−−−→* * × *(− cos t, , sin t)* * dt

.&3 !VO 0 E (x, y, z) Y\ & $ M x= A, .8 R# *O&0 T = xy+z 0 0 C = ( , , )( , − , ) y B$&U & $ 0 & Z$ . 1 #& S Bc# y N# .$ . *M [& . #& & $ 0

R# . #$ ('&" *( h =

E 0 5F E (x, y) *M [& . .&j' 0 *Z#0 C g F 0 h E ( =&M *!' 9ZM .&3 . g F C

/ + y/

:

x/ + y / = /

:

r(t) =

−−−−−−−−−−−−→!

cos t, sin t ;

≤ t ≤ π

&0 ( 0 0 .&j' [&) ('&" .$ 0 A=

h ds C

+ sin t

*−−−−−−−−−−−−−−−−−−−−−−−−→* * !* * ×* * − sin t cos t, cos t sin t * dt

0 π

=

= =

0 0

π

−−−−−−−−−−−−−→

∂U ∂U ∂U F= , , ∂x ∂y ∂z

* >. T R# .$ M ( RO. E !T&' 0{&) 5 !"& 3 '() * ) B&Q . $ 5F E x, y, z Y\ M ( , , )(, , ) *M [& . ( δ = x + y (x , y , z ) 5&V .$ m 4) 0 Y\ P ZM x= *!' m @ 0{&) 5 !"& 3 >. T R# .$ *O&0 E ( >.&[, C U (x , y , z ) = √ 0 Gm (( − t) + t) dt = ( − t − x ) + (t − y ) + ( − z ) )/ √ 0 = Gm t − (x + y − )t −/ +(x + y +( − z ) ) dt

0 = Gm (t−y − +x )

sin t cos t dt

π/

sin t cos t dt =

+ (x +y − ) + (z − )

= Gm ln −y +x , + ( −y +x ) +(x +y − ) + (z − ) ÷ +y −x , + ( +y −x ) +(x +y − ) + (z − )

0 *

−−−−√ −−−−−−−−→ t ; C : r = t, t/ ,

" 0 #&+3 A ≤t≤

5&V .$ 5F ( mj 5 c M (M' .$ &+3 R# *( q = /(x + ) E &0 (x, y, z) M mj ( . \ Q " !M 3.& !VO 0 3 A;

−−−−−−−−−−−−−−−→! C : r = cos t, − sin t, t ; h=

−/ dt

≤ t ≤ π

0 0 (x, y, z) Y\ .$ 5F `&. M ( *M [& . $ ^Y !M ('&" *O&0

+ z/

&$0.&M M ( RO. 78 * 0 + 5 , 0 *$ O+ T? _&0 $. 0 ` y 9M ) #E &a

Y0&- &0 5& E X"' 0 #&+3 A& '() * −−−−−−−−−−−−→ ( 5 c S *M (M' r(t) = t, + t , t + t ! (, V ( (0&f N# A M O&0 q = AV 0 0 5F =j

hB E &+3 ( mj !M 5 c ( &+3 hB *M oL6 . t = & t = 9Z#0 C &+3 " 0 q E ( =&M *!' 0

!VO 0 c= A

−−−−−−−−−−−−−√ −−−−−→ t/ ; C : r = t cos t, t sin t,

≤ t ≤ π

5F E (x, y, z) 5&V .$ V# VB .&0 B&Q M ( . R# 0 $ ) .&0 !M . \ *( q = x E &0 *M [&

Q=

q ds C

* * 0 , *−−−−−−−−−−→* * = A ( ) + (t) + ( + t) *( , t, + t)* * dt 0 =A

( t + t + ) dt = A

.& U#C g F !VO 0 ,&1 &0 .&j' A, .8 Y\ .$ `&. M ( .& .$ x + y = A!VO /

/

/

&F Z#0 h .$ ( ) R# &0 . C S *$ + .&3 5

(O 5 #E !VO 0 . −C −C : m(t) =

−−−−−−−−−−−→! sin t, cos t ; −π/ ≤ t ≤ π/

Y0&- &0 h : [−π/; π/] −→ [; π] S M $ O ) ;*7 !VO 0 *m(t) = r(h(t)) &F O&0 h(t) = π/ − t *$ O )

A `GF !F $4 5 4 #&0 &z \ 4$ ` y G#U 0 *ZM KY . . $0 U0& &q= .$ . $0 5 N# E . h * v e0& #$ 5&0 0 *F : R −→ R !VO 0 ( M . $ $ ) 5&Q F B e0 4&0 F(x, y)

=

−−−−−−−−−−−−−−−−−−−−−−−−→! P (x, y, z), Q(x, y, z), R(x, y, z)

=

P (x, y, z)i + Q(x, y, z)j + R(x, y, z)k

5 #E !VO 0 . T .$ . $0 5 0&6 >. T 0 $ + G#U

}

N# 0 ( ) 9;*7 !VO >. T $ 0 . C = (, , )( , , ) y .&3 A, .8 D. #E .$ 0 F E &# F 0 E $M . ) 5

9Z# $.F . $ 5$M .&3 −−−−−→! −−−−−→! C : r(t) = ( − t) , , + t , , −−−−−→! = t, , t ; ≤ t ≤ −−−−−→! −−−−−→! C : m(t) = ( − t) , , + t , , −−−−−−−−−−−−→! = − t, , − t ; ≤ t ≤

# U U# *m(t) = r( − t) (B&' R# .$ M $ O ) h(t) = − t Y0&- &0 h : [; ] −→ [; ] h $. .&3 *( (, , ) ( , , ) / g. &0 da C ZM x= A .8 U R# 0 ( V C >. T R# .$ *O&0 (, , ) 0 . C !B$ R+ 0 *$ O E& .&3 !1 ' 0 M *A$ O ) *7 !VO 0C $M . ) 5 >. T = C = −C C = −C C = −C M $ O h'?

*C = −C }

da N# 0 G L & ) 9*7 !VO 0 $ *

F(x, y)

=

−−−−−−−−−−−−→! P (x, y), Q(x, y)

= P (x, y)i + Q(x, y)j

e0 $. .$ T& H M x= * . T .$ A *** #n3t 6 3 hC ( v Q &B M ( H (T& . $ F . $0 5 Zg S

*O&0 H (T& . $ 5F M M x= 0 U!U) 2 7OR * N# E . h O&0 &q= .$ N# C : r(t) ; a ≤ t ≤ b #n3!" #$ #n3/ VU ( U0& C 0 I tJ >. T R# .$ h : [c; d] −→ [a; b] &

C : m(t) : = r(h(t)) ; c ≤ t ≤ d

( #n< VU h 5 Q *Z & h -+ C I tJ . ([a . + &# ( . + &# (c, d) 0 h (t) R# 0&0 u< I tJ h O&0 ([a (a; b) 0 h (t) S *O&0

7 % I tJ . h O&0 (c; d) 0 h (t) S 7 % %&L C 0 i jL0 .&3 N# l&Q *Zg S L3 5 $S0 ( ) & .&3 # U E M Z E&) $ O !+, *Z & % !( N# . C &z'?YT $ O $& ( ) &0 C *Z & ! % . 0 ( ) %&L & . Q.&. T $ 0 *$ + .&3 5 t#@ 0 . R# *C : x + y = M x= A& '() * −−−−−−−−−−−→! t ∈ [; π] M r(t) = cos t, sin t >. T 0 .

k

0 ) C

0 = a

) )

Y0&- &0 h

(a F + a F ) • dr

0

0

C +C

F • dr =

0

−C

C

0 F • dr + a 0

F • dr +

0

C

F • dr = −

C

F • dr

F • dr C

F • dr C

0 ` y M $ O ) 78 * * "0 4$ ` y M B&' .$ $. "0 ) *$. $ 5F 0 4& &0 F = P i + Qj + Rk l&Q 78 1 * (O 5 F dr C

•

0

P dx + Q dy + R dz C

V# . $0 $& & "M S ?U0 &0 &F Z#0 v = (cos α, cos β, cos γ) . (O 5

T = r /r 1 C F • dr

0

0 F • dr =

(P cos α + Q cos β + R cos γ) ds

−−−−−−−−→! C : r(t) = cos t, sin t ;

≤ t ≤ π

E ( >.&[, C 0 F .$

0 π

0 F • dr

=

C

=

0 π −−−−−−−−−−→! −−−−−−−−−−→! − sin t, cos t • − sin t, cos t dt 0 π

=

! −−−−−−−−−−→! F cos t, sin t • − sin t, cos t dt

C : r(t) ; a ≤ t ≤ b

>.&[, &0 O&0 O $ $ ^- .&3 Q y C S . Q R# & .&[, ` + _&0 &" .$ ( . (+ 9ZM $& 0

0

F • dr =

−−−−−−→! 0 . F = −−xy, yz, zx . $0 5 A, .8 −−−−−→! C : r = , t, t ; − ≤ t ≤

0

F dr +

C

C

r(t)

−−−−−−−→ = (x(t), y(t))

0 F • dr =

0

F dr + · · · + C

F dr Cn

M . T .$ &j L 5&0 0 −−−−−−−→ Q(x, y)) &F F = −(P−−(x,−−y),

0 b ! ! P x(t), y(t) x (t)+Q x(t), y(t) y (t) dt a

C

−−−−−−−−−−→ 5 r(t) = −(x(t), y(t), z(t)) M . T .$ ?U0 −−−−−−−−−−−−−−−−−−−−−→! Z#. $ *O&0 F = −−P−(x, y, z), Q(x, y, z), R(x, y, z) 0

0 F • dr = C

dt = π

! F r(t) • r (t) dt

( O x= &# .$ M ZM G#U

γ

*( O . ) &aa ( ) 0 M O&0 ' # $ C *M [& . C 0 F ( &aa 5F ( ) C : x + y = 5 Q *!' Z"#

b

a

C

C

−−−−−−−−−−−−−−−−−−−−−−→! F = − y/(x + y ), x/(x + y )

$ 5&6 A

*( .&3 # U h(t) = arcsin t C : r(t) = −−sin−−t,−−cos−−t,−→t! , −π ≤ t ≤ π &S A; ]< ( .&3 v N# h(t) = πt M $ 5&6 *"# 0 . h y C .&3 # >. T Q 0 . R .$ C : x + y = q0 A *M 5&0 . N# $M . ) 5

y M ( x + y = [B $ C M x= A Q 0 . C * O ) z = z = x >&T *M .&3 . N# $M . ) 5 >. T M ( x + y + z = M E +"1 S M x= AH Q 0 . C *O&0 5F [B C $. $ . 1 Z 6 N# .$ *M .&3 . N# $M . ) 5 >. T ( . ) N# C M x= % * ` yC >. T R# .$ *O&0 . $0 F !VO 0 . C 0 F A4$

0

β α M $ O ) *( ` y N# M . −z . −y . −x &0 T M " # E X 0 *$E&

M x= A& '() *

: [− ; ] −→ [−π/; π/]

! P x(t), y(t), z(t) x (t) a ! +Q x(t), y(t), z(t) y (t) ! +R x(t), y(t), z(t) z (t) dt b

C C . $0 5 F F F S & * >. T R# .$ O&0 LB$ $ , a a C

*M [&

Z#. $ G#U N+M 0 *!'

*M [& C 0 . F $M .&3 . C $. .$ 9$ O ) RV+ & ) 0

F • dr C

−−−−−−−→ ) F = (z, y, x), C : x + y + z = , z = , ≤ y

) F = yi + (x + y)j,

) F = (x + y + z)

C

=

D : x ≥ , y ≥ , x + y ≤

(, , ) ( , , ) / g. &0 da A .8 ( , , ) . $0 $ . ( ) &0 M ZM . ) . @ . ! 5F 0 F = −z,−−x−−+−y,−−−→ z . $0 5 E *O&0 .&SE& *#0 >. T 0 . . Mn Δ da #&0 GB *7 !VO 0 h *!' >. T R# .$ & *ZM #c Δ = i j + j k + k i

−−−−−−−−−−−−→! ) F = x, y − z, x + z , S : y ≥ x , z = x + , y ≤

) F =

−

t dt =

(, , )

0 . F $M . j S &# D [B . C $. .$ 9$ O ) RV+ & ) 0 *M [&

) F = x j,

−−−−−→! −−−−−−→! F , t, t • , , t dt

−

≤y≤

D : x + y ≤ , ≤ x

−

0

! i+j−k ,

) F = yi + j,

=

0 −−−−−−→ ! −−−−−−→! t, t , t • , , t dt =

C : y = x , y ≤

C : x = y = z,

0

0

−−−−−−−→ −−−−−−−−→ /x, /y), C = ( , )(, )

) F = (

−−−−−−−−−→! √ z , x + y, z , S : x + y + z ≤ , y = z

−−−−−−−−→! − t, t, ; −−−−−−−−→! : r(t) = , − t, t ; −−−−−−−−→! : r(t) = t, , − t ;

i j : r(t) =

≤t≤

jk

≤t≤

ki

}

0

0 Δ

0

≤t≤

&0 ( 0 0 h $. 0

F • dr =

F • dr + F • dr + F • dr ij jk ki 0 −−−−−−−−→! −−−−−−−−→! F − t, t, • − , , dt

=

+

0 −−−−−−−−→! −−−−−−−→! F , − t, t • , − , dt

0 −−−−−−−−→! −−−−−−−→! F t, , − t • , , − dt +

. h 0 & 5$ + .&3 9*7 !VO 4$ ` y [&

0 −−−−−→! −−−−−−−−→! , , • − , , dt

=

+

A `GF !F N (4 5F e[ 0 N#c= .$ , &$0.&M 4$ ` y *$ O M{ &# .$ & F E $. & M $. $ .$ M x=

L . !) 2 L *

C : r(t) ; a ≤ t ≤ b

. $0 F ( 5& E X"' 0 M (M' "

t +Δt & t & E E&0 S *$$S G#U C 0 M ( Z $ O &M d0 (M E V# 50 Z#0 h .$ . r(t + Δt ) & r(t ) E Z\ " " 0 w M ZM x= i

i

i

i

i

i

0 −−−−−−−−→! −−−−−−−→! t, − t, t • , − , dt

0 −−−−−−−−−−−−→! −−−−−−−→! − t, t, − t • , , − dt + 0

0 dt +

=

(t − ) dt +

0

9#0 C 0 F E $. .$

(t − )dt =

0 *

−−−−−−−−→! −−−−−−→! ) F = x, x − y , C : r(t) = t+ , t −t ; ≤ t ≤

) F =

−−−−−−→! −−−−−−−−→! x, yz, xyz , C : r(t) = t , t , t ; − ≤ t ≤

) F = yi + xj,

C : r(t) = ti + t j ; − ≤ t ≤

) F = zi + yj + xi, C : r(t) = sin ti + cos tj + tk ;

≤ t ≤ π

0 π =

n

=

n

0 π

0 π =

n

F a sin t, a cos t

! −−−−−−−−−−−→! • a cos t, −a sin t dt

−−−−−−−−−−−−−−−−−−−−−−−→!

•

=

dt = nπ

=

⎧ ⎨ x + y = : y=z ⎩ ≤ x, y 4 −−−−−−−−−−−−−−−−→! r(t) = cos t, sin t, sin t : ≤ t ≤ π/

−−−−−→! −−−−−→! : r(t) = ( − t) , , + t , , ; −−−−−→! −−−−−→! : r(t) = ( − t) , , + t , , ;

≈

M B&' .$ O 4& .&M N#c= t0&Y V# ^- *( $ %qT&' 0 0 ( (0&f 5 Y 5&V v S_ q1 E A;C .$ *( AC . $0 .$ 5&V v . $0 t +Δt t R0 $, ξ M Z# $M $& . $0 e0 0 O 4& .&M !M &+B D. 0&0 R# 0&0 *O&0

0 0 F 5 f& ( w y i

i

i

=

π/

=

0

π/

≤t≤ ≤t≤

•

−−−−−−−−−→ C : r(t) = ( + t , t, t ) ; − ≤ t ≤

F = zi + (x − y)k

W

= =

= =

C

F • dr +

C

0 =

+ t , t, t

! −−−−−−−→! • t, , t dt

−

−

t + t − t

!

dt =

# $ $S 0 .&0 n M M x= A, .8 . $0 5 f& ( Q &aa ]V, ( )

x + y = a

F • dr

0 −−−−−−−−−→ −−−−−−−→ (, t − , ) • (−, , ) dt +

−

F

0 −−−−−−−−−−−−→ ! −−−−−−−→! t , , + t − t • t, , t dt 0

0

0

C

F • dr 0

. $0 5 f& ( M (M' *M [& . O 4& .&M *$ $ . 1 Z#. $ B&" &$ $ 0 ) &0 *!'

F dr =

=

(cos t − ) dt = −π

C

F • dr C

0

−−−−−−−−−−−−−−−−−−→! − sin t, cos t, cos t dt

0

! F r(t) • r (t) dt =

*Z#0 C 0 F dr E ( =&M ` + .$ ]3 *( " 0 M M x= A& '() *

0

=

ΔWi

a

−−−−−−−−−−−−−−→! sin t, − cos t, •

Δti →

=

C

0

n /

lim

n→∞ i= 0 b

F • dr

= =

W

i

W

.$

0 W

≈

()

9( y .&3 $ `&+ ) C

C

−−−−−−−−−−→ −−−−→! r(ti )r(ti + Δti ) • F r(ti ) ! ! r(ti + Δti ) − r(ti ) • F r(ti ) ! F r(ti ) • r(ξi )Δti ! F r(ti ) • r (ti )Δti

()

ΔWi

−−−−−−−−−−−→! a cos t, −a sin t dt

Y\ E C " E .&[V# M A .8 0C C " E #$ .&0 $. B = (, , ) Y\ 0 x= 0 . $. .$ O 4& .&M A$ O ) % *7 !VO *M [& F = yi − xj + k ( q0 C " *!'

C

i

a sin t + a cos t, a cos t − a sin t a

A = (, , )

C

*( F(r(t )) 0 0 UY1 R# 0 c F M (M' 0 0 UY1 R# .$ O 4& .&M X R# 0

0 −−−−−−→ −−−−−→ (t, , ) • (, , ) dt

dt =

$ O ) O 4& &.&M >& 0 0 *

. $0 5 0 . **7 &a E (+"1 A *M !' F = xi + j + zk

! F = (x + y)i + (y − x)j /(x + y )

*M [& . w y O 4& .&M *$ $ . 1 >. T 0 . R# *C : x + y = a ZM x= *!' −−−−−−−−−−→! C : r(t) = a sin t, a cos t ;

≤ t ≤ π

.&0 n w 5 Q *$M .&3 5 / VU ( ) .$ . ! &M .$ N# .$ O 4& .&M ( =&M ( Q R# 0&0 *ZM %- n .$ . !T&' ]< $ + [&

0

W

=

F • dr

n C

3.& $ .$ . [, 5&#) . \ A, .8

−−−−−−−−−−−−−→ C : r(t) = ( cos t, t, sin t) ;

≤ t ≤ π

*M [& . F = zi + j − xk . $0 5 y &0 ( 0 0 h $. .&O . \ G#U N+M 0 *!'

0 Fn =

F • dr C

0 π = =

cos t, t, sin t

! −−−−−−−−−−−−−−−→! − sin t, , cos t dt •

0 π −−−−−−−−−−−−−−→! −−−−−−−−−−−−−−−→! sin t, , − cos t • − sin t, , cos t dt 0 π

=

C

F

−dt = −π

0 % *

9M [& .

−−−−−−→! −−−−−→! ) F = x, y, z , C : r(t) = t , t , t ; − ≤ t ≤ −−−→! , ) F = x, y / x + y , C : x + y =

) F =

−−−−−−−−→! yz, xz, xy , C : y = x = z , y ≤

) F =

−−−−→! x, y , C : x/ + y / =

U!U) < : 9 IG 8 . !) 2 L & *

0 M ( . $0 F M x= !"# .$ N# C : r(t) ; a ≤ t ≤ b O G#U Ω '& *ZM [& . C E F y .nS .&O *( Ω C &0 $ + $& &+B D. E . h R# 0 E V# 50 *Z#S h .$ . r(t + dt) & r(t) E UY1 >.&[, UY1 R# M ZM x= Z $ O &M d0 (M *dr(t) = r(t + dt) − r(t) 9( r(t + dt) & r(t) E y .&3 E (0&f dr(t) 0 F 5 M ZM x= Z Rl+ y dr(t) E .nS .&O!X R# 0 *( F r(t)! 0 0 0 $ +, $ .$ F r(t) # j @ &0 ( 0 0 F(r(t)) Z#. $ #$ >.&[, 0 dr(t) @ .$ %- dr(t) dFn

≈

dr(t) dr(t)

⊥ dr(t)

= F(r(t)) • (dr(t))⊥ −− −−−−− !−−−−−−→ ! −−−−−−−−−−−→! = P r(t) , Q r(t) • d y(t), −d x(t) ! ! = P x(t), y(t) y (t) − Q x(t), y(t) x (t) dt

9E ( >.&[, F y C E . [, .&O !M R# 0&0 0

0

Fn =

P dy − Q dx

dFn = C

C

−−−−−−−−−−−−−−−−−→! cos t, sin t, t + ; −π ≤ t ≤ π

C : r(t) =

. F f& C y O 4& .&M $. .$ O&0 M [&

GB C F = xi + j + k %C F = x + yj + zk ! :C F = −x,−−x−−+−y,−−x−−+−y−−+−→ z $C F = (x + y + z) i + j + k! <

$ .$ F .nS 5&#) $. .$

−−−→! F r(t) •

!VO 0 3.& C M x= A;

C)

Y R

. !)

2 L *

'& .$ M ( . $0 F M x= U!U) N# C : r(t) ; a ≤ t ≤ b M x= *( O G#U Ω C $ .$ F .nS 5&#) Z *( Ω .$ e1

F 5&#) E +' Q M ZM oL6 U# *ZM [& . 4& &+B D. 0 . .&M R# *M (M' C $ .$ .$ r(t + dt) & r(t) E UY1 C !M &0 ZM x= *Z$

ZM x= Z d0 (M E O &M 50 *( .& dr(t) = r(t + dt) − r(t) ( y .&3 N# UY1 R# M .$ *O&0 F(r(t)) 0 0 (0&f UY1 R# 0 F ?U0 .$ F(r(t)) # j @ 0 0 UY1 $ .$ 5&#) >. T R# #$ 5&0 0 *( dr(t) @ .$ %- dr(t) $ d Fn = F(r(t)) • dr(t)

E ( >.&[, C $ .$ 5&#) !M .$ 0

Fn

b

d Fn

= a

0

b

=

! F r(t) • dr(t)

0a F • dr

= C

$ .$ 5&#) . \ A&

'() $ *

−−−−−−−−−→! C : r(t) = t, t + , t ; ≤ t ≤ −−−−−−−→! F = x , y , z

. $0 5 y *M [& . &0 ( 0 0 h $. 5&#) . \ G#U N+M 0 *!' 0

Fn =

F • dr C

0 = =

F

,t

! −−−−−−→! dt • , t,

0 −−−−−−−−−−−−−−−−→ ! −−−−−−→! t , t + t + , t • , t, dt 0

=

t, t +

t + t + t + t

dt =

& F & A & F 0 M O&0 Ω .$ $ C M M y O 4& .&M &#F M ( R# * " M ( M y O 4& .&M 0 0 M (M' C 0 y O 4& .&M &#F j L 5&0 0 M (M' C 0 "0 " %&L 0 . $ . 1 F f& ( M Ω .$ & M

0 F Zg S O&0 R# &3 l&Q &# $. $ *Z#E $3 4 R# t1$ UB&Y 5&0 0 ?z#{ *( #&\0 Ω B

M ( . $0 F M x= $ * ( !9 Ω 0 F Zg S . T .$ *$$S G#U Ω 0 e1 C C $ Ω E B A Y\ $ Ǳ E&0 M 0 F 0 0 C 0 F B & A 0 &0 Ω .$ *A$ O ) GB H*7 !VO 0C O&0 C

E .nS .&O A&

x + y = −−−−−−−−−−−−−−−−−−−−−→ F = x/(x + y ), y/(x + y )

.

'() * *

. $0 5 y *M [&

>. T 0 . R# *!'

r(t) =

−−−−−−−−−−−→! cos t, sin t ;

≤ t ≤ π

E ( >.&[, C E .nS .&O .$ *ZM .&3 0

Fn =

P dy − Q dx

0

C π

( cos t) d( sin t) ( sin t) + ( cos t) ( sin t) d( cos t) − ( sin t) + ( cos t) 0 π ( cos t)( cos t dt) − ( sin t)(− sin t dt) =

=

0 π

*O&0 Ω 0 . $0 5 F M x= $ *

Ǳ E&0 M ( !9 Ω 0 F Zg S . T .$ 0 0C F dr = 9O&0 T C 0 F Ω .$ C "0 *A$ O ) GB H*7 !VO }

=

( + sin t) dt = π

E .nS .&O A, .8 −−−−−−→ C : r(t) = ( − t, t ) ; − ≤ t ≤

•

C

*M [& . F = x i + yi . $0 5 y Z#. $ J**7 0 ) &0 *!' 0

Fn =

P dy − Q dx C

=

#&\0 5 G#U GT 9H*7 !VO *B$&U ; G#.&U $ * O&0 g&\0 Ω 0 G#U .$ [U 0 F ZM x= R# .$ *ZM %&L . Ω .$ e1 C LB$ "0

ZM %&L C 0 . B A !a >& Y\ $ S >. T M C = C + C &F Z &0 C C . !T&' UY1 $ O E&bF B E C M 3 Z B 0 O E&bF A E C E −C C $ .$ *M 3 Z A 0 & F 0 F nB M 3 Z B 0 O E&bF A R# 0&0 *( &"

0

0

F • dr = C

0

F • dr − C

−C

−

0 ( − t) (t) − (t )(− ) dt = − =

−

y C E . [, 5&#) $. $ .$ 0 * 9M [& . F . $0 5

−−−−−−→! −−→! *C : r(t) = −−t −sin−−t,t−−cos t ; ≤ t ≤ π F = y ,−x A + ( ) &0 C : x + y = F = −(x,−−→ y)/ x + y A; *&aa

−−−−−−−−−−−→ ( ) &0 C : x + y = F = x + y,x + y! A *&aa

−−−−−→ −−−−−−−−→ *C : r(t) = t,sin t!; −π ≤ t ≤ π F = xy,x +y! A /

F • dr =

*O&0 g&\0 Ω 0 ; G#U .$ [U 0 F nB $ O&0 g&\0 Ω 0 ; G#U .$ [U 0 F ZM x= A 0 &0 Ω .$ e1 C C $ Ω E B A Y\

0 ( − t) d(t ) − (t ) d( − t)

/

# )' +4 , + 0 M ( . $0 5 N# F M x= C " Ω E Y\ $ B A ZM x= *$ O G#U Ω

O $.F H !j= .$ ?z[1 AC A;C 4&V' 5$ 0 $&U

() ( $&U () &0 ( ) M ZM >&[f &# .$ *( *( $&U () &0 c e0& $ 0 g&\0 Ω 0 F ZM x= ( ) ⇒ () f (A) M ZM G#U >. T R# 0 Ω 0 . f \\' . \ &0 A 0 H E HA ( . y .&3 0 F y O 4& .&M 0 0 #$ >.&[, 0 O&0

>. T R# .$ *ZM %&L LB$ >. T 0 B & R# 0&0 ( Ω .$ "0 N# C = C − C

.$ *( T C 0 F F • dr −

0 f (A) :=

F(( − t)H + tA) dt

A .$ 0 &0 Ω .$ LB$ N# C ZM x= &' C = HA+C −HB >. T R# .$ *O&0 B .$ & .$ *( T 5F 0 F nB ( "0 Ω .$ 0

F • dr

= C

0 0 F • dr + F • dr − F • dr HA C HB 0 = f (A) + F • dr − f (B) 0

=

C

*O >&[f AH*7 C &" R# 0&0 C C O&0 . 10 AC ZM x= () ⇒ ( ) A .$ 0 &0 $ R# .$ O&0 Ω .$1 e1 B .$ & 1 w 6 . \ &0 F dr F dr $ >. T *( . 10 AC nB 0 0 f (B) − f (A) T A;C .$ Ω f F ZM x= () ⇒ () $ Ai = , &0C C : r ; a ≤ t ≤ b ZM x= *M >. T R# .$ *O&0 w 6 & 0 &0 Ω .$ LB$

•

C

•

C

i

0

0

b

F • dr = C

0 + 0

i

P (r(t)) x (t) dt

a b

0

b

Q(r(t)) y (t) dt + a b

R(r(t)) z (t) dt

a

dx ∂f dy ∂f dz ∂f (r(t)). + (r(t)). + (r(t)). dt ∂y dt ∂z dt a ∂x 0 b b d () f (r(t)) dt = f (r(t)) = dt a a = f (B) − f (A). =

*( O $& t 6 E ,&1 E AC .$ V# ^- *( . 10 AC R# 0&0 Y\ *O&0 . 10 AC ZM x= () ⇒ () & 5F 0 N#$c .&"0 Y\ Ω E A = (x, y, z) LB$ S >. T R# .$ *Z#S h .$ . B = A + (ε, , )

0

0

C

0

F • dr = C

C

F • dr =

*O&0 g&\0 Ω 0 G#U .$ [U 0 F B 4&0 q1 0 E& . $0 5 5$ 0 #&\0 t\ 0 )*TFC K?YT q1 R# .$ 9Z#. $ #&\0 & q1 *( O ^#6 *;*H .$ M $. $ $ ) >?@ + !VO .& , + Ω M x= $ * *O&0 #n<\ 6 Ω 0 M ( . $0 F = −−P,−−Q,−−R→! 9B$&U #E 4&V' >. T R# .$ U# ( g&\0 Ω , + 0 F . $0 5 A " E !\ " Ω .$ e1 & 0 F E B *( Ω 0 F dr = P dx + Q dy + R dz " #$ >.&[, A; $. $ $ ) 5&Q Ω 0 #n<\ 6 f U0& U# ( ! &M #$ >.&[U0 *df = F dr M ∂f ∂f ∂f = P, = Q, = R. A*7 C ∂x ∂y ∂z $&U 5&0 0 &# 0 0 0 f = P dx, f = Q dy, f = R dz. A;*7 C 2

•

•

(0 . f 5 R# . \ "#&\ &0 M *$.F Ω 0 F dr = P dx + Q dy + R dz " #$ >.&[, A >.&[U0 *( T Ω 0 5F !" #$ U# ( t1$ #$ ∂Q ∂P ∂R ∂Q ∂R ∂P = , = , = . A*7 C ∂y ∂x ∂z ∂x ∂z ∂y •

0 M $. $ $ ) S 0 Ω 0 #n<" #$ f U0& A B & A 0 &0 Ω .$ C LB$ E Z#. $ 0

F • dr = f (B) − f (A) C

A*7 C

*Z & F L >E I . _&0 .$ f e0& S M O&0 Y\ H $ 0 !VO .& Ω ZM x= !T y' .&3 ZM !T 5F 0 . A !a Ω E LB$ Y\ *$0 &) Ω .$ &z &+

0 M $. $ $ ) S 0 Ω 0 #n<" #$ f U0& A B & A 0 &0 Ω .$ C LB$ E Z#. $ 0

-&#. E S_ q1 0&0 &F O&0 B 0 A E y .&3 C M $. $ $ ) 5&Q ε I R0 δ $,

+, ε

0

F • dr

AH*7 C

F • dr = f (B) − f (A) C

y

∂P = ey − y ∂y

! y

= ey −

y

!

= xey − x +

x

∂Q ∂x

=

E ( >.&[, 5 R# !"& 3 X R# 0 f = xey − xy − y +

#E (" #&\0 F =

(0&f

−−−−−−−−−→! x + y , xy

5 A, .8

! ! ∂P ∂Q = xy x = y = y = x + y y = ∂x ∂y

*y = y R# 0&0 y = 5F 0 M . −x 0

5 A .8

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→! F = yz + xy+z + ,xz +x + yz + ,xy+ xz +y +

G#U Ω 0 F ( !VO .& R _z #E ( #&\0 R 0 &f .$ $ O

∂P ∂y

= = =

∂P ∂z

= = =

∂Q ∂z

= = =

yz + xy + z +

! y

f (B) − f (A)

=

∂f (x + δ, y, z).ε ∂x

>. T 0 . C =@ E

*Z & F L >E I . _&0 .$ f e0& Y0&- &0 F . $0 5 A& '() % $ * −−−−−−−−−−−−−−−→! R 0 F _z #E *( #&\0 R 0 e − y, xe − x − &f .$ $ O G#U

=

C

r(t) = ( − t)A + tB = (x + tε, y, z)

.$ * ≤ t ≤ M $ + .&3 5

0

0

F • dr

=

C

(=&# < δ

P (x + tε, y, z) dt

= P (x + ε, y, z) − P (x, y, z)

N# < ε E 0 M O (0&f ]3 M $ O

< ε

P (x + ε, y, z) − P (x, y, z) =

∂f (x + δ, y, z).ε ∂x

(x, y, z) E 0 M Z#S t 6 #& V# !B$ 0 (x, y, z) = P (x, y, z) Z#. $ Ω E $ 0&6 >. T 0 * ∂f ∂x 2 *$$S A*7 C .$ #$ &" O&0 T .$ . $0 5 S M $ O ) 9$ + 5&0 5 #E >. T 0 . H*H*7 q1 &F !VO .& , + Ω M x= $ $ * −−−−→! Ω 0 M ( . $0 F = P, Q Oxy T .$ 9B$&U #E 4&V' >. T R# .$ *O&0 #n<\ 6

U# ( g&\0 Ω , + 0 F . $0 5 A " E !\ " Ω .$ e1 & 0 F E B *( ! &M Ω 0 F dr = P dx + Q dy " #$ >.&[, A; M $. $ $ ) 5&Q Ω 0 #n<\ 6 f U0& U# ( #$ >.&[U0 *df = F dr •

z + x xz + x + yz +

!

yz + xy + z +

!

y + z

x

!

xz + x + yz +

!

xy + xz + y +

∂Q ∂x

•

∂f ∂f = P, = Q. ∂x ∂y

z

xy + xz + y +

x + y

=

x

=

∂R ∂x

z

! y

=

∂R ∂y

E ( >.&[, 5 R# !"& 3 f = xyz + x y + y z + z x + x + y + z +

(0&f

0 f=

0 P dx, f =

$&U 5&0 0 &#

Q dy.

(0 . f 5 R# . \ "#&\ &0 M *$.F ( t1$ Ω 0 F dr = P dx+Q dy " #$ >.&[, A #$ >.&[U0 *( T Ω 0 5F !" #$ U# •

∂Q ∂P = . ∂y ∂x

=

−xy

# = ∂P ∂z

#

= = =

∂Q ∂z

= = =

(" #&\0 F =

(x + y + z )/ y + x + y + z

" = "x

x + = x + y + z z −xz (x + y + z )/ # " z + = x + y + z x # " y + = x + y + z z −yz (x + y + z )/ # " z + = x + y + z y

∂Q ∂x

∂P ∂y ∂P ∂y ∂Q ∂z

= = f

= = =

f

= = =

=

y

∂R ∂y

! ∂P = x+y y = ∂y

x

=

f

= 0

=

x

∂Q ∂x

! x + xy + A(y) x + y dx =

0 f

!

P dx

=

+ dx x + y + z , x + y + z + A(y, z) 0 Q dy

= x−y

E ( >.&[, c F !"& 3 ?U0

0

x

=

Q dy 0

! y + B(x) x − y dy = xy −

=

R# 0&0 * LB$ U0 B A M

y

+ dy x + y + z , x + y + z + B(x, z) 0 R dz

f=

z + dz x + y + z , x + y + z + C(x, y)

0 (,−) ( ,)

x

+ xy −

y

+

(0&f Z#. $ [ .$ 0

(x + y) dx + (x − y) dy =

= f (, −) − f (, ) = −

Z#. $ $. R# Ǳ"#&\ &0 R# 0&0 (0&f

F • dr C

Ǳ [ E ( , , )0& (, , ) E " M . T .$ A4 .8 x dx + y dy + z dz + *M [& . *$.n x + y + z 0 (, , ) E LB$ N# C B&" R# .$ *!' E 0 . R# *$.nS+ Ǳ [ E M ( ( , , ) . 1 ≤ x &q= Z .$ ?zM &# M $M #c 5 & E Z R# E N# 5 Q *x ≤ &q= Z .$ ?zM &# . $ x dx + y dy + z dz F = + . $0 5 VO .& &&q= x + y + z 5 Q c ( #n<\ 6 & F E N# 0 ( , , )

( , , )

.$ 0 (,,) x dx + y dy + z dz + x + y + z ( , , )

= = y = xy

( , )

P dx

, f = x + y + z +

!

∂Q ∂x ! ! ∂R x − z z = −z = = z x = ∂x ! ! ∂R xy z = = z y = ∂y x − z

( ,− )

0 =

=

!

. $0 5 A .8 #E

*( #&\0 . −x 0 & 5 U# y = z = VF

0 . x + y! dx + x − y! dy . \ A .8 *M [&

( LB$ (, −) & (, ) E C " &# .$ *!' ! 0 F #E *( #&\0 5 R# *( F = −x−−+−−y,−x−−−−→ y Rl+ ( #n<" #$ R !VO .& , +

∂R ∂x

0 *( #&\0 F R# 0&0 * LB$ U0 C B A M 9ZM !+, #E D. 0 F !"& 3 5$.F (0 f

=

−−−−−−−−−−−→! x − z , xy, z

=

f ( , , ) − f (, , )

=

−=

5 M M oL6 $. .$ 0 & $ * $ 0 #&\0 F l&Q *( #&\0 , + Q 0 O $ $ 9M oL6 . 5F !"& 3

∂P ∂y

#

=

x

+ x + y + z

"

= y

*$ O $.F G#U Q M x= % * Zg S . T .$ *O&0 O 5&0 .&3 N# & &0 . "0 Q `&+ ) *r(a) = r(b) M ( )E C # L . T .$ . C *Z & "0 c & $ c0 M $ + .&3 . @ F 5 0 M Zg S !J": $ 6 $ +3 .&0 $ .&3 N# y Y\ .&3 *$ 6 $ +3 c .&3 $ y Y\ ?U0 *O&0 UY1 $ 50 "0 M Zg S Lk . T .$ . C C : r(t) ; a ≤ t ≤ b

.$ " 5 $.W c b a $. J*7 !VO .$ '() % * & F M ( 5F g f d !V6 * " g f e d M B&' *(" "0 (e) . $ UY1 $ }

−−−−−−−−−−−−−−−→ ) F = x + y , x , y + y −−−−−−−−−−−−−−−−−−−−→ , ) F = /y − y/x , /x − x/y / x + y , , ) F = (xi+yj) / x +y ) F = (yi − xj) / x + y

) F = (x + y)i − xyj ) F =

)

0

)

5.$ &0 5 $.W N# C M x= % * M ( ) C + 7$8 7 % E . h *( D !VO 0 *$. $ . 1 D , + w |Q (+ . + 5F .$ *$ O ) *7

(,−)

0 (,)

5 $.W b 5 $.W 9J*7 !VO

) A

x dx − y dy + z dz

0 (,)

)

)

0 /&+ b X E E&bF &0 `&UO ⇔ X ∈ Ext(C) A% Int(C) *M eY1 Y\ :E $ U .$ . C , + C 7*7 !VO 0C Z & C L . Ext(C) C L . 0 Y\ N# X .$ Y\ N# X $ O ) *A( C

( ,,)

( , ,)

)

0 /&+ b X E E&bF &0 `&UO ⇔ X ∈ Int(C) AGB *M eY1 Y\ $= $ U .$ . C , + C

−−−−−−−−−−−−−−→ , xz + , xy)

) F = (yz −

Q M M oL6 *M [& . #E & B E N# *( E& O & & 0 E "

)

.$ 5 $.W N# C M x= Yk % * `&+ ) >. T 0 . T !M >. T R# .$ *O&0 T M . @ (O 5 C ∪ Int(C) ∪ Ext(C) c , +

−−−−−−−→ x, y , z

) F = (x/y) i + j

( ,)

0 (,

(xy − ) dx + (x + ) dy

(x − y)(dy − dx)

)

(−,−)

0

( ,, )

( , , )

)

0 (, ) x dx + y dy + x + y (, )

(x + xy ) dx + (x y − y ) dy

(x − yz) dx + (y − xz) dy

0 (,−,)

+(z − xy) dz

(x + y − z) dx + (x + y − z) dy +(x + y + z) dz / x + y + z + xy ( ,,)

0

(−,,)

(,,)

y − + y z

x x + dx+ z y

xy dy − dz z

cM &0 !VO .& , + Ω S M $ 5&6 A&5 X ∈ Ω 0Ǳ E&0 &F O&0 #&\0 Ω 0 F O&0 0 ! F • dr = F ( − t)A + tX • (X − A) dt AX 0 ! f (X) = F tX • X dt A=O

f (X) =

*

&F O&0 [

S

aB D 04 & q1 Z+U R#S q1 M (S 5 .&j 0 y 5 q1 R# N+M 0 *( 8**7 #&\0 *$ + !#[ " 5F ! $ 0 &S$ 0 . "0 " 0 %&"' $ B&O /&S ]M &#&q1 + 0 q1 R# .$ F q1 $ *$ !V6 . . $0 !" #$ ( 4E_ q1 R# >. T 5&0 E ![1 * F !j=

}

. O $ $ 5.$ $. .$ 0 % % * *M 4 U . 5F 0 $. & ([a ( ) $ + oL6

) |x| + |y| =

)

x + y =

)

” max{|x|, |y|} = ” ∪ ” max{|x|, |y|} = ”

)

”x + y = ” ∪ ”x + y = x” ∪ ”|x| + |y| = ”

[B C M x= *$ O =U D , + $. .$ C 0 $. & ([a ( ) $M oL6 . C 5.$ *( D *M oL6 . )

x + y ≤

) x + y ≤

)

≤ x + y ≤

Int(C) : x + y < , Ext(C) : x + y >

, |x| + |y| ≥

0 *( &aa ( ) 5&+

) ≤ x , ≤ y, x + y ≤ )

0 Y\ A% *.$ Y\ AGB 97*7 !VO .$ *C : x + y = M x= A& '() $ % * ?U0 ( 5 $.W N# C >. T R#

” ≤ x + y ≤ ” ∪ ”(x − ) + y ≤ ”

0 (M' ([a ( ) *$ O ) GB I*7 !VO C

∪”(x + ) + y ≤ ”

)

x + y ≤

, ≤ y , ≤ x , (x − ) + (y − ) ≥

0 0 D $ 0 5 $.W C S & % * C = ∂D Z"# >. T R# .$ *O&0 C 5.$ &0 C `&+ ) ([a ( ) &0 5 $.W

C S *Z & D [B . C @ 0 $& F dr $&+ E F dr &0 &F O&0 $. & *ZM

T .$ , + D M x= 0 G * % * ([a ( ) C 0 ( ) O&0 5F [B C 5 $.W ( #n<\ 6 F = P i + Qj M x= Rl+ *O&0 $. & >. T R# .$ O&0 3 D 0 •

•

C

C

00

@

P dx + Q dy = C

∂Q ∂P − ∂x ∂y

dA

}

([a ( ) 5 $.W N# 5.$ 9*7 !VO 5F 0 $. & (− , − ) / g. &0 e0 ! $ D M x= A, .8 x + y = / # $ :.& ( , ) ( , − ) (− , ) *( 5 $.W N# C >. T R# .$ *O&0 D [B C ( `?- 0 $. & ([a ( ) *$ O ) % *7 !VO 0 m? ! $ # $ 0 ( ) & ( &aa ( ) 5&+ e0

&# .$ *( &aa

Int(C)

D

Z D M ZM >&[f B&' 0 . q1 0 z .&j . #&, + RQ *Zh −y Z ( Zh −x !VO .$ D '& ZM x= R# 0&0 *Z & rZh s 9(O 5 0 #E >. T 0 . GB *7

Ext(C)

: − <x< : ”|x| >

,−
, |y| > ”

, x + y > /

&# x + y <

/

R : a ≤ x ≤ b , f (x) ≤ y ≤ g(x)

u#&+3 5 #E >. T 0 . D '& E >. T R# .$ 9$ + C

= AEB + BF A

−BF A

:

r(x) = (x, f (x)) ; a ≤ x ≤ b

AEB

:

r(x) = (x, g(x)) ; a ≤ x ≤ b

([a ( ) 5 $.W & 5.$ 9I*7 !VO 5F 0 $. &

}

9ZM Z"\ Zh ' 0 . −y . −x D=

n <

0

@ P dx

Di

P (x, g(x)) − P (x, f (x)) dx a ) 0 b (0 f (x) ∂P (x, y) dy dx − ∂y a g(x) 00 ∂P dA − ∂y

=

i

=

i

=

D

9(O 5 0 c #E >. T 0 . D ZM x= 0&6 >. T 0 R : c ≤ y ≤ d , m(y) ≤ x ≤ n(y)

C

i=

P dx + Q dy

Ci

*O&0 Zh , + & C E N# 5.$ ( 0 0 2 *( (.$ LB$ 0 R#S q1 R# 0&0 }

u#&+3 5 #E >. T 0 . D '& E >. T R# .$ 9$ +

i

C

= EBF + F AE

EBF

:

r(y) = (n(y), y) ; c ≤ y ≤ d

−F AE

:

r(y) = (m(y), y) ; c ≤ y ≤ d

0

@ Q dy

EBF 0 d

= = =

M $$S h'? X R# 0 0

Q dy −

=

C

* *8*7 &a AGB 9;*7 !VO *; *8*7 &a A%

P dx −BF A

AEB 0 b

i=

n @ /

0

P dx −

=

C

R# E N# C E 5 M *$ O ) % *7 !VO 0 ( ) &0 & F ( ) M ZM . Q . @ . D &, +

w 6 !j= .$ >. T R# .$ *O&0 .&SE& C 0 0&L , + d' E M $. $ $ ) $ &, + R# E & $ 5 R# 0&0 *& ( ) h E B " V# .$ *( T & F `&+ ) 0 F =A (P, Q) . $0 ` + &0 P dx + Q dy

M $$S h'? X R# 0

Q dy −F AE

Q(n(y), y) − Q(m(y), y) dy c ) 0 d (0 n(y) ∂Q dx dy c m(y) ∂x 00 ∂Q dA ∂x D

. $0 5 0 . R#S q1 A& '() % * −−−−−−−−−−−→ ( , ) (, ) / g. &0 C da F = x + y, x + y! *M t\ (, ) E y = y ( , ) (, ) p&\ E M ZM ) *!' y (, ) ( , ) p&\ E x = y (, ) (, ) p&\ R# 0 *A$ O ) GB ;*7 !VO 0C $.nS x + y = ([a ( ) ( 5 $.W N# Ada C C X (, ) E U# ( &aa ([a ( ) 5&+ 5F 0 $. & E ,&+ ) >. T 0 . C *(, ) ]< ( , ) 0 Z"# y .&3 C = C + C + C

≤t≤ − t)(i) + t(j) ; ≤ t ≤ − t)(j) + t(0) ; ≤ t ≤

C

: r(t) = ( − t)(0) + t(i) ;

C

: r(t) = (

C

: r(t) = (

E i& ` R# 0 R#S q1 _&0 >.&[, $ e+) &0 5 M *$$S >&[f D ' }

Zh , + AGB 9*7 !VO ,&+ ) >. T 0 . $.W 5.$ A% (O 5 Zh &, + E O&0 .& .$ C LB$ 5 $.W N# M x= &' E p Y E $& &0 . D , + *Z &0 D . 5F 5.$

0

0 F • dr +

=

F • dr

C

C

0 π −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→ ! +sin t−cos t, − cos t+ + sin t+sin t = −−−−−−−−−→! cos t, − sin t dt 0 π −−−−−−−−−−−−−−−−−−−−−−−−−−−−→ ! + cos t − sin t, − sin t + cos t

E ( >.&[, R#S = |Q (+ .$ @

@

C

•

•

0 π

0 π

= −

=

+ 0

sin t = −π

[B C M ZM ) R#S = ( . (+ [& 0 !-& >. T 0 F M ( D : x ≤ x + y ≤ \' M (O 5 D = D − D D : x + y ≤ x D : x + y ≤ R# 0&0

D

D

=

00

−

(t +

00

D :

00

⎡

! x + y dA

= −

0

0 $ 0 =

!

dt

− x)

00

dA =

xy − y

x −

D

x −

!

%y=(−x)

x −

!

dA

) dy dx

dx

y=

! − − x dx =

−−−−−−−−−−−→! F = x − y , x − y

+

x x − x dx

$(x − x − )+x − x + arcsin(x − −π

% )

O&0 x + y = ax # $ E +"1 C M . T .$ A .8 $ O $ +3 (, ) 0 &aa ( ) .$ (a, ) E M 1 . I = e sin y − y! dx + e cos y + x! dy *M [&

. C $M $& R#S q1 E 5 0 V# 0 *!' !#[ "0 N# 0 (a, ) & (, ) E C y .&3 y >. T R# .$ *$ O ) GB *7 !VO 0 *ZM

*( D : x + y ≤ ax, ≤ y # $ Z [B C + C 5 $.W 9$ + .&3 5 #E >. T 0 . C ?U0 x

t−

. $0 5 0 . R#S q1 A, .8

x−x

= − =

0

−

≤ y ≤ (

(−x)

=

0 0 x−x ⎣ √ = − x dy ⎦ dx −

,

=

⎤

√

≤x≤

∂Q ∂P − ∂x ∂y ( 0 0

x dA

D

dt −

E ( >.&[, R#S = ( . (+ R# 0&0 *(

dA

D

00

t − ) dt =

!

( RO. ZM oL6 . C 5.$ 0 U0 ' .$ , + [B C M

D

! x + y dA −

D

=

!

t − t +

t dt + 0

=

0 −−−−−−−−−−−−−−→ ! −−−−−→! − t, (t − ) • , − dt 0

=

x − (−y)

F • dr C

0 −−−−−−−−−−−−−−−→ ! −−−−−−→! • + t + , t − t + − , dt

! − sin t+cos t−sin t dt

00 ∂Q ∂P − dA = ∂x ∂y D 00 ! x + y dA =

F • dr +

0 −−−→ ! −−−→! t, t • , dt

−−−−−−−−−−−−−→! − sin t, cos t dt

00

0

C

C

cos t+ sin t− sin t cos t

=

F • dr

P dx + Q dy = 0 0 = F • dr +

C

x

C

−−−→ −−−→ C : r(t) = ( − t)(, ) + t(a, ) ;

≤t≤

c x + y

# $ `&+ ) E !T&' C *M t\ *x + y = O&0 # $ R0 '& 5F 5.$ ( 5 $.W C *!' Rl+ &aa ( ) m? .$ . x + y = x .$ 0 *ZM . ) #&0 &aa ( ) .$ . x + y = !VO 0 . C .$ *$ O ) % ;*7 !VO 9ZM .&3 #E >. T 0 O C = C + C ` +

@

C

:

C

:

=

x

−−−−−−−−−−−→! + sin t, cos t ; ≤ t ≤ π −−−−−−−−−−−→! r(t) = cos t, sin t ; ≤ t ≤ π r(t) =

E ( >.&[, R#S = ( . (+ .$ 0

F • dr

P dx + Q dy = C

C

= = =

−

0 π

−−−−−−−−−−−−→ F(R cos t, R sin t) • (−R sin t, R cos t) dt

Z#. $ .$

0

−−−−−−−−−−−−−−→ 0 π

−−−−−−−−−−−−→ − − sin t, cos t • (−R sin t, R cos t) dt R R 0 π − dt = −π

I

F • dr

= @

0

C

F • dr −

= C+C

00 D

−

−−−−−−−−−−−−−→ F = (xy − x , x + y )

M ( "0 C A *O&0 x = y y = x + $ 0 $ '& E

−−−−−−−−−−−−→ F = (x − y, x + y )

A; e1 `?- &0 ( U0 C *y = y = x = x = p Y 0 y) A *C : x + y = F = −(x,−−x−−+−→ *C : |x| + |y| = F = (x + y) i − (x + y )j A (, ) (, ) / g. &0 da C F x + y

=

x # $ $ `&+ ) C

−−−−−→ = (y , x )

*( (, )

AH

−−−−−→

F = (y, x ) A8 *O&0 x + y =

−−−−−−−−−−−−→ F = (−y sec x, tan x)

(, ) / g. &0 ( aa C AI *(π/, π/) (π/, ) 0 $ # $ e0. E C F = (y − x y)i + xyj A *O&0 4 e0. .$ e1 x + y = −−−−−−−−−−−−−→

p Y 0 $ da C F = (y − x , x + y) A; *( y = x x = y = '& E C F = arctan(y/x)i + ln(x + y)j A *( [Y1 T .$ ≤ θ ≤ π ≤ r ≤

=

0

Area(D) =

=

! − ex cos y − dA

! −−−→! F at, • a, dt

0 −−−−−−−−at→! −−−→! dA − , at + e • a, dt

D

πa

Ǳ [ E .nS @b 5 $.W N# C M x= A .8 0 . I(C) = x xdy −+ yydx . \ ( >&j L

*M [& C G L & U- /& 5.$ .$ Ǳ [ AGB 9Z#S h .$ (B&' $ *!' O $ . 1 C 50 .$ Ǳ [ A% O&0 O . 1 C *O&0 T I(C) R#S q1 N+M 0 GB (B&' .$ R# 0&0 ∈/ D 5 Q &F O&0 C 5.$ D S #E ( ( #n3t 6 D 0 F = −y/(x + y), x/(x + y) C

00 4

−−−−−−−−−→

p Y 0 $ '& [B C F = (x − y, xy) AJ *O&0 x + y = x + y = y = x = max &|x|, |y|' = e0 `&+ ) C F = −(x−−+−−y,−→ y) A7 *( x + y = # $ E R#S q1 E $& &0 & >&#+ .$ 9#0 O $ $ 0 F . $0 5

+ + C F = x + y i + (xy + y ln(x + x + y ))j A *( , ) (, ) (, ) ( , ) / g. &0 ( U0

00

!

ex cos y +

=

$. .$ . R#S q1 7 & .$ 0 % * 9M t\ O $ $ 5

F • dr C

I(C)

=

y − x y − x − (x + y ) (x + y )

5 dA

D

00

=

dA =

}

D

* *8*7 &a AGB 9*7 !VO * *8*7 &a A% C 5.$ ( 5 $.W N# C 5 Q A%C (B&' .$ & 5 R & $, .$ O&0 E&0 , +

. 1 D 5.$ .$ ?z &M R `&UO O cM 0 # $ M . @ (=&# R# C `&+ ) E !T&' C + C 5 $.W 5 M *$S

(" tU C + C 5.$ 0 o #$ *Z#S h .$ . # $ *$ O ) % *7 !VO 0 *$$S E&0 AGB C (B&' 0 nB &0 ( 0 0 I(C) .$ @

0

F • dr = C

0

F • dr − C+C

F • dr C

, + R# E *$ + 0 0 $ . !T&' $M [& . M C = C + C + C E ( >.&[, −−−−−−−−−−−−−−→ # " t + t + C : r(t) = ,t ; t + t +

≤t≤

−−−→ −−−→ C : r(t) = ( − t)( , ) + t(, ) ;

≤t≤

−−−→ −−−→ C : r(t) = ( − t)(, ) + t( , ) ;

≤t≤

+ ((t + ) )

$ =

− + (t + )

@

√

+

π √

(t −

=

(t − t +

=

C

!VO g F '& ('&" A& '() % * *M [& . x + y ≤ a E ( >.&[, C 5 $.W &# .$ *!'

)

/

%

−t

x dy @ − y dx C @ x dy − y dx C

) t + dt (t − t + ) #√ "

+ =

Area(D) =

(t − t +

arctan

C

0 0 (t + ) dt + dt + dt (t + ) (t − t + ) 0

=

,

9Z#E $3 R#S q1 E $0.&M M{ 0 5 M 5.$ M O&0 T .$ '& D S % * E ( >.&[, D ('&" >. T R# .$ ( C 5 $.W

&0 ( 0 0 Area(D) R# 0&0 *$ O ) *7 !VO 0 0

@ ,

. T .$ . x + ydx + y x − x + y dy A `&UO Ǳ [ cM 0 # $ ! $ D M M [&

$& 5 R#S q1 E &#F *O&0 5F [B C $ 0 Q $ +

)

/

/

x/ + y / = a/

}

−−−−−−−−−−−−→

.&3 5 r(t) = a cos t, a sin t! !V60 . 5F M 0 0 O ('&" .$ * ≤ t ≤ π M $ + &0 ( @

Area(D)

= =

y M e0. E +"1 9*7 !VO ( O ) x + y = x + y '& ('&" R#S q1 N+M 0 0 % * 9M [& . #E & E N# 0 $

) (x − x ) /a + (y − y ) /b = −−−−−−−−−−−−−−−−−−−−−−−−→! ) r(t) = cos t − cos t, sin t − sin t ,

≤ t ≤ π

− #−−−−−−−−−−−−−−→ " t − t(t − ) ) r(t) = , , − ≤t≤ t + t + −−−−−−−−−−−−−−−−→! cos t − sin t, sin t ≤ t ≤ π

)

r(t) =

)

(x/a)/ + (y/b)/ =

)

(x + y ) = a (x − y )

)

x + y = axy

)

(x + y)n+m+ = axn y m ,

x dy − y dx C

0 π

a cos t

− a sin t

0 π

= =

a 0 a

!

π

!

a cos t sin t dt

!

− a sin t cos t dt

!

! sin t cos t dt sin (t) dt =

πa

x + y = x + y 0 $ ('&" A, .8 *#.F (0 . &j L &.

.$ 5$ $ . 1 E ]3 ZM $& y = xt .&3 v E *!' *y = tt ++ t x = tt ++ #F (0 O $ $ B$&U

ZM x= Z " 5.&\ y x 0 ([" B$&U 5 Q R# .$ * ≤ t ≤ &# ≤ xt ≤ x U# * ≤ y ≤ x ≤ x O '& ('&" &0 >. T D :

≤ x , ≤ y , x + y ≤ x + y

'& Gj ('&" 5

< a, n, m ∈ N

D :

≤ x , ≤ y ≤ x , x + y ≤ x + y

y C 5 $.W E .nS .&O & % * 5 # S &0 ( 0 0 F = P i + Qj . $0 5

9C 5.$ D '& 0 F 00

@ P dy − Q dx = C

D

∂P ∂Q + ∂x ∂y

dA

M x= 0 G 3 4 . * % * x= *O&0 O G#U T E Ω '& 0 F . $0 5

R `&UO 0 # $ N# C ( LB$ Y\ X ∈ Ω M .nS .&O 5 c *$. $ . 1 Ω .$ &z &+ M ( X .$ cM C ! $ '& ('&" 0 Z"\ C $ .$ F y . M ! T 0 R M & C 0 ([" F uQ . 0 *Z$ 5&6 Curl(F) $&+ &0 & X .$ F uQ #$ 5&0 C $ .$F .&O Curl(F) := lim C ! $ ('&"

@

@ X

=

P dx + Q dy ÷ C

)

n

n

,

< a, b, n ∈ N n−

(x/a) + (y/b) = (x/a)

n+

+ (y/b)

,

b , n ∈M N x= A `&UO 0 # $ 5.$ . r `&UO <0a, # $ Z# &Q DcvB 50 An ∈ N MC R = nr . R# 0$ ('&" *( O !T&' C * & g V 3 . RQ *M [&

#&\0 i j .$ 8**7 q1 5 R#S q1 N+M 0 #E G#U 0 Z+U R# K@ 0 *$ $ Z+U . & 5$ 0 *( E& . T .$ . T E D , + #E % * N# y 5 0 F E Y\ $ _z M Zg S ! $F 5.$ &f .$ $ + !j Z 0 & Y0

) H*7 !VO 0 *$0 . 1 D .$ &z &+ D .$ e1 5 $.W *$ O

−y dx + x dy C

Y0 . >. T R# .$ O&0 !" #$ F S M $ O (0&f . R#S q1 5 M *( . 10 Curl(F) = ∂Q/∂x − ∂P/∂y 9$ + [U 5 #E >. T 0 .$ F . $0 5 y .nS .&O % * '& 0 F uQ &0 ( 0 0 C 5 $.W & . 9C 5.$ D 00

@ P dx + Q dy = C

(x/a)n + (y/b)n =

R→

lim R→

)

D

∂Q ∂P − ∂x ∂y

dA

58' 9 / 24 ;H7 T 0 !< . c= 4 E $& >& \ &6 0 *$ O U) doc.pdf !#&= &# B&V 5 M x= '+ Z= 'BC & * 0 r := vector ( [X, Y, Z] ) >. T 0 M C f(x, y, z) >. T 0 t $ c O&0 .& .$ ( O .&3 t X"' XB&Y . V E S ) 0 >. T R# .$ *O&0 a ≤ t ≤ b

} #&\0 5 9H*7 !VO C S A8**7 )*;: `FJ X $ % * 4E_ pO >. T R# .$ O&0 D . [+ , + [B Y0 . M ( 5F F = P i + Qj 5 5$ 0 #&\0 0 =&M *O&0 . 10 Int(D) 0 ∂Q/∂x = ∂P/∂y . [+ B O&[ !VO .& D ( RV+ R# 0&0 *O&0 . 10 8**7 q1 ZV' 5&l+ $ 0 M x= 0 G 3 4 . % % * x= *O&0 O G#U T E Ω '& 0 F . $0 5

R `&UO 0 # $ N# C ( LB$ Y\ X ∈ Ω ZM C E .nS .&O 5 c *$. $ . 1 Ω .$ &z &+ M ( X .$ cM

([" X F !3 . C ! $ ('&" 0 Z"\ F y 5&0 0 M ! T 0 R M B&' .$ ZM G#U C 0 #$ C E F y !M .&O div(F) = lim C ! $ ('&"

@

@ R→

=

P dy − Q dx ÷

lim R→

C

−y dx + x dy C

>. T R# .$ O&0 #n3!" #$ F S M $ O (0&f >. T 0 . R#S q1 5 M *div(F) = ∂P/∂x + ∂Q/∂y 9$ + [U 5 #E

O .&3 t X"' 0 r:=vector([X,Y,Z]) >. T 0 M C .$ *O&0 a ≤ t ≤ b >. T 0 t $ c O&0 .& .$ ( 9ZM G#U XB&Y . V E S ) 0 >. T R#

LinIntFirTyp:=proc(f,r,a,b) local INT ;

LinIntSecTyp:=proc(F,r,a,b) local INT ;

INT:=linalg[norm](map(diff,r,t))*INT ;

INT:=subs({x=r[1],y=r[2],z=r[3]},F) ;

return(int(INT,t=a..b)) ;

INT:=linalg[dotprod](INT,map(diff,r,t)) ;

end :

return(int(INT,t=a..b)) ; end :

. $ E C 0 F [& 0 5 M LinIntSecTyp(F,r,a,b)

*ZM $&

9ZM G#U INT:=subs({x=r[1],y=r[2],z=r[3]},f) ;

. $ E C 0 f [& 0 5 M LinIntFirTyp(f,r,a,b)

5 G#U 0 2 Y ) & * . $ E !< y .$ F = −−P,−−Q,−−R→! . $0 F:=vector([P,Q,R])

# /.$F .$

78 & *

http://webpages.iust.ac.ir/m_nadjafikhah/r2.htm

*( O $.F MnB = d'&[ 0 p 0 e0& & B&a

*ZM $&

*ZM $&

5 M x= Q+ Z= 'BC & * [x,y,z] &v &0 F:=vector([P,Q,R]) . $0

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S : z = f (x, y) ; (x, y) ∈ D

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S : z = −

,

− x − y ; x + y ≤

% * !VO 0 *( `&UO k cM 0 M g&3 + *$ O ) }

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b, 7 &#. Xb *( U0 &q= .$ U0 $ ǱO N# #. . @ 0 * " XT Z") ).& ^Y E +"1 &# 4&+ [) &#. 9ZM Z"\ M $ $ 0 . &#. M * .&3 &#. S

( v U0& f M x= .&j 0 ( c $, 0 h f e0& E ^Y S : f (x, y, z) = c

0 $ 0 #n<" #$ f M Z & $% )* . S . T .$ E 0 f (X) = 0 l&Q *f (X) = 0 X ∈ S Ǳ E )* . S O&[ . 10 S 0 e1 & &# p&\ 0 c . V [) #. Q `&+ ) *Z & )? $% *Z & )? $% )* V [) &#. E #& B&a **H (+"1 .$ N# . $ M &[) #E & B&a *( O KY

*$. $ + (#& 0 ?z+,

;** &a E ; (+"1 9* !VO $ U0& f M x= z + x : 2 = $ U0& f M x= y + x : 2 = 5.$ 0 f S *O&0 f $ E , + #E D $ 0 v 5.$ 0 f S *O&0 f $ E , + #E D $ 0 v

;;

f =

+

D : y ≤ , y − z ≤

, S : x=

, + &F O&0 #n<\ 6 D , +

9$ O ) #E + y − z M x= A& .8 >. T R# .$

S : y = f (x, z) ; (x, z) ∈ D

$ O x= S M Q *( V [) #. N# S : g(x, y, z) = (O 5 &F g = f (x, z) − y −−−−−−−−−−−−−→ $. 0 + 5 , 0 *g = −(∂f /∂x, − , ∂f /∂z) = 0 9$ O ) #E f = x − z M x= A& .8

+ y − z , y ≤ , y − z ≤

T−yz 0 5F # j M ( V [) #. N# *$ O ) GB * !VO 0 *( D 3 B Bn R# .$ *D : y ≤ , z ≤ f = y M x= A, .8 E +"1 S : x = y , y ≤ , z ≤ #. >. T D !Y " T−xz 0 5F # j M ( x = y + *$ O ) % * !VO 0 *O&0

}

D : − ≤x≤

,− ≤z≤

#. >. T R# .$ S : y = x − z , − ≤ x ≤

. −y & . .$ y = x −z B Bn 5 S+ E +"1 GB ;* !VO 0 *( D T−xz 0 5F # j M ( *$ O ) f = sin(x + z) M x= A, .8 D : − ≤ x ≤ ,

** &a E ; (+"1 9* !VO f M x= 9fC) a2 < : p? $ O&0 &q= E , + #E Ω O&0 #n<\ 6 v U0& M x= ?U0 *S : f (x, y, z) = a , (x, y, z) ∈ Ω c E & $ U &# p&\ E & $ U c0C + Ǳ E 0 M pO M $ O ) *O&0 T GB&L f t 6 AS .$ &

S E Q 0 p&\ &# S E Y\ Q .$ f = 0 . O&[ . 10 f = 0 pO 5F Ǳ E 0 M @&\ *O&[ . 10 S >. T R# .$ *Zg S `g . p&\ #& & j? a .$ f E ^Y eY\ E ( >.&[, ( V #. N# 9$ O ) #E $. 0 + 5 , 0 *Ω , + &0 Ω : x +y +z ≤ f = x +y −z M x= A& .8 >. T R# .$ *a = −

,− ≤z≤

≤z≤

>. T R# .$ S : y = sin(x + z ) , − ≤ x ≤ , D

≤z≤

T−xz 0 5F # j M ( V [) #. N# *$ O ) % ;* !VO 0 *O&0

}

S

:

x + y − z = − , x + y + z ≤

:

−x − y + z =

, x + y ≤

Ǳ [ cM 0 Q.&3 $ 5 SB Bn N# E +"1 M # $ T−xy 0 5F # j M ( . −z 5.&\ .

*$ O ) GB * !VO 0 *O&0 x + y ≤ + Ω = R f = ( x + y − ) + z M x= A, .8 0 [) #. S : f (x, y) = a >. T R# .$ *a = cM N# `&UO 0 # $ 5 .$ E M ( % N# !VO

** &a E ; (+"1 9;* !VO $ U0& f M x= z + y : 2 = 5.$ 0 f S *O&0 f $ E , + #E D $ 0 v

, + &F O&0 #n<\ 6 D , +

S : x = f (y, z) ; (y, z) ∈ D

$ O x= S M Q *( V [) #. N# S : g(x, y, z) = (O 5 &F g = f (y, z) − x −−−−−−−→ $. 0 + 5 , 0 *g = −(−−−−, −∂f−−/∂y, ∂f /∂z) = 0

ZM G#U S A −

−−−−−−−−−→ ∂x ∂y ∂z , , ru = ∂u ∂u ∂u

−

−−−−−−−−−→ ∂x ∂y ∂z , , rv = ∂v ∂v ∂v

ZO&0 O $ (u, v) ∈ IntD E 0 &F S &z'?YT (B&' R# .$ *r (u, v) × r (u, v) = 0 *( Zh IntD 0 r M $ O

*O&0 N[V# IntD 0 r AH N# $0 `&+ ) 0 0 M , + * , + *$ O & I )* O&0 6\ Q &# P'S N# &O 3 . S #. & F $0 M #&6\ I . #. N# 0 ]@ RU !+, *$ O & S *Z & S L 0 $ pL Ω M x= A& + '() 5F ).& ^Y S $ 0 z = z = x + y &#. H* !VOC ( #. N# S M Z$ 5&6 Z *O&0 *AGB *ZM Z"\ S S UY1 $ 0 . S . h R# 0 *!' &F D : x + y ≤ S u

S

:

S

:

v

z = ; (x, y) ∈ D , z = x + y ; (x, y) ∈ D

−−−−−+ −−−−−−−→ r(u, v) = u, v, u + v

6\ $0 S M ( RO. $ .$ M O&0 &h(u,'v) = −(u,−−v,−−→) 6\ $0 S S 0 "@ A = r, h >. T R# .$ *(u, v) ∈ D 5 r &0 ?za *( \ 6\ %&L [B *O&0

−−−−→ M r(u, v) = −(u−−cos−−v,−−u−sin v, u) 9$M $& #$ 6\ E * ≤ u ≤ , ≤ v ≤ π 5F .$ XUV ).& ^Y S M x= A, .8 Ω :

≤x≤

, ≤ y ≤ , ≤ z ≤

!VO 0C ( #. N# S M Z$ 5&6 Z *( uO 0 Ω M ZM ) . h R# 0 *A$ O ) H* z = y = x = z = y = x = T 9ZM Z"\ e0 uO 0 . S /& R+ 0 *( $

M S = S ∪ S ∪ S ∪ S ∪ S ∪ S ;

≤y≤

, ≤ z ≤

S : x = ;

≤y≤

, ≤ z ≤

S : y =

≤x≤

, ≤ z ≤

S : x =

;

*$ O !T&' . −z ' T−xy .$ e1 (, , ) .$ *$ O ) % * !VO 0 }

H** &a E ; (+"1 9* !VO N# #E $. E N# M $ 5&6 0 % *$$S Z #. !VO $ O U *( V [) #. 9#0 N+M !< E .&M R# 0 ) z = xy, − ≤ x ≤ , − ≤ y ≤ + ) y = x + z , y ≤ , ≤ x

) (x + y ) = x − y , z ≤ ) x + y = z , −

≤y≤

) z = x + y , z ≤

)

y = x + z , x + z ≤

) x =

)

x = z , x ≤

)

xyz =

,

+

− y − z

) x + y + z = ) x/ +y / +z / =

, y ≤

)

) (x + y + z ) = x y

|x| + |y| + |z| = & ' ) max |x|, |y|, |z| =

) x + y + z = xyz

)

√ √ √ x+ y+ z =

B$&U N+M 0 #. N# G#U 5&V $. E 0 .$ .&3 &0. E $. RQ .$ Xb *$. $ ) *Z#. $ E& 6\ G#U 0 5F E ![1 *$ O $& !VO 0 ( U0& )9 E . h & r(u, v) = Z"# 0 Z &0 D . D $ S M r : R → R −−−−−−−−−−−−−−−−→! v), y(u, v), z(u, v) &F −x(u, *O&0 . M "0 D $ A D 0 z y x v $ e0 U# *O&0 3 D 0 r A; *O&0 3 0 z y x v $ e0 U# *O&0 . + IntD 0 r A *O&0 #n3 t 6 IntD

D :

+ { − v sin(u/)} sin u j + v cos(u/) k

S : y = ;

≤x≤

, ≤ z ≤

≤ u ≤ π , −

S : z =

;

≤x≤

, ≤ y ≤

S : z = ;

≤x≤

, ≤ y ≤

≤v≤

$. $ . N# & M U R# 0 ( #n3& ( ) #. R# *$ O ) ** 0 *A QC }

** &a E (+"1 98* !VO >. T R# −.$−−−*−S−−:−−z−→= x + y M x= A .8 M f = (−x, −y, ) f (x, y, z) = z − x − y . #. R# *( #. S ]3 *( T GB&L . + U#C 5 M0 #. M Z# & q0 5 S+ ?z[1 x = x= &0 *A$ O ) GB J* !VO 0C ( A$ & S R# 0&0 *z = u M Z#S [ y = u sin v u cos v &F D : ≤ u , ≤ v ≤ π $ O x=

−−−−−−−−−−−−−→ S : r(u, v) = (u cos v, u sin v, u ) ; (u, v) ∈ D

O .&3 !VO 0 U# O I S #. X R# 0 *O Z#. $ >. T R# .$ *S : y = x M x= A4 .8 GB&L 6+ M f = −(−−−−x,−−−,−→) f (x, y, z) = y − x . #. R# *( #. S + R# 0&0 *( T −−−−−→ 5 #E S : r(u, v) = (u, u, v) ; (u, v) ∈ D >. T 0 B$&U R =&# 0 *A$ O ) % J* !VO 0C $ + .&3 & . $0 X = (−, , ) Y\ .$ S 0 /&+ T

−−−−−−−−→ n = f (X ) = (−x, , )

X

−−−−−→ = (, , )

h $. T B$&U >. T R# .$ *Z#S h .$ . Y\ .$ S 0 $ +, y B$&U *( x + y + = $& &# x + = y − = z − E ( >.&[, c X *rx + = y , z = s

}

** &a E ; (+"1 9H* !VO 9$ + .&3 5 #E !VO 0 . & F E N# −−−−−→ S : r (u, v) = ( , u, v) −−−−−→ S : r (u, v) = (u, , v) −−−−−→ S : r (u, v) = (u, v, )

−−−−−→ S : r (u, v) = (, u, v) −−−−−→ S : r (u, v) = (u, , v) −−−−−→ S : r (u, v) = (u, v, )

R# 0&0 * ≤ u ≤ ≤ v &≤ & F + .$ M' *O&0 S 0 ]@ N# A = r , r, r, r , r , r #. 0 $ ).& ^Y S M x= A .8 Z *O&0 z = z = >&T x + y = 0 . S . h R# 0 *( #. N# S M Z$ 5&6 9ZM Z"\ (+"1 S = S ∪ S ∪ S

S : z = ; x + y ≤

S : z = ; x + y ≤

S : x + y = ;

≤z≤

0C ZM ' @ #E KO 0 #&6\ , + R# 0 9AGB 8* !VO −−−−−→ r (u, v) = (u, v, ) ; u + v ≤ −−−−−→ r (u, v) = (u, v, ) ; u + v ≤ −−−−−−−−−−−−−→ r (u, v) = ( cos u, sin u, v) ; ≤ u ≤ π ,

≤v≤

*( S 0 "@ A = {r , r , r} >. T R# .$ .$ 0 !VO !Y " nb&M . S * / 0 . A .8 Z 0 F $ ]< &Q .$ Z N# F O&0 .& mU / 0 . 4&0 M $$S !T&' VO ZM !T .&3 0 D. N# *A$ O ) % 8* !VO 0C ( 9( #E 6\ E $& 5F 5$M r(u, v) = { − v sin (u/)} cos u i

}

)

Ω : x + y + z =

)

Ω : x + y =

)

Ω : x + y = , z = , z =

)

Ω : |x| + |y| + |z| =

)

Ω : x/ + y / + z / =

)

Ω : x + y =

)

Ω : x = y , y = x , z = , z =

)

Ω : x = , y = , z = , x + y + z =

)

, x + z =

, z = , x + y + z =

Ω : x = , x =

, y = z , z =

S : r(u, v); (u, v) ∈ D Z#S >. T R# .$ *X = r (u , v ) (u , v ) ∈ IntD &+ S 0 X Y\ .$ r (u , v ) r (u , v ) &. $0 *O&0 $ +, S 0 X Y\ .$ r (u , v ) × r (u , v ) nB .$ *#0 h .$ . S 0 e1 h(t) = r(t, v )

>. T R# v

u

u

** &a E 8 H (+"1 9J* !VO >. T R# .$ −*−S−−:−−x−−+→y + z = M x= A5 .8 ]3 *f = (x, y, z) f (x, y, z) = x + y + z x = y = z =s U 0 rf (X) = X ∈ S s pO ]3 *O&0 1& M ( rx + y + z = x = cos u cos v x= &0 *( . M #. N# S M ρ = &0 M >&j L C z = sin u y = cos u sin u 9(O 5 Aθ = u ϕ = v

v

4

h (u ) = ru (u , v ).

h(u ) = r(u , v ),

*( /&+ X = r(u , v ) .$ S 0 r (u , v ) R# 0&0 /&+ X = r(u , v ) .$ S 0 r (u , v ) 0&6 >. T 0 2 *( #0 4 XY *( X ∈ S S : f (x, y, z) = c Z#S MC >. T R# .$ *Af (X ) = 0 U#C ( Zh Y\ *( Zg&1 X .$ S 0 f (X ) 0 ?za . f (x, y, x) = c #. mU B$&U ( =&M >. T 0 . S #. ]< *z = h(x, y) 9$ + !' z X"' >. T R# .$ *ZM .&3 r(x, y) = (x, y, h(x, y)) u

v

rx (x , y ) = ry (x , y ) =

−−−−−−−−−−−→ −−−−−−−−−−→! fz , , , hx (X ) = , , − fx X −−−−−−−−−−−→ −−−−−−−−−−→! , , hy (X ) = , , − fz . fy X

S :

−−−−−−−−−−−−−−−−−−−−−−−−−→ r(u, v) = ( cos u cos v, cos u sin v, sin u) (u, v) ∈ D

!VO 0 *D

: −π/ ≤ u ≤ π/ ,

≤v

≤

π 5F .$ M

*$ O ) GB 7* >. T−−R# −−−.$−−−*S−−−:→z = x + y M x= A0 .8 f = (−x, −y, z) R# 0&0 f (x, y, z) = z − x − y V #. N# S ]3 *$$S T (, , ) .$ & S 0 M S pL z = u y = u sin u x = u cos v x= &0 *O&0

!VO 0 .

4

S :

−−−−−−−−−−−−→ r(u, v) = (u cos v, u sin v, u) (u, v) ∈ D

r = z = u &0 >&j L C $ + .&3 5

*( D : u ∈ R , ≤ v ≤ π $ 5F .$ M Aθ = v *$ O ) % 7* !VO 0 }

.$

rx (x , y ) × ry (x , y ) = =

− −−−−−−−−−−−→ − fx fy , , fz X fz X fz (X )

f (X )

** q1 E ZV' 5 M *( f (X ) . $0 E M 2 *$$S uL0 .$ 5F q1 R# $0.&M i j .$ #& B&a

*( O $.F H !j= E

** &a E 7 J (+"1 97* !VO 0

90&0 S 0 "@ *( Ω ).& ^Y S $. .$ ) Ω : x + y + z = x

-

−

+

=

, fx + fy + fz dxdy =

|fz |

! `GF b, !F 7

fx fy + − dxdy fz fz

=

|fz |

f (X ) dxdy

*&[f !0&1 0&6 >. T 0 $. #&

2

#. S

M x= f (x, y, z) Av e0& C B&V 5 O&0 O .&3 ` ^Y >. T R# .$ *$ 0 O G#U 5F 0 !VO 0 . S 0 f : r(u, v) ∈ D

00

00

f dσ := (S)

! f r(u, v) ru × rv dudv

u

D

#. !M (= 0 0 . \ R# #$ 5&0 0 *ZM G#U r(u + du, v) r(u, v) / g. &0 NQ M UY1 S #E ( S Z#0 h .$ S 0 . r(u + du, v + dv) r(u, v + dv) `?- r(u, v) .$ / . N# &0 `?-_ E 5F &0 &F M $ + x= c (=S h .$ 5 . r dv r du . \ .$ *O&0 f (r(u, v)) 0 0 (0&f 5F 0 f ('&" %qT&' &0 ( 0 0 S E NQ M UY1 R# (= ('&" & *r(u, v) Y\ .$ f e0& . \ .$ `?-_ E

&. $0 ).& %qT&' E 0 0 `?-_ E R# R# 0&0 *O&0 5F E& v

dQ

#. N# 0 B&V 5 N# ` ^Y ` y Z+U # E R# *( A^Y &#C *O&0 &S $ 4 Z+U #$ E ( M Z#. $ E& ('&" 5&+B 4 0 R# G#U 0 *Z#E $3 5F ^#6 0 O .&3 #. N# S M x= . S 7E LFH >. T R# .$ O&0 r(u, v) 6\ y 5 .&j 0 *ZM G#U dσ = r ×r dudv !V60 .$ S 0 /&+ NQ M `?-_ E ('&" dσ M (S " r dv r du E >.&[, 5F e- $ M ( r(u, v) Y\ *A$ O ) * !VO 0C u

}

#. N# ('&" 5&+B 9* !VO !VO 0 i& $. .$ . ('&" 5&+B 9$ + [& 5 #E &F S : f (x, y, z) = S AGB dσ =

0 O&0 O !V6 (+"1 Q E S #. l&Q (0 $ , ]< $ + [& &S ) & F E N# 0 . *ZE e+) . F f

dσ =

dσ =

(S)

D

&F S : 0f0(x, y(x, z), z) = a , (x, z) ∈ D S A%

00

h dσ = (S)

D

! f dxdz h x, y(x, z), z |fy |

&F S : 0f0(x(y, z), y, z) = a , (y, z) ∈ D S A:

00

h dσ = (S)

! f dydz |fx |

h x(y, z), y, z D

+

,

&F S : 0f0(x, y, z(x, y)) = a , (x, y) ∈ D S AGB ! f dxdy h x, y, z(x, y) |fz |

f f f dxdy = dxdz = dydz |fz | |fy | |fx | ,

v U0& h #. S M x= >. T R# .$ *$$S G#U S 0 M O&0 h dσ =

u

u

! = ru du × rv dv f r(u, v) ! = f r(u, v) ru × rv dudv

00

v

dσ = z

&F S : z = f (x, y)S A%

+ (fx ) + (fy ) dxdy

&F S : y = f (x, z) S A:

+ (fx ) + (fz ) dxdz

&F S : x = f (y, z) S A$

+ (fy ) + (fz ) dydz

X"' 0 . f (x, y, x) = c #. mU B$&U l&Q >. T 0 . S #. *z = h(x, y) 9ZM !'

−−−−−−−−−→! r(x, y) = x, y, h(x, y)

.$ *$ + .&3 5

*−−−−−−→ −−−−−−→* * * dσ = rx × ry dxdy = *( , , hx ) × (, , hy )* dxdy , *−−−−−−−−−→* * * = *(−hx , −hy , )* dxdy = + hx + hy dxdy

# $ Z T−xz 0 S # j M R# 0 ) &0 D : x + z ≤ ,

.$

y = x + z

00

00

(S)

r cos θ sin θ + r

= D

(0

0

= =

! + r + r

cos θ sin θ +

r

0

π

+

r +

+

dθ dr dr

π

( O x= AC .$ M 9( T−rθ

r +

D : −π/ ≤ θ ≤ π/ ,

pL E +"1 S M . T .$ A .8 e0 T−yz 0 5F # j x ≤ 5F .$ M O&0 ,− ≤y≤

*M [& D 0 . h = x ( .$ f = y + z − x E ( >.&[, S #. mU e0& *!' dσ

= =

+ 00 , (y + z ) + y + z + dydz h dσ = − y + z | − y + z | D (S) √ 00 ! = y + z dydz 00

D

=

( √ 0 0

−

√ 0

−

00

00 h dσ =

−

) (y + z ) dy dz

+ z

dz =

√

+ (fx ) + (fy ) dxdy

&F S : y =! f,(x, z) , (x, z) ∈ D S A

+ (fx ) + (fz ) dxdz

h x, f (x, z), z

(S)

D

00

00

&F S : x =! , f (y, z) , (y, z) ∈ D S A + (fy ) + (fz ) dydz

h f (y, z), y, z D

$& ;*;* q1 ^Y G#U E ( =&M 2

*$ O

M #_&0 + S +M . T .$ A& '() $ S 0 . h = z x + y O&0 x + y + z = *M [&

f = x + y + z E >.&[, S mU e0& B&" R# .$ *!' .$ ( dσ

= =

−−−−−−−−→ (x, y, z) f dxdy = dxdy |fz | |z| + x + y + z dxdy = dxdy z z

Z#S z X"' 0 f = B$&U +!' &0 =@ E ( >.&[, T−xy 0 5F # j M z = − x − y *;* E AGB C (+"1 0&0 .$ *D : x + y ≤ E Z#. $ 00

00

z

h dσ = (S)

D

=

−−−−−−−−−→ f (−x, y, z) dydz = dydz |fx | | − x| + x + y + z dydz |x|

( D e0 E >.&[, T+ −yz 0 S # j V# 0 ) &0 (+"1 0&0 .$ x = − y + z Z#. $ S #. 0 c Z#. $ *;* E A:C

=

D

√

≤r≤

x = y + z

D : − ≤z≤

(S)

(S)

)

&F S : z =!f,(x, y) , (x, y) ∈ D S A$

h x, y, f (x, y)

h dσ =

dr

u (u − ) du =

.$ D # j D u =

dxdz

r drdθ

−π/

√

π

=

r +

π/

−π/

()

+ !

0 $ + %π/ r r + (θ − cos θ)

0

=

Z#. $ S 0 V# 0 ) &0 c O&0

Z#. $ *;* E A%C (+"1 0&0

D

00

00 h dσ =

≤x

!+ xz + (x + z ) x + z +

h dσ =

00

00

, x + y

z

dxdy

, ( − x − y ) x + y dA

D

=

00

( − r )rr dA

D

#0

" #0

π

=

=

(π)

dθ

−

" ( − r )r dr

= π

q0 5 S + E +"1 S M . T .$ A, .8 h = xz + y e0& *y ≤ ≤ x M O&0 y = x + z *M [& . S 0 .$ *f = x + z − y E ( >.&[, S #. mU e0& *!' dσ

= =

−−−−−−−−−→ (x, − , z) f dxdz = dxdz |fy | |− | + x + z + dxdz

= # " > > −y ? = − y (z − y) + + +() dydz − y D ) 00 0 (0 = (z − y) dydz = (z − y) dy dz 00 ,

−

D

0

=

T E +"1 0 h = x +x +y +y z e0& E A .8 *#0 $. $ . 1 Z 6 N# .$ M x + y + z = B&" R# .$ M $ O h'? *!' S : z=

z dz =

T−xy 0 S # j D : ≤ x ≤ , ≤ y ≤ − x M Z#. $ *;* E A$C (+"1 0&0 .$ *(

M 0 h = z / e0& E

A5 .8 *#0 9ZM .&3 M >&j L N+M 0 . S M *!'

S : x + y + z =

00

h dσ = (S)

−−−−−−−−−−−−−−−−−−−−−−−−−→! r(ϕ, θ) = cos ϕ cos θ, cos ϕ sin θ, sin ϕ

00

x+y x + y + ( − x − y)

=

) &0 *D* : −π/* ≤ ϕ ≤ π/ , ≤ θ ≤ π 5F .$ M . dσ = *r × r * dϕdθ ('&" 5&+B #&0 0 *;* 0 9ZM [&

θ

D

=

ϕ

*−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→! dσ = * − sin ϕ cos θ, − sin ϕ sin θ, cos ϕ −−−−−−−−−−−−−−−−−−−−−−−−→!* × − cos ϕ sin θ, cos ϕ cos θ, * dϕdθ *−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→!* = * − cos ϕ cos θ,− cos ϕ cos θ,− sin ϕ sin θ *dϕdθ = cos ϕ dϕdθ

√ 00

(x + y) dxdy =

00 h dσ

=

(S)

D

#0

( sin ϕ) cos ϕ dϕdθ π/

" #0

sin ϕ cos ϕ dϕ

=

=

× × π = π

−π/

"

π

+

( √ 0 0 −x

( − x)

) (x + y)dy dx √

dx =

S : y = x − z ; (x, z) ∈ D

.$ *( T−xz 0 S # j D : x + z ≤ z i1 M Z#. $ *;* E AC (+"1 0&0 00

00

+

h dσ = (S)

D

00 =

S = S + S + S + S + S + S ,

+ (− ) + (− ) dxdy

+"1 0 h = z/ + x + z e0& E A .8 y M y = x − z B Bn 5 S+ E *#0 ( O ) x + z = z B&" R# .$ M $ O h'? *!'

dθ

>&T 0 $ XUV Ω M . T .$ A0 .8 ^Y S $ 0 z = y = x = >&T &j L

*M [& S 0 . h = xyz O&0 Ω ).& 9$ O !V6 !VO e0 UY1 uO E #. R# *!'

,

D

√ 0

= x( − x) +

Z#. $ .$ 00

− x − y ; (x, y) ∈ D

00

z dz = D

0

π

(0

+

z + x + z

+ x + z dxdz

r sin θ.r drdθ D

sin θ

=

) r sin θ dr

dθ =

0

π

sin θ dθ = π

E +"1 0 h = x(z − y) e0& E +A4 .8 O ) z = z = >&T y M x = − y *#0 ( B&" R# .$ M $ O h'? *!'

S : z =

; (x, y) ∈ Dxy , S : z = ; (x, y) ∈ Dxy ,

S : y =

; (x, z) ∈ Dxz , S : y = ; (x, z) ∈ Dxz ,

S : x=

S : x =

; (y, z) ∈ Dyz , S : x = ; (y, z) ∈ Dyz ,

!Y " E >.&[, T−yz 0 S # j D

,

T−xy 0 S # j E >.&[, X 0 D D D M 9 " T−yz T−xz yz

Dxy : Dyz :

≤ x, y ≤ ≤ y, z ≤

. .

Dxz :

xz

≤ x, z ≤

D :

xy

,

00

≤z≤

,− ≤y≤

Z#. $ *;* q1 E AC (+"1 0&0 .$ *( h dσ =

(S)

− y ; (y, z) ∈ D

9#0 S 0 h E $. .$ 0 % z = −x −y q0 5 S+ E +"1 S f = z A *$. $ . 1 T−xy _&0 .$ M + M E O ) VQ M (+"1 S f = z x + y A; *( z = a/ T y x + y + z = a x + y + z = M E +"1 S f = x + y + z A *( O ) x + y = x y M ( y M ( x + z = E +"1 S f = y A *( O ) y = y = >&T x + y + z = T E +"1 S f = x + y + z AH *$. $ . 1 Z 6 N# .$ M ( E ≤ x &q= Z .$ e1 V+ S f = z A8 *( x + y + z = $ 3 ).& ^Y S f = x + y + z AJ *( z = z = >&T x + y = 0 M ( x + y + z = M E +"1 S f = x A7 ) x + y = z x + y = z K Y y *$. $ . 1 ≤ x &q= Z .$ $$S

0 $ pL ).& ^Y S f = x + y + z A *O&0 x = T x = y + z T 0 $ 4 ).& ^Y S f = z AI *( &j L >&T x + y + z = + o1& pL ).& ^Y S f = y x + z A y = y = >&T y = x + z 0 $

*$. $ . 1 ≤ x &q= Z .$ O&0

|x| + |y| + |z| = ) (6 S f = |x| + |y| + |z| A; *( & F + M ( !#{ i . $ ` ^Y 9&[f !0&1 &S $ i E $& &0 ' . 0 a v e0 h h h M x= & M ZM x= ?U0 *O&0 #. S S S (0&f $ , b >. T R# .$ *O&000 T S ∩ S00 ('&"

) ∀X ∈ S : h (X) ≤ h (X) ⇒

(S)

h dσ ≤

& ' ) min h(X) | X ∈ S × Area(S) ≤

00 h dσ (S)

(S)

h dσ

Z#. $ *;* E AC AC A$C $. 0 ) &0 .$ 00

00

, xy( ) + () + () dxdy

h dσ = (S)

(S )

00

, xy() + () + () dxdy

+ (S )

00

,

+

x( )z

(S )

00

x()z

+ (S )

00

+

, ( )yz

(S )

00

+

=

()yz

(S )

00

,

Dxy

=

+ () + () dxdz + () + () dydz + () + () dydz

00

00

xy dxdy + #0

,

+ () + () dxdz

xz dxdz +

Dxz

" #0

x dx

Dyz

"

yz dydz

y dy

=

' . ≤ x ≤ π M y = sin x A2 .8 f = e0& E *$$S !T&' S #. $ $ 5 .$ . −x + *#0 S 0 / − y − z >. T 0 . S #. *!' S : y + z = sin x ; y + z ≤

' 5 .$ # E v u

ZM x= *$ + 5&0 5

Z#. $ X R# 0 *( . −x

= x

−−−−−−−−−−−−−−−−−−→! S : r(u, v) = u, sin u cos v, sin u sin v ; (u, v) ∈ D

.$ *D : ≤ u ≤ π , ≤ v ≤ π 5F .$ M dσ

* * *ru × rv * dudv *−−−−−−−−−−−−−−−−−−→! * * , cos u cos v, cos u sin v

= =

−−−−−−−−−−−−−−−−−−−→!* , − sin u sin v, sin u cos v ** dudv *−−−−−−−−−−−−−−−−−−−−−−−−−−−−→!* * * * sin u cos u, − sin u cos v, − sin u sin v * dudv + + cos u dudv sin u. ×

= =

R# 0&0

00

00

+

f dσ = (S)

D

− sin u

00 sin u ddv =

= D

0

+ + cos u dudv . sin u.

π

sin u du

#0 π

" dv

= π

60 $ U 0 5$ 0 Z = !0&1 U# XY 5$ 0

+, $ $ &$0.&M R# 0&0 ( O =S h .$ 5&S E ^Y RV+ &$0.&M + E 1 $ U & O *O&0

N# 0 0 (0&f . h l&Q 7JD) . !) ('&" U 0 S E NQ M 5&+B N# (= &F Z#0 *O&0 S ('&" U 000S !M (= .$ *( 5F *Area(S) = dσ M Z#S R# 0&0 E !T&' % ^Y ('&" A& .8 '() [& . a < b `&UO 0 # $ ' a `&UO 0 # $ 5 .$ *M R# B$&U M Z#S vv E A8C (+"1 N+M 0 *!' + 0 M ( S : x + y − b + z = a E >.&[, #. >. T

) ) )

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→! (a cos u + b) cos v, (a cos u + b) sin v, a sin u

5F .$ ( 5$M .&3 !0&1

D :

≤ u ≤ π , ≤ v ≤ π

&0 ( 0 0 5F ('&" 5&+B ?U0

* * dσ = *ru × rv * dudv *−−−−−−−−−−−−−−−−−−−−−−−−−−−−→! * = * − a sin u cos v, −a sin u sin v, a cos u ×

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→!* * × − (a cos u + b) sin v, (a cos u + b) cos v, * dudv * −−−−−−−−−−−−−−−−−−−−−−−→!* * * = *a(a cos u + b) −cos u sin v, cos u cos v,−sin u *dudv

= a(a cos u + b) dudv

&0 ( 0 0 S !M ('&" .$

00 Area(S) =

dσ = (S)

#0 π =

a

00

a(a cos u + b) dudv D

(a cos u + b) du

" #0

dv

= π ab

' C : x + z = a y = # $ 5 .$ E !VO R# M $. $ . 1 T 0 # $ R# *$ O !T&' & z .

C : x + y = b # $ 0 5F cM $.nS & z . E C y %qT&' 0 0 S ('&" V# XB&) *$. $ . 1 ( C y .$ z = x + y q0 5 S+ E +"1 ('&" A, .8 *M [& . ( $ z = a T 0 M >. T 0 . #. R# >&j L N+M 0 *!' −−−−−−−−−−−−−→! r(u, v) = u cos v, u sin v, u

00 h dσ =

(S +S )

00

(S)

dσ (S)

(S)

00

h dσ + (S )

Area(S) =

h dσ (S )

00 00 ) h dσ ≤ |h| dσ (S) (S)

4&") !M (= ` ^Y N+M 0 M . @ 0 S ?za *$ + [& 5 . U0 &q= .$ U0 $ N+M 0 &F Z 0 . #. N# E Y\ 4) B&Q 5 . #. uQ `&UO !\f cM 4) ` ^Y ` R# S.&V0 .$ M T 0 &# .$ *$ + [&

*ZM ^#6 . &0 #. N# S M x= BV K+ 5 h(x, y, z) M x= *(u, v) ∈ D M ( r(u, v) 6\ G#U S 0 M ( A v e0& C B&V (= 5&+B 0 S 0 h !M (= [& 0 *$$S

/ g. &0 UY1 ( =&M . h R# 0 *ZM [& . S 0 . r(u + du, v + dv) r(u, v + dv) r(u + du, v) r(u, v) r du &. $0 0 . `?-_ E &0 . 5F =S h .$ (0&f UY1 R# 0 h ZM x= ]< *Zc0 X#\ r dv 0 0 UY1 R# (= X R# 0 *O&0 h(r(u, v)) 0 0 &0 ( ! dQ = `?- _ E ('&" × h r(u, v) u

v

=

"

π

(S)

00

! `GF b, !F N $7

(S)

r(u, v) =

00

& ' ≤ max h(X) | X ∈ S × Area(S) 00 00 ! ah + bh dσ = a h dσ + b h dσ

=

* * *ru du × rv dv * h(r) * * h(r) *ru × rv * dudv

9( S 0 h ` ^Y 0 0 S !M (= .$ 00 Q

=

dQ D

00

=

! h r(u, v) ru × rv dudv

D

00 h dσ

= S

^Y &$0.&M E 0 5&0 0 $ .$ & &$0.&M R# %&L .$ M ( #0 *Z#E $3

00

00 m=

δ dσ , (S)

y =

m

x =

m

00

xδ dσ,

.$ *D : ≤ u ≤ a , ≤ v ≤ π 5F .$ M ZM .&3 E ( >.&[, . Mn #. ('&" 5&+B

zδ dσ.

dσ

(S)

00

yδ dσ ,

z =

m

pL E (x, y, z) Y\ Z#S A& '() % + δ = x + y + z 4) B&Q . $ z ≤ &0 z = x + y *0&0 . 5F !\f cM 4) *( + S : z = x + y ; (x, y) ∈ D >. T 0 . #. R# *!' 5F ('&" 5&+B *D : x + y ≤ M $ + ^#6 5

E >.&[, dσ

(S)

(S)

= # " # " > > x y ? = + + + + dxdy x + y x + y √ dxdy =

0 0 S 4) .$ *( 00 m

= = = =

* * *ru × rv * dudv *−−−−−−−−−−−→! −−−−−−−−−−−−−−−→!* * * * cos v, sin v, u × − u sin v, u cos v, * dudv *−−−−−−−−−−−−−−−−−−−−→!* * * * − u cos v, −u sin v, u * dudv + u + u dudv

Area(S)

=

δ dσ

00 dσ =

(S)

dv

π

π

+

" 0

π

= =

u

+ u dudv

D

#0

=

=

&0 ( 0 0 S ('&" R# 0&0

00

a

u

+

( + u )/

(a + )/ −

+ u du

a

(S)

, " √ 00 # = x +y + x +y dxdy D

=

√ 00

√ 00

! x + y dxdy =

D

#0 π √

" #0

"

r r drdθ

D

r dr

=

=

(π)() = π

dθ

√

√

v −x 0 x # U &0 5F B&Q e0& S #. 5 Q *( .$ $. $ . 1 x = T 0 S !\f cM M+ cM y o L 0&6 !B$ 0 *( T S !\f cM x o L

&0 ( 0 0 S !\f cM z o L *O&0 T c S !\f 00

z

=

=

m

m

zδ dσ (S)

00 , √ x + y (x + y ) dxdy

D

00

=

=

x + y

D

#0

!/

" #0

π

dθ

00 dxdy =

" r dr

=

r r drdθ

D

π

*( C = (, , π/) 0 0 S !\f cM R# 0&0 Y\ M 5F 0 p6 ' XUV ^Y 4) A, .8 (0 . O&0 δ = x + y + z 4) B&Q . $ 5F E (x, y, z) *#.F

0

9M [& . #E &#. E N# ('&"

y M z = x + y q0 5 S+ E +"1 A *( O ) z = z = >&T T y M y = x + z pL E +"1 A; *( O ) y = y M z = x − y B Bn 5 S+ E +"1 A *( O ) x + y = *|x| + |y| + |z| = a . 0 ^Y A *(x + y + z) = a (x + y − z) #. ^Y AH 5 .$ . −x ' . [a; b] E&0 0 y = f (x) e0& . $ + A8 *Z#.F (0 . S #. $ $ ∗

−−−−−−→! ' . C : r(t) = −x(t), y(t) ; a ≤ t ≤ b AJ *Z# $.F (0 . S #. $ $ 5 .$ . −x . ** E &a .$ O F / 0 . ('&" A7 *M [&

+ < E1n 4L) + QR . !) 2 L $

) B&Q . $ S #. E (x, y, z) Y\ M x= &+B D. S.&V0 &0 >. T R# .$ *O&0 δ(x, y, z) m 4) M $$S &6 A&S $ 0 [O ?z &MC 9( RQ >.&[, S #. C = (x , y , z ) !\f cM

00

Y\ T&= h M I := hδ dσ E ( >.&[, P ' R# .$ O&0 S 4) m S *O&0 P , + +& (x, y, z) ∈ S *Z & P ' S uQ `&UO . R := I /m >. T . −y . −x O Ǳ [ 0 0 P S i& (B&' .$ R# .$ O&0 T−yz &# T−xz T−xy . −z &0 0 0 X 0 P ' S #. & .& 6S >. T 00 P

(S)

P

P

00

δ dσ (S)

00

=

x+y+ (S )

! y + z δ dσ

00

(S)

! x + y δ dσ

Iz =

00

Ixy =

(S)

=

(S)

00

00

y δ dσ

Ixz =

Iyz =

(S)

x δ dσ

! x + y + z δ dσ

IO = (S)

00

! , + (− ) + () dxdy x + ( − x) + z z

= D

( √ 0 0

z x − x +

√ 0

x − x +

! + z dz

dx =

) dx

√

' x + y + z = R &Q M uQ `&UO A, .8 *M [& . . −z 9ZM .&3 . S =S N+M M >&j L E *!' −−−−−−−−−−−−−−−−−−−−−−−−−−→! r(ϕ, v) = R sin ϕ cos θ, R sin ϕ sin θ, R cos ϕ

E J (+"1 &+ *D : ≤ ϕ ≤ π , ≤ θ ≤ π 5F .$ M M $$S h'? H*;* * * dσ = *rϕ × rθ * dϕdθ = R sin ϕ dϕdθ

E ( >.&[, . −z ' S & .& 6S .$ 00

Iz (S) = (S)

! x + y δ dσ

00 (S )

00

!

x + y + 00

!

dσ +

=

x + z +

!

dσ

!

dσ

Dxz

y + z +

0 #0 0

+y+z

(S )

00

!

dσ

Dyz

=

! x + + z dσ

+ z dσ +

+ y + z dσ +

+

(S)

00

x+

(S )

Dxy

T E +"1 S M . T .$ A& '() D0 z = &j L >&T y M O&0 x + y = O&0 δ = z B&Q . $ (x, y, z) ∈ S Y\ ( $. *M [& . Ǳ [ ' S & .& 6S S : y = − x ; (x, z) ∈ D >. T 0 . #. R# *!' .$ *D : ≤ x ≤ , ≤ z ≤ M $ + ^#6 5

&0 ( 0 0 Ǳ [ ' S & .& 6S

(S )

+ 00

z δ dσ

! x + y + dσ

!

00

(S)

00

00 dσ +

(S )

! x + z δ dσ

Iy =

!

00 +

(S)

Ix =

=

00

m =

! x + y + z δ dσ

IO =

=

4) *Z# $ + GT H*;* E 7 (+"1 .$ . #. R# *!' &0 ( 0 0 S #.

" (x + y + ) dy

dx

(x + ) dx =

0 &

. 8** &a E ; (+"1 .$ #. !\f cM >&j L A *0&0 R+ ^Y E 6L0 !\f cM +>&j L A; #0 x + y = ax ^Y y M z = x + y *M [& . ( O Ω : ≤ x, y, z ≤ ' XUV ^Y S M . T .$ A O&0 δ = x +y +z 0 0 (x, y, z) Y\ B&Q O&0 *0&0 . S !\f cM 4) R+ M ^Y E +"1 !\f cM >&j L A z = a z = >&T y M x + y + z = R *0&0 . $. $ . 1 ≤ y &q= Z .$ ( O ) * < a < R &# .$ >&T 0 $ 4 ).& ^Y !\f cM >&j L AH M 0&0 . T .$ . x + y + z = T &j L

δ = x+y+z ) B&Q . $ 5F E (x, y, z) Y\ *O&0 Z8

+

)

+CcG

. !)

2 L *

Y\ M x= )R + < ] Y\ N# P $ 0 δ(x, y, z) ) B&Q . $ S #. E S & .& 6S >. T R# .$ *O&0 T N# &# y N# (x, y, z)

.$ *ZM .&3 D : ≤ x ≤ , ≤ z ≤ 5F = 00 > > ?

00 Area(S) =

dσ = (S)

00

D

dxdz + − x

= D

=

# +

+

" #0

#0

"

−x

− x

dz

−

+

x % $ z arcsin = π

=

0 0 . −x & mean h = S

=

=

=

=

00 Area(S)

+ ()

T&= M R# 0+ ) &0 Z#. $ ( h = y + z

S : y = ; (x, z) ∈ Dxz

S : x = ; (y, z) ∈ Dyz

≤y≤

−x

Dxz :

≤x≤

,

≤z≤

−x

Dyz :

≤y≤

,

≤z≤

−y

(S)

(S)

T dσ +

= (S )

xz

xy

dσ

00

00

(S )

00

T dσ + (S )

00 δ dσ = δ

(S)

dσ = δ Area(S) = πδ R

(S)

0 0 . −z ' S uQ `&UO R# 0&0 *( .

Iz = m

πδ R / = πδ R

.

R

0 ([" x + y + z = a R+ 3 & .& 6S A *M [& . . −z ' max &|x|, |y|, |z|' = a XUV ^Y & .& 6S A; ^Y B&Q e0& M R# x= &0 . >&j L Ǳ [

*M [& ( δ = |x| + |y| + |z| y M z = x + y pL E +"1 & .& 6S A ' . ( O #0 z = z = >&T 4) B&Q e0& M $ O x= *M [& +. −x *O&0 δ = z x + y R+ e0 & .& 6S A

∗

S : z = ,

≤x≤

,

≤y≤

O G#U S #. 0 h v e0& M x= p? >. T 0 . S 0 h y *O&0

00

T dσ ÷

S

00

< 2 a2 < =C) . !) 2 L

yz

mean T =

R

*M [& . x/a = y/b = z/c y '

T−xy 0 4 # j X 0 D D D M 0 *$ O ) I* !VO 0 * " T−yz T−xz E ( >.&[, T y X R# 00

πδ

0

− x − y ; (x, y) ∈ Dxy

,

sin ϕ dϕ =

& .& 6S S δ .$ $ ) 5.&\ !B$ 0 M $ O ) *( c $.n0 Ǳ [ E M #$ . ' S uQ `&UO *O&0 . \ R+

S = S + S + S + S

≤x≤

π

dθ

Rz =

(S)

" 0

#0 π

0 0 5F 4) ( (0&f S B&Q 5 Q =@ E m=

>&T 0 $ 4 ).& ^Y S M x= A, .8 5F E (x, y, z) Y\ M ( x + y + z = T &j L

#. R# p&\ & $ y *( T = x + y + z & $ . $ *M [& . 9$$S !V6 UY1 .& Q E 4 R# *!'

Dxy :

=

− x

h dσ

S : z = ; (x, y) ∈ Dxy

δ R

"

dx

00 dxdz ( − x ) + z + π − x D ) ( 0 0 + − x + z dz dx π − − x 5 0 4 + + + −x dx π − − x + x − x + arcsin x = π −

S : z =

(R sin ϕ)δ R sin ϕ dϕdθ

D

−

(x, y, z)

00 =

00 T dσ + (S )

T dσ

00 mean h = S

Area(S)

h dσ (S)

*$ + [& 5

0 e1 p&\ T&= e0 y A& '() T &j L >&T 0 $ x + y = *M [& . −x & . z = + .$ M S : y = − x ; (x, z) ∈ D !VO 0 . #. *!'

X =

( O x= B&" R# .$ M R# 0 ) &0 ( −−−−−→ Z#. $ v = ( , , ) (, , ) h

=

=

* i j k * √ * x y z * , √ (x − y) + (x − z) + (y − z)

÷ ()

=

I

= (S)

= =

=

=

=

T dσ +

÷

00 ,

00

+ =

− cos θ sin θ dθ =

mean I = S

I = Area(S)

=

( π × = π . \ 0 0 S ('&" M R# 0 ) &0 &0 ( 0 0 x = y = z y ' S .& 6S y R# 0&0 π 0

. z ≤ M z = x + y ^Y 0 p&\ `&. y A *M [&

y M ( x + y = T E +"1 S M x= A; ( O ) x + z = T &j L >&T *M [& . S 0 f y *f = x + y + z & F ` + M & $, z y x M x= A *M [& . & F %qT&' y *( a 0 0 &0 &Q e#& N# E & _& ( cL Ω M x= A x + y = a 0 m @ E M δ (0&f B&Q _&0 E z = x + y q0 5 S+ 0 Rg&3 E ^Y 0 .&6= y *( $ z = a T 0 *M [& . Ω ,&1

dσ +

(S )

00

dσ

(S )

,

+ (− ) + (− ) dxdy , (x + y + ) + () + () dxdy

×

00 ,

+ () + () dxdy

00 &

+

√ '

dxdy

Dxy

)

!

(S )

00 & √ ' x + y + dxdy ÷

! − cos θ sin θ r dr dθ

00

Dxy

! − cos θ sin θ r drdθ

−π/ 0 π/ cos θ −π/

T dσ ÷

(S )

dσ

+ (− ) + (− ) dxdy

Dxy

( π/ 0 cos θ

00 dσ +

Dxy

Dxy

D

D

(S )

(S )

00

00

00 dσ +

00 & ' x + y + ( − x − y)

+

h δ dσ

0

=

(S )

Dxy

+ (x − y) + x + y + + dxdy 00 ! x + y − xy dxdy

dσ +

(S )

5 S : z = ; (x, y) ∈ D !VO 0 . . Mn #. ?U0 >.&[, S .& 6S .$ *D : x + y ≤ x M $ + 5&0 E ( 00

00

00

00

) ( 0 0 −x & √ ' x + y + dy dx #

=

& √ ' ÷ + Area(Dxy ) √ √ " √ + − + = ÷

$& S S S R0 $ ) 5.&\ . AC .$ M R# ^- *Z# $M # $ 0 M O&0 T−xy E +"1 S M x= A .8 E (x, y, z) Y\ 4) B&Q O&0 $ x + y = x ' S & .& 6S y *( δ = x + y + z 0 0 5F *M [& . x = y = z y }

;** &a E ; (+"1 9I* !VO & (x, y, z) Y\ T&= M ZM ) . h R# 0 _> 0 0 v $& X &S V &0 y *−−−→ * *X X × v* h= v

) % * 0C ( O !V6 Zh b p&\ E z & S M 4&+ LB$ c b a &. $0 5 Q & *A$ O *+h }

. $ x +y+z = R M E (x, y, z) Y\ M x= AH E (+"1 5F 4) y *( δ = |xyz| 4) B&Q *M [& . $. $ . 1 4 Z 6 N# .$ M M

A `GF b, !F (7

;** E ; & +"1 9* !VO T 0 $ 4 Ω M x= A .8 0C O&0 5F ).& ^Y S ( &j L >&T 5&O 3 0 #E KO 0 6\ .& Q *A$ O ) I* !VO ) ;** E ; (+"1 0C $ + KY 5 S V.& Q 9A$ O

x+y+z =

−−−−−−−−−−−→! −−−−−→! r (u, v) = u, v, − u − v , r (u, v) = u, v, , −−−−−→! r (u, v) = u, , v ,

r (u, v) =

−−−−−→! , u, v .

, + 0 (u, v) & F & .& Q .$ M D :

≤u≤

,

≤v≤

−u

5F p&\ 4&+ S 4 e- .& Q c0 R# 0&0 *( tU

K - 0 S & B&# &O S E .&[, M p&\ #& *+h

*+h b p&\ , + #E $. E N# .$ 0 9M oL6 . O $ $ #. Zh

>&T y M ( z = T E +"1 S A *( O ) y = x = >&T &j L

>&T y M ( x + y = E +"1 S A; *( O ) x + z = z = y M (

pL E +"1 S A *( O ) x = x = >&T

5 4$ ` ^Y M (S 5 $& 5&0 0 F y S E . [, .&O . \ S . ) ^Y 0 F . $0 4 & ( 4E_ ` R# G#U E ![1 *O&0

*ZM ^#6 . ^Y N# 0 ( ) E Y\ X #. S M x= M ( `g )*"9 N# X Zg S . T .$ *O&0 5F . X M $$S (=&# 5&Q S 0 D $ &0 r & 6\ >. T 0 (u , v ) ∈ Int(D) 5 Q .$ Y\ N# E 0 .$ Y\ G#U &6 0C (O 5 0 r(u , v ) = X S 5.$ . S #. Zh p&\ , + *A$ O U) *8*7 0 *Z$ 5&6 IntS $&+ &0 & 5 S+ E +"1 S M x= A& '() O ) z = T y M ( z = x + y q0 −−−−−−−−−−→! r(u, v) = u, v, u + v 6\ N# & y . S *( .$ *D : u + v ≤ M $ + .&3 5

Int(D) : u + v < . E r y M @&\ + S 0 6\ & * " S Zh p&\ $S # j 5 U0 . S 0 u + v = O # j p&\ M (=&# 5 + .&[, S Zh b p&\ X R# 0 *$S 0 .$ Zh p&\ cM 0 # $ M S − Int(S) : x + y = , z = E * !VO 0 *( z = T .$ `&UO (, , ) *$ O ) GB ?z+, *O&0 R `&UO X cM 0 M S ZM x= A, .8 *$ + 5 S 0 AMC #&= v) 6\ (#& 0 *O&0 X .$ 0 &0 V# &U . $0 c b a ZM x= 9ZM G#U #E KO 0 6\ &' ! ! ! r(ϕ, θ) = R cos ϕ cos θ a + R cos ϕ sin θ b + R sin ϕ c D :

≤ ϕ ≤ π , ≤ θ ≤ π

E >.&[, D 5.$ 5 Q

x = y + z

*( ' XUV ^Y S A

Int(D) :

< ϕ < π , < θ < π

# j S 0 Int(D) e0 E r y M @&\ 4&+ ( S 0 # + N# & M $ O ) *+h p&\ O

)* *&O 3 5 0 .&SE& 6\ $ U y . 5F M Zg S ) 5F >&UY1 E N# 0 M ( #n< ) #. % *0 0 w 6 & $ 0 0&L >& ) ( O %&L 0 S #. 5&+ −S E . h O&0 . ) #. S S 0 ?z[1 S U# *( 5. ( ) &0 , + 5 , *ZM $& −n E −S 0 5 M O&0 O $& n E S R# .$ O&0 c V k . $ S #. S M ( M{ 0 4E_ *$ + G#U 5 S UY1 0 >& ( ) >. T M E +"1 S M x= A& '() & R# *$. $ . 1 T−xy _&0 .$ M O&0 x + y + z = S : x + y + z = , ≤ z ; (x, y) ∈ D !VO 0 . #. q1 E GB (+"1 0&0 *D : x + y ≤ M ZM 5&0 E .&[, S 0 Zg&1 V# &. $0 H** k

−−−−−−−−→! x, y, z f n = ± = ±+ f (x) + (y) + (z) −−−−−−

,−−−−−−−−−−→ ± = x, y, − x − y

!VO .$ *O&0 S #. mU e0& f = x + y + z M *( O $ $ 5&6 n ( ) &0 S #. GB * }

Y\ X ( #. S M x= . @ ( S 0 6\ N# r ZM x= *O&0 5F E Zh

&. $0 >. T R# .$ *r(u , v ) = X M ru (u , v ) × rv (u , v n = ±* *ru (u , v ) × rv (u , v

) * )*

*( N# 0 0 & F E 4 M @ $ +, S 0 X Y\ .$ U# * " r 6\ %&L E !\ " ±n M $ O (0&f >. T R# .$ $ $ ^- . X 5 0 c #$ 6\ &0 S Y\ .$ R# 0&0 *$ 0 −n n 0 !T&' &. $0 *$. $ $ ) Zg&1 V# . $0 $ & $ S #. E Zh

9( :& !0&1 ;** q1 E ' . 0 #E q1 0 i& & B&' .$ . −n n &. $0 $ 9$ + 5 #E !VO &F S :, f (x, y, z) = a S AnH −−−−−−−→! fx , fy , fz f n = ± = ±, f (fx ) + (fy ) + (fz ) −−−−−−−−→! fx , fy , − n = ±, S : z = f (x, y) + (fx ) + (fy ) −−−−−−−−→! fx , − , fz n = ±+ S : y = f (x, z) + (fx ) + (fz ) −−−−−−−−−→! − , fy , fz , n=± S : z = f (y, z) + (fy ) + (fz )

&F

S AK

&F

S Aw

&F

S A }

J** E ; & +"1 9* !VO pL E +"1 S M x= A, .8 0 . #. R# *( $. D0+z = T y M O&0 M $ + 5&0 5 S : z = x + y ; (x, y) ∈ D !VO &. $0 H** E % (+"1 0&0 5 M *D : x + y ≤ E .&[, S 0 Zg&1 V# z = x + y

n

−−−−−−−−−−−−−−−−−−−−−−→ " # y x + ,+ ,− x + y x + y = ±= # " # " > > x y ? + + + + x + y x + y −−−−−−−−−−−−−−−−−−−−−−→ # " x ± y + = √ ,+ ,− x + y x + y

6\ N# $0 0 ( ) AGB 9;* !VO .&SE& .} ) 6\ $ A% E 6\ r $ 0 #. S ZM x= % Zg&1 V# &. $0 ±n D AQ.&
r 6\ $ *A$ O ) GB ;* !VO 0C ( G#U .&SE& . T .$ . D D X 0 & $ &0 S 0 r O&0 V# r(D ) ∩ r(D ) , + 0 $ 5F ( ) M Zg S ]o % . T .$ . S #. *A$ O ) % ;* !VO 0C

&S ) . UY1 ]< $ + Z"\ V uO 0 ** . $0 n S 5 M *$ O ) % H* !VO 0 *ZM .&3 Z#. $ x= t0&Y &F O&0 S 0 ( ) i

i

n = i,

n = −i,

n = j,

n = −j,

n = k,

n = −k.

0 . S M ( M{ 5&#&O *$ O ) GB H* !VO 0 R#+ 5 , 0 $ + . ) 5 G L >. T = *M Z" $ 0 . & F E + Q 0 M ( Z 6 N# E +"1 Ω M x= A4 .8 S ( O $ z = T x + y = ZM . ) Z . @ . S *O&0 5F ).& ^Y *O&0 Ω E 50 0 . n . $0 . + M 0 #. k3 E ,&+ ) !VO 0 . S 0 . h R# 0 9Z"# #E KO S = S + S + S + S + S S : x + y = ,

≤ x, ≤ y, ≤ z ≤

S : z = , x + y ≤ ,

≤ x, ≤ y

S : z = , x + y ≤ ,

≤ x, ≤ y

S : y = ,

≤ x ≤ , ≤ z ≤

S : x = ,

≤ y ≤ , ≤ z ≤

i

i

=

= n

=

n

=

−− # −−−−−−−−−−−−→ " −x + ,− , − x ±= " ># > ? + −x + (− ) + () − x −−−−−−−−→ −−− + ∓ − x, − x ,

−−−−−−−−→! x, − ,

n = ±+ (x) + (− ) + () −−−−−−−−→! ± = + x, − , + x

. −y (+"1 0 . n ZM %&L . + l&Q . −y ([a (+"1 0 . −n ZM %&L . − S $ 0 *$ O ) GB * !VO 0 *$ 0 q0 5 S + E +"1 S M x= A .8 R# .$ *$. $ . 1 Z 6 N# .$ M O&0 x = − y − z S : x = − y − z ; (y, z) ∈ D !VO 0 . S >. T D : y + z ≤ , ≤ y , ≤ z M $ + 5&0 5

E .&[, S 0 Zg&1 V# &. $0 H** E $ (+"1 0&0 −−−−−−−−−−−−→! − , −y, −z

. $0 n S M $$S h'? H** q1 0 ) &0 5 M >. T R# .$ O&0 S 0 Zg&1 V# n

. − S $ 0 Rg&3 0 . n ZM %&L . + l&Q ) % * !VO 0 *$ 0 _&0 0 . −n ZM %&L *$ O O&0 y = x + E +"1 S M x= A .8 .$ *( O ) z = z = y = >&T y M 5 S : y = x ; (x, z) ∈ D !VO 0 . S >. T R# (+"1 0&0 D : − ≤ x ≤ , ≤ z ≤ M $ + 5&0 E .&[, S 0 Zg&1 V# &. $0 H** E A:C

n = ±+ (− ) + (−y) + (−z) −−−−−−−→! ∓ = + , y, z + y + z

. −x (+"1 0 . n ZM %&L . + l&Q . −x ([a (+"1 0 . −n ZM %&L . − S $ 0 *$ O ) % * !VO 0 *$ 0 }

−−−−−−−→! , , −

±+ = ∓k () + () + (− ) −−−−−−−→! , − , ±+ = ∓j () + (− ) + ()

.$ n M ( O 5 Q *n = ∓i n = ∓k - + & ?, E H* !VO 0 ) &0 O&0 50 0 . &' $& n n n n n &. $0 0 X 0 + + + *ZM

J** E & +"1 9* !VO *O&0 5F ).& ^Y S ' XUV Ω M x= A .8 (+ 0 . + M ZM %&L Z . @ . S 0 ( ) ( ) M $ O S &z'?YT (B&' R# .$C O&0 Ω E :.& E ; (+"1 & . S 0 . h R# 0 *A( "0

Q *$. $ . N# & #E }

R#?M Y0 98* !VO . $0 &0 . ) #. N# S M x= R# .$ $. $ . 1 F . $0 5 f& ( ( n ( ) 00 F n dσ !V60 . S 0 F 4$ ` ^Y >. T B&V 5 N# f = F n M $ O ) *ZM G#U *O&0 S 0 Av U0& U#C •

(S)

•

. $0 5 F F F M x= ('&" #n< ) &#. S S S \\' $ , b a >. T R# .$ *O&0 T S ∩ S 00

(aF + bF ) • n dσ =

) (S)

00 a

00 F • n dσ + b

(S)

)

(S)

00

00 F • n dσ =

(S ∪S )

)

00

00 F • n dσ +

(S )

F • n dσ

(S )

00

F • n dσ = − (−S)

F • n dσ

F • n dσ

(S)

5 S+ E +"1 S M x= A& '() z = T y M ( z = + x + y q0 . $0 5 *O&0 Rg&3 0 . n ( $. D0 −−−−−−−→ *M [& . S 0 F = x , y, z! S : z = + x + y ; (x, y) ∈ D >. T 0 . S #. *!' J* !VO 0 *D : x + y ≤ M $ + =U 5

>. T R# .$ *$ O ) GB −−−−−−−−−→! x, y, − n = ±, (x) + (y) + (− )

o L U#C O&0 Rg&3 0 . n ( . 1 5 Q M ?U0 *ZM %&L . (B&' AO&0 n E z

}

J** E 8 H & +"1 9H* !VO 0 . O =U & #. E N# 0 * ( ) Q M M oL6 $. .$ *M . ) LB$ *$ + G#U 5 O $ $ #. 0 G L

* ≤ z M x + y + z = x M E +"1 A *y ≤ M x + y + z = M E +"1 A; z = >&T R0 M x +y = z pL E +"1 A *$. $ . 1 z = − x + y = 0 $ 3 ).& ^Y A *x + z = z = >&T *max{|x|, |y|, |z|} = a XUV ^Y AH *|x| + |y| + |z| = a ) (6 . 0 A8 x + y = z pL 0 $ Z") ).& ^Y AJ *z = z = >&T &j L >&T 0 $ Z") ).& ^Y A7 *x + z = x + y = >&T x + y = # " & 0 $ , + A *x + z = O&[ #n< ) ( RV+ #. 78 =U . @ #. 0 .&SE& 6\ & $ U 5 U# &#. R# mU *&O <0 . #. ^Y ?z &M M $ + 9R#?M Y0 / 0 . E .&[, #n3&( ) *$ O ) % 8* !VO 0 _a \& M O&0 0 !Y " U M x= _jh " \, A X 0 . 5F / g. ( O & V _ ]) N# E . 1 BC e- 0 . @ . AD e- l&Q *Z &0 D C B Z N# 0 $ O t[Y D 0 C B 0 A M Z$ CD \' 0 . AB \' &' *$ O ) 8* !VO 0 *. \' 0 ( ) D 0 A &[Y 0 ?, M Z$ . 1 . @ 0 X R# 0 *$$S t[Y c DC \' 0 ( ) &0 AB ( #n< ) c #. R# *Z. jh " 4&0 #.

%&L . (B&' .$ O&0 . −y ([a ( ) 0 . Z#. $ ` + .$ ]3 *n = j U# ZM

00

00

−−−−−−−−−−−−−−−−−→! x − , − z, z − x • j dxdz

F • n dσ = (S)

D

00 =

(

− z) dxdz =

D ()

=

00

00 dxdz − D

.$ *dσ

D

$M $& z 0 ([" D 5.&\ z e0& 5$ 0 $= E AC .$ *( T D 0 z M Z# $ + &6

M ( x + y = T E +"1 S M x= A .8 #0 y + z = T y $. $ . 1 Z 6 N# .$ . −x ( ) 0 . n M M x= Rl+ *( O −−−−−−−−−−−−−−→! S 0 . F *F = x, x + y, x + y + z O&0 O $ *M [&

5 S : x = − y ; (y, z) ∈ D >. T 0 . S *!' !VO 0 *D : ≤ y ≤ , ≤ z ≤ − y M $ + =U

>. T R# .$ *$ O ) GB 7* √ −−−−−−−−−→! − ,− , ∓ −−−−−→! n = ±√ = , , + + , √ (− ) + (− ) + () dzdy = dzdy dσ =

&0 ( 0 0 h $.

−−−−−−−−−→! x, y, −

z dxdz

−=

+ (x) + (y) +0(− 0 ) dxdy (S)

F • n dσ

00 −−−−−−−−−−−−−−−−−→ ! x , y , ( + x + y )

• +

D

Area(D) = × × ×

=

00 = D ()

=

00

−

,

+ x + y

x + y − (

+ x + y dxdy =

+ x + y )

dxdy

( + x + y ) dxdy

D ()

=

00

−

( + r ) r drdθ

D

= =

− −

#0 π

" #0

dθ

π

!

=

" ( + r ) r dr

− π

$ 5.&\ y x e0 5$ 0 $= E AC .$ M R# ^- v E A;C .$ *T $ R# 0&0 O $& D 9O&0 D [Y1 # j D #$S $& [Y1 >&j L

*D : ≤ θ ≤ π , ≤ r ≤ }

. −x ( ) 0 . n ( . 1 M R# 0 ) &0 V# U# (B&' AO&0 n E x o L √U#C O&0 ! Z#. $ ` + .$ ]3 *ZM %&L . n = − −−,−−,−→ } ;** E ; & +"1 9J* !VO . $0 5 A, .8 F = (x − y)i + (y − z)j + (z − x)k

;** E & +"1 97* !VO 00

00 F • n dσ = (S)

−−−−−−−−−−−−−−−−−−−−−−−−→! − y, − y + y, − y + y + z

D

−−−− −−−−→ # " √−−−−− √ •

−

, − ,

√

dydz

T |x| + |z| = $. 0 E !T&' S #. 0 . ([a ( ) 0 . n ( ) . $0 M 0&0 . T .$ y = *O&0 . −y 5 S : y = ; (x, z) ∈ D >. T 0 . S #. *!' Ǳ [ cM 0 E B D *D : |x| + |z| ≤ M $ + =U

±j ±k / g. &0 *$ O ) % J* !VO 0 *O&0

−−−−−−−→! + Rl+ n = ± , − , / + +√ >. T R# .$ n ( . 1 M R# 0 ) &0 *dσ = + + dxdz

00

00

00

F • n dσ = (S)

Z#. $ ` + .$ R# 0&0

F • n dσ +

(S )

D

(S )

=

D

−−−−−−−−−−−, −−−−−−−−−−−−−→

x y a dxdy , , • a − x − y + a a a a − x − y − − − − − − − − − − − − − − − → " # 00 x + y + − y, x, a D

−−−−−−→! , x, y, −a + x + y + a dxdy x + y + a a 00 4 x + = + a a − x − y D 5 , •

a − x − y

00 − 00 4

a +

=

a − r

D

−

r

a

+

0 a ( 0 π 4 =

a − r

0 a

a − r

+ a +

π r + πr a

+

+

dσ

dr = −

π

. n ( x + y + z = M S M x= A .8 [& S 0 . F *F = −z,−−−y,−−−x→! $. $ S 50 0 *M 9ZM .&3 M >&j L N+M 0 . #. R# *!' r(u, v) =

−−−−−−−−−−−−−−−−−−−−−−−−−→! sin u cos v, sin u sin v, cos u

R# .$ .$ *D

:

≤ u ≤ π, ≤ v

dσ = r

u

=

≤

= n

π 5F .$ M

× rv dudv

(B&'

=

= dσ

ru × rv ru × rv −−−−−−−−−−−−−−−−−−−−−−−−−−→! = ± cos u cos v, cos u sin v, − sin u −−−−−−−−−−−−−−−−−−−−−−−→! × − sin u sin v, sin u cos v, ÷ ru × rv −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→! ± = sin u cos v, sin u sin v, cos u sin v ru × rv

n=±

=

=

a − r

)

y − dz dy

− y − y + dy =

. −z ([a ( ) 0 n

az , z ≤ a

−−−−−−−−−−−−−→! x, y, z − a f ± = ±+ f x + y + (z − a) ± −−−−−−−−−→! x, y, z − a a f dxdy |fz | , x + y + (z − a) dxdy (z − a) a dxdy z − a −−−−−−−−−−→! x, y, −a g ± = ±+ g x + y + a −−−−−−→! ± x, y, −a + x + y + a , g dxdy = x + y + a dxdy |gz | a , x + y + a dxdy a

=

a − r

!

R#&3 E M ( +") ).& ^Y S M x= A .8 x + y + M 0 _&0 E az = x + y 5 S+ 0 Z") 5F E 50 0 . n &) + .$ ( $ z = az ! *M [& S 0 . F = −−−−−−y,−−x,−→ z *O&0

z + az = az .$ . #V# O $ $ M 5 S + *!' .$ *z = a &0 z = R# 0&0 *M eY1 z = az &# w O .$ . # $ z = a .$ &+ Z 0 & F z = 9$$S !V6 UY1 $ E S = S + S R# 0&0 *. $

n

5 ) r cos θ r + r dθ dr − a a

aπr −

=

+

5

r cos θ

0

0 (0 −y

. −z ( ) 0 n *( D : x + y ≤ a T−xy 0 $ # j mU e0 M R# 0 ) &0 *$$S ) % 7* !VO 0 f = x + y + z − az E .&[, X 0 S S Z#. $ g = x + y − az

dxdy

r cos θ r drdθ a

+

y − dydz =

S : x + y =

D

+ a +

−

!

S : x + y + z = az , z ≥ a

! x + y − x dxdy

r cos θ

=

F • n dσ

−−−−−−−−−−−−−, −−−−−−−−−−−−−→ 00

= − y, x, a + a − x − y

+

00

= =

*O&0 S 0 Zg&1 V# . $0 n S ('&" !" #$ dσ M E :.& (+ 0 n . + M R# 0 [ & 0 ) &0 n . −z ([a ( ) 0 . n #&0 ]3 O&0 . ) S U# O&0 . −z ( ) 0 . i

n =

−−x−−−− y−−−z−−−−→ , , − a a a

i

i

−−−−−−→! x, y, −a n = + x + y + a

i

y = x +z 0 $ o1& pL ).& ^Y S A7 . ) S 5.$ 0 . n ( y = y = >&T −−−−−−−→ *F = x, y , z!

O&0 S E 50 0 . n &) + .$ #&0 M R# # ) &0 X R# 0 *ZM %&L . 00( ?, n .$ #&0 M Z#S

E ( >.&[, F n O •

00

A `GF b, !F N +7 9$$S E&0 #E B&" 0 4$ ` ^Y $0.&M R#+, ( . ) #. N# S M x= VD) $ % Y *$$S G#U S 0 M O&0 . $0

*F y S E . [, .&O . \ [& ( .&3 D $ &0 r(u, v) 6\ y . S M x= *!' !Y " 5$M # j E !T&' NQ M !Y " Z# $ + 9Z#S h .$ . S ^Y 0 U = [u; u + du] × [v; v + dv] ,?-_ E &0 . yLB !Y " R# *Σ = r(U ) r(u, v) Y\ .$ r (u, v) dv r (u, v) du &. $0 y M $ + x= 5 ?U0 *$E X#\ 5 O . F(r(u, v)) 0 0 (0&f !Y " R# 0 F . $0 5 M yLB S 5F .$ M T B&" &0 R# 0&0 *O&0

E N# S 5F .$ M Z#. $ $& B&" ( v F 0 0 . [, .&O B&' RQ .$ *( (0&f 5F 0 F `?-_ #. 0 Zg&1 $ .$ 5 # j .$ #. ('&" %qT&' Y\ .$ $ ) ( ) . $0 r(u, v) S #$ 5&0 0 *O&0

>. T R# .$ O&0 r(u, v) F

v

u

* ** * dF = *ru (u, v) du × rv (u, v) dv * *projn(u,v) F(r(u, v))* * * = F(r(u, v)) • n(u, v) *ru (u, v) × rv (u, v)* dudv = F(r(u, v)) • n(u, v) dσ

&0 ( 0 0 S E . [, .&O . \ R# 0&0 00 F=

00 F • n dσ

dF = D

(S)

5 &0 &q= .$ B& M x= A& '() $ .&O . \ *$. $ $ ) F = −z,−−−y,−−−x→! > .{ (, . $0 .$ . F y ≤ y M x + y + z = M Z E . [, . −y ([a ( ) 0 . n Zg&1 . $0 &) + M 0&0 . T *O&0 + S : y = − x − z ; (x, z) ∈ D !VO 0 . S #. *!'

D

(S)

−−−−−−−−−−−−−−−−−−−−−−−−−→! cos u, sin u sin v, sin u cos v −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→! sin u cos v, sin u sin v, cos u sin v dudv 00 & sin u cos u cos v + sin u sin v = •

D

' + sin u cos u sin v cos v dudv 0 π $ 0 π & = sin u cos u cos v + sin u sin v ' % + sin u cos u sin v cos v dv du 0 π & ' = sin u π cos u + π sin u du )π ( sin u cos u + − cos u = π =

0

9M [& S 0 . F $. .$ N# .$ M ( x + y + z = M E +"1 S A ( . −z ([a ( ) 0 . n $. $ . 1 Z 6 −−−−−−−→ *F = x , y, z! M ( z = x + y q0 5 S+ E +"1 S A; 0 . n ( O ) z = z = >&T y ! y $. $ . −z ( ) *F = −y−−+−−z,−x−−+−z,−−x−−+−→ T y M ( y = x + z pL E +"1 S A ([a ( ) 0 . n ( ≤ z $. D0 y = ! *F = −x,−−−y,−−−→ z $. $ . −y y M ( x + y = T E +"1 S A ( ) 0 . n ( $. D0 y + z = # "

! *F = −−x,−y,−−→ z $. $ . −y

y M ( x + y + z = T E +"1 S AH ( ) 0 . n ( O $ $ D0 |y| + |z| = E B ! *F = −−−x,−x−−+−−y,−x−−+−→ z $. $ . −x ([a

x = y >&T 0 $ Z") ).& ^Y S A8 S E 50 0 . n ( z = z = x = −y ! *F = −−x,−z,−−→ y $. $ x + y = 0 $ . 3 ).& ^Y S AJ S E 50 0 . n ( y = z + y = z >&T *F = −x−−−−−y,−y−−−−−z,−z−−−−x→! $. $

e0& 5 Q .$ *D : x +z ≤ M $ + .&3 5

Z#. $ ( f = x + y + z E >.&[, S mU

Z#. $ .$ *ZM .&3 n

ru × rv ru × rv −−−−−−−−−−−−−−−→! −−−−−→! − sin u, , cos u × , , ± ru × rv −−−−−−−−−−−−−−−−−→! − cos u, , − sin u ± ru × rv

±

=

=

=

n =

=

?U0 *n = j n = −j ( [1 $. (B&' M dσ

=

dσ

=

ru × rv dudv + dσ = + + dxdz = dxdz

F

00

00 F • n dσ +

= (S )

00

=

00 D

00

+

F • n dσ

D ()

(S )

00

D

0 (0 π

=

−−−−−→! −−−−−→! x, x, • , , dxdz

D

cos u dudv π

cos u du

" #0

"

−

dv

= π

}

F • n dσ

h $. .&O . \ .$

dxdz − x − y xz +

− x − z dxdz + − x − y r cos θ+ sin θ + − r r drdθ − r

=

D

#0

00

=

−−−−−−→! −−−−−−−→! x, x, − • , − , dxdz

D

=

=

−−−−−−−−−−−−−→! −−−−−−−−−−−−−→! cos u, cos u, • cos u, , sin u dudv +

F • n dσ +

(S)

− −−−−−−−−−−−−−→

−−−−−, x z + • , −x −y ,

00

(S )

D

=

D

00

(S)

00

, F x, − x − y , z

F • n dσ

=

&0 ( 0 0 F = 00

&0 ( 0 0 h $. .&O .$

00

−−−−−−−−→! x, y, z f + =± ± |fz | x + y + z −−−− −−−−−−−−−−−−→

, ± x, − x − y , z

π

sin θ + r − r r cos θ + dθ − r

)

dr

0 + r − r dr = π

( O $& [Y1 >&j L E AC .$ M R0 ^- *O&0 T−rθ .$ D # j

Z") ).& ^Y !VO 0 . N# S M x= A, .8 y = y = − >&T x + z = 0 $

5F (, &. $0 5 M $. $ . 1 $. .$ O&0

! *M [& . . [, .&O *( F = −−x,−x,−−→ y 9(O 5 #. E ,&+ ) >. T 0 . #. R# *!' S = S ∪ S ∪ S S : x + z = ; − ≤ y ≤ S : y = − ; x + z ≤ S : y =

;*H* E ; & +"1 9* !VO B$&U 0 % S M x= A .8 ,

! x + y − + z =

−−−−−−−−→! x, y, −z

?U0 $. $ S E 50 0 . n ( *M [& . F y S E . [, .&O *( M ZM x= 0 S #. 5$ + .&3 0 *!'

F =

; x + z ≤

h'? !VO 0 ) &0 *$ O ) GB * !VO 0 0 nB O&0 S 50 0 . n #&0 S $. .$ M $$S

S $. .$ n = −j #&0 S $. .$ ( $ +, . −y !VO 0 . #. R# *n = j #&0 S : r(u, v) =

−−−−−−−−−−−−−→! cos u, v, sin u ; (u, v) ∈ D

S : y = − ; (x, z) ∈ D

S : y = − ; (x, z) ∈ D

≤ u ≤ π , −

D : x + z ≤

D :

≤v≤

+

y = − x − z 0 $ S Z ^Y S AH $. $ S E 50 0 . n ( y = T *F = −x−−+−−y,−y−−+−−z,−z−−+−x→!

+

Z#. $ R# 0&0 *z = sin u x + y − = cos u 9$M x= 5 x + y = (cos u + ) x = (cos u + ) cos v

E >.&[, S !M &O 3 6\ ` + .$ ]3

c B D 07 P k )*;: 4&0 . 5F e) E 0 .$ M /&S q1 +V' M =U !?+3+ )*;: &# !3 )*;: N# 0 "0 ^Y N# 0 4$ ` ^Y !#[ 0 ( q1 R# >. T Z 0 *#. 5F ! $ 0 &S 5 N# ] W. #$ "0 #. 9Z#. $ E& #) 4 $ 0 *Z#E $3 & F 5&0 0 ?z#{ M . $0 + )*E . T .$ . S #n< ) #. % S M $ + (=&# . @ Ω & R .$ +' 5 0 M Zg S *S = ∂Ω Z"# >. T R# .$ *O&0 5F ).& ^Y O w O & F E #& $ M $& #. Q `&+ ) *$ O & )E )* c O&0

R# .$ O&0 "0 #. S M x= % Z"\ #E KO 0 , + #E $ 0 R − S p&\ >. T /&+ b `&UO M X ∈ R −S 5 Q @&\ AnH 9 O *M eY1 Y\ $= $ U .$ . S #. X E .$&T S 0 S 0 /&+ b `&UO M X ∈ R − S 5 Q @&\ AK *M eY1 Y\ :E $ U .$ . S #. X E .$&T $&+ &0 & S L . 5F ( . M , +

L . 5F ( 5 M0 4$ , + *Z$ 5&6 IntS S M ( RO. *Z$ 5&6 ExtS $&+ &0 & S U#C $6= , + Ω &F O&0 . M Ω = IntS ∪ S (O 5 R# 0&0 O&0 Ω [B S ( A. M "0 *S = ∂Ω S "0 #. 0 + 7 % E . h % V# . $0 X ∈ S Zh Y\ E 0 5F .$ M ( ) *O&0 O $ m S E 50 0 X .$ O %&L n Zg&1 N# S S * & E . n &z'?YT (B&' R# .$ 00 F n dσ $&+ E &F O&0 $. & ( ) &0 "0 #. 00 *$ O $& F n dσ &0 •

(S)

•

(S)

y = (cos u + ) sin v

,

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→! r(u, v) = (cos u + ) cos v, (cos u + ) sin v, sin u D :

≤ u ≤ π , ≤ v ≤ π

R# 0&0 ( n

ru × rv ru × rv

=

−−−−−−−−−−−−−−−−−−−−−−−−−−→! (cos u + ) cos u cos v, cos u sin v, sin v + sin u = ± ru × rv

.$ ]3 *ZM %&L 00 . (B&' 5$ 0 "0 ( ) 0 M &0 ( 0 0 F n dσ O .&O ` +

•

(S)

00 D

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→! (cos u + ) cos v, (cos u + ) sin v, − sin u −−−−−−−−−−−−−−−−−−−−−−−−−−→! cos u cos v, cos u sin v, sin v + sin u (cos u + ) dudv 00 ! (cos u + ) cos u + cos u − sin u dudv = •

D

0 π $ 0 π

=

(cos u + )

cos u

! % + cos u − sin u du dv

=

π 0 $

9M [& . F y S E . [, .&O $. .$ &q= Z .$ M O&0 z = x + y pL E +"1 S A n ( O ) z = T y $. $ . 1 x ≤ y ! *F = −−y,−x,−−→ z O&0 . −z ( ) 0 . >&T y M O&0 y = x E +"1 S A; ( ) 0 . n ( O #0 z = z = y = −−−−−−−−−−→ *F = yz, zx, xy! $. $ . −y ([a

x + y + z = M Z 0 $ Z") ^Y S A $. $ S 50 0 . n ( ≤ y &q= Z .$ e1 −−−−−−→ *F = z, y, x! x + y + z = T 0 $ 4 ^Y S A $. $ S 5.$ 0 . n ( &j L >&T −−−−−−−→ *F = x , y, z!

0 *M !+, v e0 0 +, R# *ZM G#U O&0 #n<" #$ v U0& f (x, y, z) S &a 5 , −−−−−−−−−−→

∂f ∂f ∂f , , E >.&[, f 0 ∇ !+, !T&' >. T R# .$ ∂x ∂y ∂z RO. *Z$ 5&6 ∇f $&+ &0 . #) e0& R# *$ 0 . ∇f e) E 0 .$ *O&0 f t 6 5&+ ∇f M ( *$ 5&6 grad(f ) $&+ &0 & f L3

+"1 pL # $ Z e0 A& '() % ( O $ "0 N# 0 M T E * " "0 b &#. E #& + }

9M $ + >&[f 5 t 6 i E $& &0 S$& 0 $ 0 #n<" #$ v e0 g f S & % >. T R# .$ O&0 \\' $, a ) ∇(f + ag) = ∇f + a∇g ) ∇(f g) = g∇f + f ∇g ) ∇ (f /g) = ( /g ) g∇f − f ∇g

}

*8* E ; & +"1 9;* !VO 5 N# F = −−P,−−Q,−−R→! M x= * % 5 A# S U#C P k >. T R# .$ O&0 #n<\ 6

5&6 div(F) $&+ &0 $ + G#U ∇ F >. T 0 . F . $0 9Z$

•

div(F) = ∇ • F =

∂Q ∂R ∂P + + ∂x ∂y ∂z

. $0 & G F M x= % O&0 \\' $, a #n<\ 6 v e0& f #n<\ 6

>. T R# .$ ) div(F + aG) = div(F) + a div(G)

)

div(f F) = (∇f ) • F + f div(F)

)

div(F • G) = G • div(F) + F • div(G)

*8* E & +"1 9;I* !VO #& + % @ 1 N# ^Y XUV ^Y M A, .8 E N# $. & ( ) ;I* !VO .$ * "0 &#. E *ZM h'? . K Y R# x + y + z = M $ E ,&+ ) S M x= A .8 ( "0 #. N# S >. T R# .$ *( x +y +z = . $0 *O&0 M $ R0 '& IntS : ≤ x + y + z ≤ S.c0 M 0 Ǳ [ (+ 0 . VQ M M 0 n $. & ( ) *$ O ) GB ;* !VO 0 *$. $ Ǳ [ E 50 (+ 0 . 5F .$ M Ω = Ω ∪ Ω ∪ Ω − Ω M x= A .8 Ω :

≤ x + y ≤ , − ≤ z ≤

Ω :

≤ x + y ≤ ,

Ω :

≤ x + y ≤ , − ≤ z ≤ −

Ω :

≤ z + y ≤ , − ≤ x ≤

≤z≤

).& ^Y S M x= *$ O ) % ;I* !VO 0 &6 . S 0 $. & ( ) % ;* !VO .$ *O&0 5F . ?0& +, ] W. #$ G#U . h 0 5 M *M

*( E& B G#U Q 0 5F E u3 *ZM =U

5 N# F = −−P,−−Q,−−R→! M x= $ % e0 M ( #n<\ 6 F Zg S . T .$ *( . $0 *O&0 #n<\ 6 R Q P v

>. T 0 . ∇ h +, % % −−−−−−−−−−→

∂ ∂ ∂ ∂ ∂ ∂ , , i+ j+ k ∇ := = ∂x ∂y ∂z ∂x ∂y ∂z

=

00 R(x, y, f (x, y)) − R(x, y, g(x, y)) dxdy D

00 (0

f (x,y)

= D

000

= (S)

g(x,y)

) ∂R dz dxdy ∂z

"0 #. S M x= F+G % O&0 Ω , + ).& ^Y ( n $. & ( ) &0 #n<\ 6 S 5.$ 0 F . $0 5 M x= *AS = ∂ΩC >. T R# .$ O&0 3 Ω 0 00

∂R dV ∂z

>. T R# .$ *F = −−P,−−Q,−−R→! ZM x=

00

00

00

F • n dσ = (S)

P.i • n dσ +

(S)

000 div(F) dV =

Ω

00

&# .$ *;**

a

00

,

a − x − y ; (x, y) ∈ D

.$ *(

000

(S)

F • n dσ =

Ω

= =

D

⎡ ⎤ 0 a+√a −x −y ⎣ dz ⎦ dA (x +y )/a

5 + y x dA a + a − x − y − a D 5 00 4 + r r dA a + a − r − a D 5 " #0 " #0 4 π a + r r dr dθ a + a − r − a

(π) − =− π 00 4

=

00

dV =

=

=

div(F) dV Ω

000

,

000

∂Q dV + ∂y

(S)

000 (S)

∂R dV ∂z

&# .$ *;** &a E H (+"1 $ !' A, .8 ∂ ∂ ∂ (z) + (y) + (x) = div(F) = ∂x ∂y ∂z

∂P dV, ∂x

=

(S)

(S)

00

000

Q.i • n dσ

∂ ∂ ∂ ( − y) + (x) + (z) = ∂x ∂y ∂z

(x +y ) ≤ z ≤ a+

(S)

000

P.i • n dσ

'() %

, + ).& ^Y S Ω :

P.k • n dσ

. #E & #&" Z ] W. #$ q1 >&[f &0 R# 0&0 9ZM >&[f

}

div(F) =

(S)

∂P dV + ∂x

00

P.j • n dσ +

(S)

000

;;* !VO &a E (+"1 $ !' A&

div(F) dV Ω

(S)

*$$S >&[f 0&6 >. T 0 #$ $. $

2

000

F • n dσ =

∂Q dV, ∂y

=

(S)

(S)

00

000

R.i • n dσ (S)

∂R dV. ∂z

= (S)

*ZM >&[f .

( =&M .$ * " Z [O M Z 0 S + S >. T 0 . S #. ZM x= . h R# 0 M Z"# 0 S : z = f (x, y) ; (x, y) ∈ D, _&0 0 . n , S : z = g(x, y) ; (x, y) ∈ D, R#&3 0 . n . Z#. $ X R# 0 *$ O ) ;;* !VO 0 n dσ

=

n dσ

=

−−−−−−−−−−−→ (z − f ) dxdy, dxdy = − fx , −fy , |(z − f )z | −−−−−−−−→ (g − z) dxdy = − gx , gy , − dxdy. |(g − z)z |

00

00 R.k • n dσ =

(S)

00 = D

(S )

00 R.k • n dσ +

.$

R.k • n dσ

(S )

−−−−−−−−−−−→ dxdy R(x, y, f (x, y)).k • − fx , −fy , 00

+ D

−− −−−−−−→ R(x, y, g(x, y)).k • gx , gy , − dxdy

.$ *Z#S . −z ([a ( ) 0 . . S 0 n . $0 , + S 5.$ ZM x= 5 M *n = k e1

00 F • n dσ

00 F • n dσ =

(S)

(S+S )

000 Ω

=

000 Ω

=

D

=

div(F) =

x +y

)

00 dA −

x dxdy

D

00

dxdy

r cos θ − r sin θ − r cos θ

r drdθ

D

0 (0 π

=

)

r cos θ− r sin θ−r cos θ dθ dr

π

0

− r

dr =

=

R# 0&0

00

(S)

000 =

'& ).& ^Y S F = xi+(x+y)j+(x+y +z)k A T _&0 .$ e1 x + y + z = M 0 $

*( z = −−−−−−−−−−→

0 $ '& ).& ^Y S F = x , x y, xz! A; *( z = z = >&T x + y =

Ω

Vol(Ω) = π × × =

π

div(F) dV Ω

! x + y + z dV = " #0

#0 π

=

=

π

Ω :

0 . /&S q1 7 & >&#+ .$ 0 % *M t\ O $ $ #. F . $0 5

dV

000

Ω

*8* &a E H (+"1 9;* !VO

div(F) dV =

5F `&. .$ %- ,&1 ('&" 0 0 Z' #E *O&0

x + y + z = z M ).& ^Y S M x= A .8 −−−−−−−→ *M [& S 0 . F *F = x , y, z! ( $ O x= S ( "0 #. N# S &# .$ *!' 0&0 .$ *S = ∂Ω &F Ω : x + y + z ≤ z Z#. $ /&S q1 F • n dσ =

}

000

Ω

=

y − y (x + y ) − x

=

∂ ∂ ∂ (x) + (x) + (y) = ∂x ∂y ∂z

(S)

F • k dxdy

D

=

π × = π

000 F • n dσ

00

y dz

Ω

Vol(Ω) = ×

00

F • n dσ

D

00 (0

00

F • n dσ

(S )

y dV −

dV

(S )

00

div(F) dV −

=

R# 0&0

000

&# .$ *;*H* &a E ; (+"1 $ !' A .8

00 F • n dσ −

Ω

=

R# 0&0 *O&0 Ω [B S + S U# *O&0

div(F) dV =

=

(S)

Ω : z = x + y ; (x, y) ∈ D

00

000

000 Ω

π/

(0 cos ϕ

dθ 0

π/

ρ ρ sin ϕ dV )

"

ρ sin ϕ dρ dϕ

cos ϕ sin ϕ dϕ = π

&# .$ M

≤ θ ≤ π , ≤ ϕ ≤ π/ , ≤ ρ ≤ cos ϕ

*( M >&j L .$ Ω # j z = x + y 5 S+ E +"1 S M . T .$ A .8 ( ) 0 . n ( $. D0 z = T y M O&0 −−−−−−−→ . S 0 F *F = z, y, x ! O&0 . −z

*M [&

M z = T E +"1 S N+M 0 . #. R# *!' "0 #. N# 0 ( O ) x + y = y ZM x= S ]3 *$ O ) ;* !VO 0 *ZM !#[ &F D : x + y ≤ S S

: z = x + y ; (x, y) ∈ D : z = ; (x, y) ∈ D

( ) n S M $ 5&6 >$ T R# .$ *r = r &F O&0 S 0 $. & 00 r π (, , ) ∈ Ω S n dσ = (, , ) ∈ Ω S r M x= F+G 3 4 . % G#U &q= E Ω '& 0 F #n<" #$ . $0 5

X ( "0 #. N# S M x= *X ∈ Ω O *$. $ m Ω E 50 0 . n O&0 5F E .$ Y\ E ( >.&[, F y S E . [, .&O00 y >. T00R# .$ M & . . \ R# ' (B&' * F n dσ ÷ dσ X .$ F # S &# ] W. #$ M ! X Y\ N 0 Ω 9Z &

•

(S)

•

(S)

#00 div(F) =

"

lim

Ω→{X }

(S)

F • n dσ

(S)

#00 ÷

" dσ

(S)

9$ + 5&0 5 #E >. T 0 . /&S q1 /& R+ 0 Z' ).& ^Y S S F+G % . $0 5 y S E . [, .&O 5 c &F O&0 Ω 0 0 Ω Z' 0 F 5 # S !M . \ &0 F #n<" #$ *( /MJ . !) F+G 2 L $ %

5F ).& ^Y S ( U0 Z") N# Ω M x= !VO 0 . Ω Z' >. T R# .$ O&0

00 Vol(Ω) =

(S)

−−−−−→! x, y, z • n dσ

$ &0 r 6\ N# & y S l&Q *$ + [& 5

>. T R# .$ *$$S GT D 00

Vol(Ω) =

r, ru , rv dA

D

9( &. $0 &S %- r, r , r OM M "#& 5& $ #$ >.&[U0 *r, r , r = r r × r ! * " r r r . $0 5F &Y w ( 5 .$ E !T&' ^Y ('&" A& '() % % . . −x ' a ≤ x ≤ b &0 y = f (x) ([a e0& . $ + *M [&

LB$ Y\ l&Q *Z &0 S . h $. #. ZM x= *!' ' θ E &0 . O $ $ e0& . $ + E X = (x , f (x ), ) $ O !T&' X = (x, y, z) ∈ S Y\ $ $ 5 .$ . −x . R# & X T&= 0 0 +. −x & X Y\ T&= &F $ R# o L −x ?U0 * x + y = f (x ) U# ( u

u

v

u

v

v

•

u

v

`&UO Ǳ [ .$ cM 0 M #_&0 + 0 $ Z' Ω A Rl+ ( Ω ).& ^Y S ( T−xy ; *F = −x−−−−−y,−y,−−z−−−−x→! y = x + z 5 + $ 0 $ Z' Ω A Rl+ ( Ω ).& ^Y S ( y = −x −z −−−−−−−−−−−−→ *F = xz, yx , −yz! T z = x + y Zg&1 pL 0 $ Z' Ω AH Rl+ ( Ω ).& ^Y S ( z = *F = −−−x−−−−y,−−z,−z−−−−x→! z = >&T x +y = 0 $ Z' Ω A8 ^Y S $. $ . 1 4$ Z 6 N# .$ M ( z = −−−−−−−→ *F = x , y, z! Rl+ ( Ω ).& ! z AJ 0 $ '& ).& ^Y S F = −x,−−−y,−−−→ *( z = z = >&T x + y = M ).& ^Y S F = (x +y +z)(xi+yj+zk) A7 *( x + y + z = /&S q1 N+M 0 S 0 F H & >&#+ .$ *M [&

M ).& ^Y S F = x i + yj + zk A *( x + y + z = T c &j L >&T 0 $ '& Ω AI ( Ω ).& ^Y S ( x + y + z = −−−−−−−−−−−−−−−→ *F = xy + z, y, x − y! Rl+ -

).& ^Y S F = xa + yb + zc −−,−−,−→! A *( xa + yb + zc = 5 Sq0 −−−−−−−−−−→

0 $ '& ).& ^Y S F = ax , by, cz! A; *( z = ±c y = ±b x = ±a >&T −−−−−−−−−−−−−−−−−−−−→ ).& ^Y S + F = yz + x , xz + y, xy + z! A y = − x + z pL $ 0 $

'& + *( y = x + z |x−y +z|+|y −z +x|+|z −x+y| = ).& ^Y S A ! y ( *F = −−x−−−−y−+−−z,−−y−−−−z−+−−x,−−z−−−−x−+−−→ S = ∂Ω ( U0 Z' N# Ω M x= AH r = (x, y, z) M x= Rl+ *(, , ) ∈ S

* * 00 * (a cos u + b) cos v (a cos u + b) sin v a sin u * * * * − a sin u cos v − a sin u sin v a cos u * dudv = ** − (a cos u + b) sin v (a cos u + b) cos v ** 0D0 ! ! a = a cos u + b b cos u + a dudv

*z/y = tan θ Rl+ *x = x R# 0&0 O&0 V# #&0 Y\ .$ y = f (x) cos θ, z = f (x) sin θ.

>. T 0 . S #. R# 0&0

D

0 0 ! a π π = ab cos u + (a + b ) cos u + ab du dv

= πa b

−−−−−−−−−−−−−−−−−→! r(x, θ) = x, f (x) cos θ, f (x) sin θ

5F .$ M $ + .&3 5

0 $ Z' A .8 (x/a)

/n

/n

+ (y/b)

/n

+ (z/c)

D : a ≤ x ≤ b,

=

*( $= $, n M M [& . E >.&[, #. R# 0 $ Z' *!' /n

/n

Ω : (x/a)

+ (y/b)

+ (z/c)

/n

≤

9( RQ ('&" 5&+B X R# 0

dσ

=&# Z+U M >&j L N+M 0 . S = ∂Ω #. *( x = a(sin u cos v) ZM x= U# ZM .&3 !M &0 6\ .$ *z = c cos u y = b(sin u sin v) E ( >.&[, S

* * * * *rx × rθ * dxdθ *−−−−−−−−−−−−−−−−−−−→! * * , f (x) cos θ, f (x) sin θ

= =

−−−−−−−−−−−−−−−−−−−→!* , −f (x) sin θ, f (x) cos θ ** dxdθ *−−−−−−−−−−−−−−−−−−−−−−−−−−−→!* * * * f (x)f (x), −f (x) cos θ, −f (x) sin θ * dxdθ , ! f (x) + f (x) dxdθ ×

n

n

n

=

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→! r(u, v) = a sinn u cosn v, b sinn u sinn v, a cosn u

*D : ≤ u ≤ π , ≤ v ≤ π E ( >.&[, 5F $ M 0 0 r, r , r R# 0&0 u

* * * * * * * * * * *

a sinn u × cosn v

b sinn u × sinn v

a cosn u

na cos u × sinn− u cosn v

nb cos u × sinn− u sinn v

−na sin u × cosn− u

n

−na sin u × sin v cosn− v

n abc

=

00 Vol(Ω) =

=

=

=

()

=

nb sin u × cos v sinn− v

sin u cos u sin v cos v

!n−

n abc

sin u

0 0 Ω Z' .$ ( r, ru , rv dudv

00 !n− sin u dudv sin u cos u sin v cos v D

0

π

| cos u|n− (sin u)n du

#0 π n−

#0

× n+

B

n n+ ,

&0 ( 0 0 h $. #. ('&" .$

00 Area(S) =

00

dσ = (S)

0 π (0

= =

,

f (x) D

,

b

f (x)

π

0

a b

,

f (x)

+

| sin(v)|

S :

dv

B

,n

"

!

f (x)

) dx dθ

! + f (x) dx

a

, x + y − b + z = a

!VO 0 . 5F M (

" n−

! + f (x) dxdθ

RU N+M 0 N#

+, -&#. .$ = R# [B *( O >&[f c .$ cM a `&UO 0 # $ 5 .$ E S % M x= A, *( O !T&' . −z ' T−xz .$ e1 (b..) *M [& . S Z' E >.&[, #. RQ B$&U *!'

n abc t−+n/ ( − t)(n−)/ dt × n+ #0 " −/ n− × s ( − s) dv

n abc

=

* * * * * * *= * * * *

D

n abc

× ()

n

v

≤ t ≤ π.

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→! r(u, v) = (a cos u + b) cos v, (a cos u + b) sin v, a sin u D :

≤ u ≤ π , ≤ v ≤ π

E ( >.&[, S Z' .$ *$ + .&3 5

00 Vol(Ω) =

(S)

−−−−−→! x, y, z • n dσ =

00

D

r, ru , rv dudv

&q= E , + #E Ω M x= & Ω 0 F . T .$ *$$S G#U 5F 0 F . $0 5 M Ω .$ e1 S S . ) ^Y $ Ǳ E&0 M ( !# ZO&0 O $ O&0 V# A. )00 N# 5 ,00 0C & F [B ^Y $& 5&0 0 * F n dσ = F n dσ *O&0 O $ "0 ^Y [B 0 & Ω .$ ( !VO .& , + Ω M x= & ) *;*H 0 !VO .& , + G#U &6 0C #n<" #$ Ω & 0 F = −−P,−−Q,−−R→! . $0 5 A$ O 9B$&U #E 4&V' >. T R# .$ *O&0

*( 6Q Ω 0 F AGB *( T S 0 F Ω .$ S "0 #. E 0 A% *Ω 0 div(F) = A: &F ω = P dy ∧ dz + Q dz ∧ dx + R dx ∧ dy S A$ 0C O&0 ! &M R# 0&0 ( t1$ Ω 0 ω 4= ; *A$ O ) ;*;*H 0 ! &M t1$ 4= G#U $ ) 5&Q G = −−f,−g,−−h→! #n<" #$ . $0 5 A E 0 M U R# 0 *F = Curl(G) M $. $ ∂h ∂g ∂h − − P = (x, y, z) ∈ Ω Q = ∂f ∂z ∂x ∂y ∂z ∂f ∂g − *R = ∂x ∂y O $ O&0 6Q Ω 0 F l&Q & *Z & Ω 0 F H . G &F F = Curl(G) ZO&0 !VO .& , + Ω M x= & O&0 6Q Ω 0 F = −−P,−−Q,−−R→! . $0 5 $ 0 R# .$ *F = Curl(G) U# O&0 F B G = −−f,−g,−−h→! >. T •

(S )

f (x, y, z) =

•

(S )

0 $ t z Q(tx, ty, tz)

% −y R(tx, ty, tz) dt + ϕ(x, y, z), g(x, y, z) =

0 $ t x R(tx, ty, tz) % −z P (tx, ty, tz) dt + ψ(x, y, z),

h(x, y, z) =

0 $ t y P (tx, ty, tz) % −x Q(tx, ty, tz) dt + η(x, y, z).

−−−−−−−−−−−−−−−−−−−−−−−→! H = ϕ(x, y, z), ψ(x, y, z), η(x, y, z)

5 N# 5F .$ M Z#. $ #$ 5&0 0C ( LB$ ! &M #n<" #$ . $0 *ACurl(H) = 0

( O x= AC .$ M R# ^- *( 9( #$S $& & 0 e0& E A;C .$ s = cos (v) *B(a, b) := 1 t ( − t) dt #E &#. E N# 0 $ Z' 0 & % 9M [& /&S q1 N+M 0 .

t = cos u

a−

b−

) x + y + z = R

)

(x + y + z ) = x + y − z

)

(x + y + z ) = x + y

x + y + z = R , < a ≤ z ≤ b ≤ R

x−x y−x z−z ) + + = a b c x / y / z / + + = ) a b c

)

)

/ i=

ai xi

/ / + ai xi + ai xi = i=

i=

*( T GB&L 5& $ &0 × ]#& A = [a ] M ij

. C

−−−−−−−→! : r(t) = x(t), y(t) ; a ≤ t ≤ b

"0 A7 *Z# . S #. 0 $ $ 5 .$ . −x ' ^Y ('&" M $ 5&6 * < a < b M x= A . −x ' xa + yb = q0 5 .$ E !T&' + 0 0 e 5F .$ M ( π( − e + e arcsin e) 0 0 + *( O x= a a − b 0 M ^Y E O ) ^Y ('&" M $ 5&6 AI pL Z y x + y + z = ay B$&U

+ aπ *( +( +A)( 0 0 y = Ax + By + B) 0 M ^Y E O ) ^Y ('&" M $ 5&6 A q0 5 S+ y x + y + z = ay B$&U

aπ *( √AB 0 0 y = Ax + By

PT 3F' 27 & F 0 ([" y M #&\0 . $0 & &+ G#U 5 . 6Q & (O "0 " 0 $. $ "0 ^Y [B 0 & & F .$ ^Y M $ + 5F 0

=

0 t x(y t − xzt ) − z(z t + x t ) dt

−−−−−−−−−→

) t t t t − x z − z − zx xy

( =

xy

=

−

x z

−

z

0 h − η(x, y, z) = t y P (tx, ty, tz) − x Q(tx, ty, tz) dt =

0 t y(z t + x t ) − x(x t + zt) dt ) t t t t + yx −x − xz yz

( =

yz

=

+

zx

+

z

+

yx

−

−

y

yz

+

+

x

−

xz

f − ϕ(x, y, z) =

E ( >.&[, G X R# 0

xyz

yx

−

i+

x

xy

−

xz

−

x y

−

z

j

k+H

−−−−−−−−−−−−−−−−−−−−−→! . $0 5 N# H = −−ϕ(x, y, z), ψ(x, y, z), η(x, y, z) M *( LB$ ! &M #n<" #$ ( x + y + z = M E +"1 S M x= A .8 0 . S 0 n ( ) . $0 $. $ . 1 z = T _&0 .$ M −−−−−→! S 0 . F = z, y, x . $0 5 *$. $ Ǳ [ E 50 *M [&

( 6Q Ω = R 0 F ]3 div(F) = 5 Q *!' M S #. 0 F &0 S 0 F . \ R# 0&0 5&"V# O&0 S [B 0 0 A. ) N# 5 , 0C 5F [B S #. [B 5 Q *( S #. Z R#0&0 ( C : x + y = , z = ) O $ $ M y M Z#0 z = T E +"1 . *ZM .& Z n = k . 5F 0 ( ) ?U0 ( #$S T−xy 0 S # j D : x + y ≤ S X R# 0 &F O&0 00

00

F • n dσ (S)

00

F • n dσ =

= (S )

00

=

R# .$ *F = xy, z, xy ! M x= A& '() $ & 6Q F . $0 5 R#0&0 div(F) = y = >. T *(" −→ M x= A, .8 >. T R# .$ *F = −(−−/y,−−−−/z,−−−/x) *( &j L >&T `&+ ) U M DF = R − U #E Ω ⊂ DF ZM x= *(" !VO .& , + R# .$ *$S . 1 DF .$ &z &+ M O&0 !VO .& , +

0 *( 6Q Ω 0 F R#0&0 div(F) = >. T R# *ZM $& *J* E G = −−f,−g,−−h→! B 5$.F (0 Z#. $ >. T R# .$

−−−−−→! z, y, x • k dxdy

D

x dxdy =

0 t =

$ y y y % − dt = t − t = − t tx x x 0 g − ψ(x, y, z) = t x R(tx, ty, tz) − z P (tx, ty, tz) dt z z z − dt = t − t = − t ty y y 0 t y P (tx, ty, tz) − x Q(tx, ty, tz) dt h − η(x, y, z) = 0 t =

=

0 t

t

−

$ x x % dt = t − t = zt z

−

x z

E >.&[, G X R# 0 −−−−−−−−−−−−−−−−−−−−−→ G = ( − y/x , − z/y , − x/z) + H −−−−−−−−−−−−−−−−−−−−−→! 5 N# H = −−ϕ(x, y, z), ψ(x, y, z), η(x, y, z) M ( *( LB$ ! &M #n<" #$ . $0 −−−−−−−−−−−−−−−−−−−−→

*F = z + x, x + z, y − xz! M x= A .8 ( !VO .& , + M Ω = R >. T R# .$ *( 6Q Ω 0 F R#0&0 *div(F) = x + − x = $& *J* E G = −−f,−g,−−h→! B 5$.F (0 0 9ZM

f − ϕ(x, y, z) =

0 t z Q(tx, ty, tz) − y R(tx, ty, tz) dt

0 = t z(x t + zt) − y(y t − xzt ) dt (

D

. $0 5 M $ 5&6 $. .$ 0 % & 9M oL6 . 5F B ]< ( 6Q Ω , + 0 F ! & F = −−y−−'−−x−+−−z,−−x−−−−y−+−−z,−−x−+−−y−+−−−→ z A *Ω : max |x|, |y|, |z| ≤

0 t z Q(tx, ty, tz) − y R(tx, ty, tz) dt

=

) t t t t zx +z −y + xyz

+

z

y xyz − + 0 g − ψ(x, y, z) = t x R(tx, ty, tz) − z P (tx, ty, tz) dt =

zx

. $[B &#. E #& + ;H* !VO .$ '() * O $ $ 5&6 & F [B 0 $. & ( ) + 0 . ) *( 5 N# F = −−P,−−Q,−−R→! M x= * . F xI &# .y >. T R# .$ *( #n<" #$ . $0 9ZM G#U #E !VO 0 i j k ∇ × F = ∂/∂x ∂/∂y ∂/∂z P Q R −−−−−−−−−−−−−−−−−−−−−→! Ry − Qz , Pz − Rx , Qx − Py

Curl(F) :=

=

f

! ) Curl a F + b G = a Curl(F) + b Curl(G) ! ) Curl f F = f Curl(F) + ∇f × F ! ) div Curl(F) = ! ) div F × G = G • Curl(F) − F • Curl(G)

. ) #. N# S M x= ;L=C $ * *O&0 $. & c C 0 ( ) ( C [B &0 . $[B 0 #n<\ 6 S ^Y 0 F . $0 5 M x= Rl+ >. T R# .$ O&0 3 C

00

F • dr = C

Curl(F) • n dσ (S)

>. T R# .$ *F = −−P,−−Q,−−R→! ZM x= @

@ F • dr

@

=

P dx +

C

@ Q dy +

C

C

R dz, C

00

00 Curl(F) • n dσ = (S)

*Ω :

(S)

≤ x,

≤ y,

F=

≤z −−−−−−−−−−−−−−−−−−−−−−→! y z − xz , x y, z − x z

*Ω : x + y + z ≤

#. N# [B 0 $. & ( ) 9;* !VO . ) . $[B #. N# S M x= * C .$ $. & ( ) E . h *( C [B n ( ) . $0 &0 . 1 w |Q (+ S . . + 5F @ M ( ) y O G#U ( ) &0 C ( ) $&U 5&0 0 &# $. $ !VO 0 *( .&SE& n &. $0 . E ( . ($ ,&1 *$ O ) ;* }

Curl(R.k) • n dσ.

. #E & #&" Z ]M q1 >&[f &0 R# 0&0 9ZM >&[f 00 P dx C

Curl(P.i) • n dσ,

= (S)

A

& (B&' 0 R#S q1 U[@ Z+U ]M q1 0 . &q= .$ 4$ ` y 5F N+M 0 ( #&q= >. T Z 0 *$ + !#[ 5 4$ ` ^Y Z & F 5&0 0 ?z#{ M Z#. $ E& #) G#U $ 0 5F *( $3 }

(S)

@

A;

−−−−−−−−−−−−−−−−−−−−−−−−→ F = −(x/y − x/z, y/z − y/x, z/x − z/y) A

00

Curl(Q.j) • n dσ +

+

*Ω = R

Curl(P.i) • n dσ (S)

00

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→! x − xy − xz, y − xy − yz, z − xz − yz

%NG D 47

. $0 & G F M x= * >. T R# .$ * \\' $ , b a v U0&

@

F=

;*7* &a 9;H* !VO

⎧ ⎨ x + y + z = : z= ⎩

C

: r(t) =

−−−−−−−−−−−−−→! cos t, sin t, ,

≤ t ≤ π

F • dr = C

0 π −−−−−−−−−−−−−→! −−−−−−−−−−−−−−−→! , cos t, sin t • − sin t, cos t, dt

=

0 π =

cos t dt = π

,

− x − y ; (x, y) ∈ D x + y ≤

: z=

D

:

&F Z#0 S mU e0& . f = x + y + z S R# 0&0 n =

=

−−−−−−−−→! x, y, z ±f = ±+ f x + y + z −

−−−−−−−− ,−−−−−−−−−−→ x y ± , , −x −y

(B&' O&0 . −z ([a ( ) 0 . n ( . 1 5 Q M ?U0 *Z##n3 . + x + y + z dxdy f dxdy = dσ = |fz | |z| dxdy = + − x − y i j k −−−−−→! Curl(F) = ∂/∂x ∂/∂y ∂/∂z = , , z x y 00

E ( >.&[, ]M = ( . (+ .$ Curl(F) • n dσ =

(S)

00 =

−−−−−→! , , •

−

−−−−−−−− ,−−−−−−−−−−→ x y , , −x −y

D

dxdy − x − y

+ 00 = D

00

= D ()

=

x+y+ +

+

00

R dz

Curl(R.k) • n dσ.

= (S)

*ZM >&[f . B ( =&M .$ * " Z [O M r : D → S >. T 0 . S #. ZM x= . h R# 0 0 0 r y D E # j r(u, v) = (x, y, z) M ZM .&3 Z#. $ X R# 0 *C = r(∂D) 9O&0 C @

@

P dx = P (r(u, v)) dx(u, v) ∂D @ ∂x ∂x = du + dv P. ∂u ∂v ∂D " 00 # ∂x ∂ ∂x ∂ () P. − P. = dudv ∂u ∂v ∂v ∂u D " 00 # ∂x ∂P ∂x ∂x ∂P ∂x . + P. − . − P. dudv = ∂u ∂v ∂u∂v ∂v ∂u ∂v∂u D " 00 # ∂P ∂x ∂P ∂x = . − . dudv ∂u ∂v ∂v ∂u D 00 # ∂P ∂x ∂P ∂y ∂P ∂z ∂x = . + . + . . ∂x ∂u ∂y ∂u ∂z ∂u ∂v D " ∂P ∂x ∂P ∂y ∂P ∂z ∂x − . + . + . . dudv ∂x ∂v ∂y ∂v ∂z ∂v ∂u 00 # ∂P ∂z ∂x ∂z ∂x () . . − . = ∂z ∂u ∂v ∂v ∂u D " ∂P ∂x ∂y ∂x ∂y . . − . dudv − ∂y ∂u ∂v ∂v ∂u " 00 # ∂P ∂(z, x) ∂P ∂(x, y) = . − . dudv ∂z ∂(u, v) ∂y ∂(u, v)

C

R# 0 *ZM [& . ]M = ( . (+ &' 9ZM .&3 . S 0 . h

S

(S)

@ C

Curl(Q.j) • n dσ,

=

C

&aa ( ) &0

&0 ( 0 0 ]M = |Q (+ X R# 0

@

Q dy

⎧ ⎨ x + y = : z= ⎩

&aa ( ) &0

00

@

R# 0&0 *$ O ) GB ;8* !VO

− x − y dxdy

− x − y

(x + y) dxdy + + − x − y

00 dxdy

D

+ Area(D) = π × = π

D

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(S)

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(S)

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z +

, z = x + y

.$ . %& & #$&\ B N T &. $0 *$ O G#U *0&0 (x, y, z) LB$ Y\ N# ∂z ∂x ∂y × × >.&[, . \ F (x, y, z) = x= &0 A ∂z ∂x ∂y *0&0 . ^ Y E + " 1 ( '& " [ & ( % Y AH *$. $ . 1 x + y = ! $ M z = x − y [ & 0 00(( % Y ) A8 * xa + yb + zc dxdydz q1 .$ F(x, y, z) = zi + x j + yk &S AJ x + y + z = M 0 _&0 E M Y 0 . ] W. #$ *M t\ ( . j z = x + y 5 + 0 R#&3 E >_$&U 0 C &S A7 R

C : x + y = a , z = x + y

. #E y ]M q1 N+M 0 O&0 O $ $ 9M [&

@

xdx + (x + y) dy + (x + y + z) dz C

] M q 1 N + M 0 . ( x + y + z = *M [&

Z#S h .$ . F = x i + yj + zk . $0 e0& A ).& q 0 ^ Y . 0 . e 0& R # ^ Y &# ] W. #$C /&S q1 N+M 0 x + y/ + z = *M [& AV$ S

\H ) ' x + yz = K Y 1? E C A $ 5& 6 *( O ! T& ' x + y + z =

Y\ E A = ( , , ) Y\ .$ C 0 /&+ y *$.nS B = (− , , ) e0& 3 A; 4 f (x, y) =

xy x +y

z=

0

t

sin(u ) du

y=

0

t

sin(u ) du

t≥

0

t

cos(u ) du

hB & t = hB E O @ (=&" ( % Y AGB *t = *0&0 t = Y\ .$ . = " Ǳ& `&UO A% >_$&U y M ( y x &v E U0& z A uC ( O G#U z = u + v x = u + v y = u + v [& ( % Y *A#0 h .$ y x E U0 . v ∂z ∂z * ∂y ∂x 000 , z x + y dxdydz [& ( % Y AH y = >&T x + y = x 0 D 5F .$ M *( O . j Z 6 N# .$ e1 z = a > z = pL ^Y E +"1 ('&" RU ( % Y A8 $ O ) x + y = ax y M z = x + y *Aa > M $ O x=C z = − x − y ( < z) ^ Y S M x = AJ . $ 0 5 0 . ] M q 1 .$ *( *M t\ ^Y R# F = −zi + xj + yk D

x

y

− x /a +y /b

R

( , , )

( , , )

(x, y) = (, ) (x, y) = (, )

*M .0 (, ) Y\ .$ . (M' #E 5 &1 &0 O $ $ " N# . 0 M A 9M

x=

O& 0 ! \ " & v y x S A [& ( % Y &F uu++vv++xx ++yy == *u v 00 ( ) R 5F .$ M e dxdy ( % Y AH *( x /a + y/b = B$&U 0 q0 ! $ '& x +y+z = a K Y R0+. j XT Z") Z' A8 *(a < b) 0&0 . z = x + y x + y + z = b 0 z = x + y pL E O ) ^Y ('&" AJ *#.F ($ 0 . z = z = >&T 0 [& . yzdx + xzdy + xydz y A7 *M 0 . ] W. #$ q 1 .$ A >& T 0 $ Z " ) :.& S ^ Y . $ 0 5 z = y = x = x + y + z = *M t\ F = (x + y) i + (y + z) j + (z + x) k

\P? ) ' Y\ E M "# 0 . T B$&U Aa x + y − z = − >_$& U 0 T $ e @& \ ! *$.n0 x − y + z = *M oL6 . xyz = ^Y ` Ab 90&0 . #E e0& !"& 3 e0& A;

(, − , )

yex + ez

. \ h(x, y) =

zey + ex j + xez + ey k f (x + y) + f (x − y) & S A *0&0 . ∂ h − ∂ h >.&[, i+

∂y

∂x

e 0& 3 4 , & # 3 .$ A ; (x, y) = (, ) (, ) Y \ .$ f (x, y) = ; (x, y) = (, ) *M d0 . j . $4) '& N# x . 0 ([" & .& 6S AH e0& 5$ 0 (0&f x= &0 . y + z = B$&U 0 q0 0 *0&0 (δ = ) B&Q R '& 0 . f (x, y, z) = xyz e0& &S A8 . x + z = x + y = & R0 . j

*M [&

−y x @ .$ . F = x + y i + x + y j e0& y AJ N+M 0 5 . = &#F *M [& &aa # $ Q $ + [& R#S q1 0 x (z − y) dx + y (x − z) dy + y A7 (, , ) / g. & 0 a a C 5F .$ M z (y − x) dz B$& U 0 T 0 e 1 (, , ) (, , ) xy x +y

C

( % Y *r(t) = e cos t, e sin t, e ! M x= A *X = r() Y\ .$ A5& 0C Ǳ& # $ %& Ǳ& S : z = x S : x + y = M x = A; oL6 R .$ . &VO R# E N# ` *C = S ∩ S *M Z >&j L & $ N# .$ . & F $ + 9#.F (0 . #E ]#& # &. $0 #$&\ A ⎛ ⎞ t

⎜ ⎜ ⎝

−

t

t

−

9M d0

⎧ sin(xy) ⎪ ⎨ x + y f (x, y) = ⎪ ⎩

R

C

⎟ ⎟ ⎠

\PH ) '

0 #E e0& 3 .$ A

e0& 3 A 4

(x, y) = (, ) ∂z ∂z +y =z x ∂x ∂y

#) &v X"' 0 . Y0 . AH 4 + 4 +#cM& #$&\ A8 *M "# E&0 v = xy w 6 !j= q0 . . f (x, y, z) = x + y + z e0& *0&0 x − z = T z = x + y pL

4 5&#&3 0 $ ^Y ('&" [& ( .% Y . A * e0. .$ e1 xa + yb = 000 N# .$ e1 Z' V 5F .$ M z dV . \ A; #E x + y + = z T _&0 >&j L Z 6 *M [& . ( z = T 5 Sq0 0 $ '& 0 . |xyz| e0& A *M [& xa + yb + zc = M E + " 1 4 ) [ & ( % Y A O ) x + y = x y M x + y + z = δ = x + y + z 0 0 (x, y, z) Y\ .$ B&Q ( *( F = . $ 0 5 0 . R # S q 1 AH (, ) / g. & 0 a a C (x − xy)i + (xy − y )j *M .0 (, ) ( , ) 5 0 . ] M q 1 A8 C F = (y + z)i + (z + x)j + (x + y)k 9M t\ ! u=x

V

cos t, sin t, cos t + sin t

; ≤ t ≤ π F = − xi + yj + zk r , r = x + y + z

•

(S)

(x, y) = (, )

C : r(t) =

^ Y ] W. #$ q 1 E $& 0 0 & 0 A7 F = xi + yj + yzk . $0 5 0 . F n dσ *M 3 |x|, |y|, |z| ≤ XUV ).& S ^Y B$&U 0 $& . + "0 '& [B C M x= A x y y . \ [& ( % Y *O&0 + = @ * −yxdx ++xydy

M O& 0 M . T .$ AJ '& 0 . ] W. #$ q1 5F .$ *M .0 ≤ x + y + z ≤ M $ 0 $

f (x, y) =

(x, y) = (, )

√ xy

x +y

(x, y) = (, )

*M .0 R !M 0 . >_$& U z = e cos t y = e sin t x = te S A; T v &. $0 &F O&0 M .{ (M' " .&3 t = hB .$ . 5F 0 Zg&1 T B$&U " κ Ǳ& *0&0 & #. 0 O . j Z " ) Z ' A *0&0 . z = x + y z = − x − y 0 ([" M 0&0 5&Q z = xy + ^Y 0 Y\ A

*O&0 T&= R# & M . $ >&j L Ǳ [

00 x−y cos dxdy [& ( % Y AH x+y (, ) (, ) (, ) ( , ) / g. &0 \E{ U 5F .$ M *( xOy T .$ e1 f (x, y, z) = xy + yz + zx e0& A. )C # t 6 A8 $ .$ xy + yz + zx = ^Y E LB$ Y\ .$ . *0&0 grad (f ) ^ Y [ & 0(0 % Y AJ x + y + z dσ E ! T& ' Z " ) ^ Y S M *( z = a z = >&T x + y = 1? . $ 0 5 0 . ] M q 1 .$ A7 (+"1 E C M M t\ . T .$ F = xyi + yzj + zxk x = >&T y x + y + z = T E O ) *O&0 z = y = t

t

t

U

(S)

\3F ) ' 4 5&

5$( [12] Dieudonne, J., Linear Algebra and Geometry,

[1] Adams, R. A., Calculus of Several Variables,

Hermann, Paris, 1969. [13] Douglass, S. A., Introduction to Mathemat-

Addison Wesley, Canada, 2000. [2] Adler,

ical Analysis, Addison Wesley, Massachusetts,

of

1996.

tial

Lectures

M.,

Several

Variables

Forms),

on

integration

(Using

Internet

Diﬀeren-

Version,

http://www.physics.nus.edu.sg/˜ [14] Gelbaum, B. R., and Olmsted, J. M. H., Theorem and Conterexamples in Mathematics, Springer Verlag, New York, 1990. [15] Garvan, F., The Maple Book, Chapman &

[3] Agarwal, D. C., Advanced Integral Calculus, Krishna Prakashan Media Ltd., India, 1997. [4] Apostel, T. M., Calculus, 2 vols., Blaisdell Pub., 1969. [5] Buck, R. C., Advanced Calculus, McGraw-Hill,

K. M., Maple V Learning Guide, Waterloo Maple Ins., 1988.

New York, 1978. [6] Udriste, C. and Balan, V., Analytic and Diﬀerential Geometry, Geometry Balkan Press,

[17] Hardy, G. H., A Course Pure Mathematics,

Bucharest, 2000.

Cambridge Univ. Press, Cambridge, 1948. [18] Ilyn, V. A. and Pozniak, E. G., Linear Al-

phy-

teoe/teaching/mm4

Hall/CRC, New York, 2002. [16] Heal, K. M, Hansen, M. L. and Rickard,

URL:

[7] Budak, B. M. and Fomin, S. V., Multiple Integrals, Field Theory and Seies, Mir Pub.,

gebra, Mir Pub., Moscow, 1986.

Moscow, 1978. [19] Ilyn, V. A. and Pozniak, E. G., Foundamen-

[8] Cain,

tal of Mathematical Analisis, 2 vols., Mir Pub.,

Herod,

J.,

Multivari-

//www.math.gatech.edu /˜ cain /notes /calculus.html, 1997.

[20] Israel, R. B. , The Maple Calculus, Addison

[9] Chen,

W.

,

Linear

Algebra,

URL:

http://www.maths.mq.edu.au/˜ wchen/In.html

[21] Kay, D. C., Tensor Calculus, Schaum’s Outline Series, McGraw-Hill, New York, 1988.

and

able Calculus, Internet Version, URL: http:

Moscow, 1982.

Wesley, New York, 1996.

G.

[10] Chen, W. , Multivariable and Vector Analysis,

[22] Klambauer, G., Aspects of Calculus, U.T.M.,

URL:

http://www.maths.mq.edu.au/˜

wchen/In.html

Springer Verlag, New York, 1986. [11] Davis, H. F. and Snider, A. D., Introduction to Vector Analysis, Wm. C. Brown, New Delhi,

[23] Knopp, K., Inﬁnite Sequences and Series,

1988.

Dover Pub., New York, 1956.

;8

[35] Olver, P. and Shakiban, C. Applied Mathematics, Lecture Notes, Preprint, 2000.

[24] Kurosh,

A.,

Higher Algebra,

Mir Pub.,

Moscow,

[36] Shakarchi, R., Problem and Solutions for Undergraduate Analysis, Springer Verlag, New York, 1998.

[25] Lang, S., A First Course in Calculus, U.T.M., Springer Verlag, New York, 1986. [26] Lang,

[37] Spigle, M. R., Vector Analysis, Schaum’s Outline Series, McGraw-Hill, New York, 1959.

S.,

Calculus of Several Variables,

U.T.M., Springer Verlag, New York, 1987. [27] Landesman, E. M. and Hestenes, M. R.,

[38] Yaglom, A. M. and Yaglom, I. M., Challenging Mathematical Problems, 2 vols., San Fran-

Linear Algebra for Mathematics, Science, and Engineering, Prentice-Hall, New Iersey, 1992. [28] Livesley, R. K., Mathematical Methods for

sisco, 1964.

Engineers, John Wiley & Sons, Canada, 1989.

G#O UT &6 $ */ & 6 O [29] Loomis, L. H. and Sternberg, S., Advanced *7I 5 P&Q E ![1 Calculus, Jnes and Bartlett Pub., Boston, 1990. G#O UT &6 $ */ & 6 O I [30] Maron, I. A., Problems in Calculus of One *7I 5 P&Q E ![1 Variable , Mir Pub., Moscow, 1988. *4 [31] Myskis, A. D., Introductory Mathematics for *7I 5 &0 R+ 0 Engineers, Mir Pub., Moscow, 1978. *4 ; [32] Nikolski, S. M., A Course of Mathematical *J 5 &0 R+ 0 Analisis, 2 vols., Mir Pub., Moscow, 1987. *P *% # $ [33] Piskunov, N., Diﬀerential Calculus and Inte*J V" gral Calculus, 2 vols., Mir Pub., Moscow, 1981. *P *% # $ [34] Pogorelov, A., Geometry, Mir Pub., Moscow, *8 5 [M .&# O c#3 +) 1987.

!"#$

% & '( )

!"#$

'( & %

'62 4= ;H t1$ ;H ! &M 4= ; ;8 t1$ ;8 ! &M

;J 4$ ` ` ^Y ;; &+B D. J &S 7; >&j L v 7J Y >&j L v 7H M >&j L v &+B D. JH Zh −xy , +

8 & .V &S$ J &S J; Ǳ& . $0 N# E J $ &[Y . $0 I $ EF Y ?\ ; &U ;7 ^Y N# 0 ( ) $ R0 # E 4 E @ \

T N# & ; # $& I "+ ; V# ; # . $0 $& . $0 ; &q= .$ y JI 0 /&+ V# . $0 ;I R#?M Y0 8 q0 H 5 Sq0

5 EF HI 4$ ).$ ` RU H 4$ t 6

8 . $0 e0& cB&F &. $0 Y ?\ HH q0 H8 + H8 B Bn 4 "M J S = ;I p6

v Q e0& U-

7 S = 4 "M B&" !' Z #. B &S$ V Zh −x , +

y ` ;I8 4$ ` H &S$ H Y >&j L v H [Y1 >&j L v H &+B D. Zh −x >&UY= 0 , + N# Z"\ D. Zh −x , + N# $' #&&O D. H >&j L v q1 Zh −y , +

8 & ^Y ;;7 ` ;88

# j T 0 y ; #$ . $0 0 . $0 N# # j ;I8 .&3 # U ;I8 5 $S0 ( ) ;I8 ( ) =&' v v ;J & = .$ I8 t 6 .$ >&j L v 7H M >&j L v 7; &S .$ H &S$ .$ Y >&j L v &S V ; &q= .$ y T 8 . $0 e0& N# B e0 ; U0& "0 e0 ;I T j % ) ;I8 ) ( ) ;H "# #. N# $. & ;H 5 $.W N# 0 $. & ([a

;I8 ( ) %qT&' 8 ).& 8 .&M$ J #& H & "#&

' 7 v Q e0& y ; T $ $. 0 E !T&' J \\' T .$ $& . $0 &S V ; &q= .$ ; x= Y\ $ E .nS T .$ y .&3 >_$&U

8H AZC .&3 8 U[@ .&3 HJ H 5$ + .&3 H7 q0 H + H B Bn H q0 H7 5 Sq0 H # $ H + H7 q0 5 S+ H7 B Bn 5 S+ HJ M H7 pL

H B Bn H7 Q.&3 $ 5 SB Bn H7 Q.&
V# &U

;H 4= N# !& 3 ;; ; . $0 5 !"& 3 ;H ul3 77 3 77 v Q e0& 3 %& J N# e0& J v −n J 8 #n
7 i& & \

. $0 e0& 8 RU 8 RU & 8 B e0 8; 3 ' 8 R .$ 8 t 6

78 M &# [Y1 >&j L 0 ' !#[ ;7 Y !#[ ;8 ]#& $&

;; z y E U0& . $ + . $0 $ R0 # E ;; ]#& #E I (O& N# 0 M W ;H , + !VO .& J (, ;; ^Y 7; E ^Y J + HH q0 5 S+ HH B Bn 5 S+ J %& O J Zg&1 J &+

J; Ǳ& `&UO 7 !VO T J '?T 7 1 & . $0 J 5& 0 7 .&M$ J Zg&1 . $0 B$&U

&M B$&U

J /&+

87 N# / 1 @ . $0 N# @ T&= ; p&\ R0 I y & Y\ ; " #$ 4= 7 1 &q= . $0 &q= U0 ; #&3 I t 6 E ,&1 JI T Zg&1 JI 0 4$ Zg&1

.&3 . $0 B$&U

&M B$&U

&q= .$ y ; $& . $0 ; &S V ; .&3 . $0 B$&U

J /&+ y 8H Z H # $ J; /&+ # $ 8 5& $ I R0 M W 7 , + N# 5.$ Y >_$&U & $ ;8 4= !" #$ ; v Q e0& N# !" #$ ;8 ] W. #$ ;; ]#& N# [. D. H &S$ N+M 0 &+B Zh −x >&UY= 0 , + N# Z"\ 7H S' [) Zh −x , + N# $' #&&O ; M (+O 4 S &+B D. ;; ` ^Y .$ &+B D. #. ;H "0 ;7 #n< ) ;7 . ) ;; #. ;;H .&3 [) V [) H 4$ ).$ ;; v e0& N# E ^Y ; . M ;;H 6\ ;; Zh Y\ ;; y x E U0& . $ + ;; z x E U0& . $ +

; Z' [& .$ /&S q1 $0.&M & &M 8 q0 7 B Bn ;H M H M $& & "M JI = kM 5&#$ S 7 S 7 S ]#&

;8 $& H %qT&' ;; [. Y ;8 5.&\

U0

H #n< VU

;7 Y !#[ u#&+ H &+ 4 +#cM&

J , + N# 0 e0& U-

, +

Zh −x JH Zh −xy Zh −y 7 E&0 J "0 7 $6= 7 . M ;; . [+ ;H !VO .& HI >&j L

7J Y H [Y1 H pL

J , + N# E

J; Ǳ& cM

; Y !\ "

t 6

q1 ;H ]M ;J ;H /&S q1 ;8 .&V 3 IH +- e0& ;; S_ =&# Z+U H &S$ .$ >&j L v H [Y1 >&j L v . ;H 5 $.W ;H & "#& 5$M Y1 ;H R#S ;I S_ ; R & 8H / 1 ` y $0.&M ;I !\f cM 4) [& .$ ;I uQ `&UO & .& 6S [& .$ ;I N# 0 e0& N# y [& .$ ;I Y\ N# 0 N# 0{&) [& .$ 4$ ` y $0.&M ;I N# $ .$ 5&#) [& .$ ;I7 .&M [& .$ &S$ $0.&M 8H U0 $ Z") N# 4) [& .$ 8 U0 4&") Z' [& .$ 87 & .& 6S [& .$ 88 !\f cM >&j L [& .$ 8I m&T ' ('&" [& .$ 8 yLB K Y ('&" .$ ` ^Y $0.&M #. N# uQ `&UO & .& 6S [& .$ .$ ; . ) ;H ^Y N# 0 e0& N# y [& .$ ;; ('&" [& .$ ; !\f cM .$ ; 4$ ` ^Y $0.&M &S $0.&M ; !\f cM 4) [& .$ XT 4&") Z' [& .$ H & .& 6S [& .$ B&V 5 N# y [& .$

I v $ e0& . $ + Z H #$ . $0 0 . $0 # j 8I #$ ǱO 0 ǱO N# # j I ]#& . $0 G#U J7 . $0 e0& G#U 7 v Q e0& G#U ;;; . $0 5 G#U 8I ǱO $ $. 0 RU 8I ǱO $ R0 # E RU 8I Z E ǱO $ T&= RU I S ' I Y >_$&U & $ !' H y !" #$ I Y [) . c= 4 E . H (+O 4 S D. H . $0 $ R0 # E R0 M W 8I T ;H $& E@ ;; ` y [&

;;; 4$ ` y [&

J; &S$ [&

7 &S [&

I " ' [&

t 6 [&

gc) t 6 [&

J7 . $0 e0& &# t 6 [&

+- t 6

I # . $0 . \

7 v Q e0 0&#. \

H Y\ ; #&\0 5

. $0 5

;H 6Q ;8 #n<\ 6

4 +

J , + N# 0 e0& U-

&" & O M ; aa

. $0 N# 4

I $ v Q e0& IH +- e0& I8 v v #c) I 0 M W I E ,&1 Y >_$&U

B$&U

T .$ y .&3 . $0 ; &q= .$ y .&3 . $0 T . $0 T .$ y .&3 ;I v m Y T .$ y &M T &M ; # . \

& \

77 e0& 77 !+, H @L eY\

H $. & 8H

7 E I [) I V [) ;I8 . ) H 4$ ).$ 8J &+B$ ;H 5 $.W I . M 8J >&V"+B J Y"

!<

;H #&OF H &. $0 0 &+, J7 . $0 e0 0 B +U &+, 8I . $0 N# E &0 ǱO N# &\ H . $0 H Y\ $ R0 !T . $0 I E K Y Z J7 Z I E & Z

H8 4$ ).$ &#. E& & Y\ ;; #. N# 0 Zh

Y\ RV 7 ["Q 7 .$ H #E J E

7 v $ e0& . $ + ;;8 / 0 . ; Y "0 ;8 # S 7 B Bn HH Q.&3 $ 5 SB Bn H Q.&
0

au du =

0

au + C, ln a

cos x = cos¾ x − sin¾ x

sin u du = − cos u + C,

0

sin x = sin x cos x,

0 tan u du = ln sec u + C,

tan x =

cot u du = ln sin u + C,

sin¾ x =

du = ln | sec u + tan u| + C, cos u

cos¾ x =

du = ln | csc u − cot u| + C, sin u

c = ,

0

0 0 0 0

− sin¾ x = cos¾ x − ,

=

cos u du = sin u + C,

0

+

du

+

du

a¾ − u¾

0

u¾ ∓ a¾

u+a

+

u¾ ∓ a¾ + C,

= ln u +

− cos x

,

,

,

+ cos x

(uv) = u v + v u,

u + C, a

= arcsin

− tan¾ x

(u + v) = u + v ,

u − a ln + C,

a

tan x

(cu) = cu ,

u du = arctan + C, a a u¾ + a¾ du = u¾ − a¾

cot x cot y ∓ , cot y ± cot x

cot (x ± y) =

u v

u v − v u , v¾

=

(nn ) = nu nn−½ ,

(sin u) = u cos u, + ¾ ¾ du u + u ∓ a + = − ln , (cos u) = −u sin u, a u ¾ ¾ u u ∓a 0 √ + tan¾ u , (tan u) = u dx ax + b

0

√

0

√

ax + b

=

,

ax + b =

(ax + b)

a

a

−a

f (x) dx =

0

+ C,

(au ) = (ln a) u au , (eu ) = u eu ,

a

f (x) dx,

¼

f (−x) = −f (x) ⇒

0

0 ¼

0

a

−a a

¾

cos x dx = π/¾

¼

¼

π/¾

¼

¾

sin x dx =

0

π/¾

¼

tan−½ u

a

,

sin¾n x dx

0

¼

π/¾

− u¾

sin¾n+½ x dx

× × · · · × (n) , × × · · · × (n + )

f (x) dx + C,

0

0

0

v dx + C,

0 u dv = uv −

0

u dx ±

un du =

,

0

(u ± v) dx =

0

v du + C,

un+½ + C, (n = − ) n+

' !G' *T

= − cos−½ u

u = − cot−½ u + u¾

=

0

0

u

= +

af (x) dx = a

× × · · · × (n − ) π . , × × · · · × (n)

cos¾n+½ x dx = =

sin−½ u

0

a

cos¾n x dx = =

0

f (x) dx = ,

0

u , u

(ln u) =

¿

f (−x) = f (x) ⇒

0

+ C,

a

,

sin¾ x + cos¾ x = , sin x , cos x cos x , cot x = sin x tan x =

sec x = csc x =

cos x

,

sin x

,

sin (x ± y) = sin x cos y ± cos x sin y,

du = ln |u| + C, u

cos (x ± y) = cos x cos y ∓ sin x sin y,

eu du = eu + C,

tan (x ± y) =

tan x ± tan y , ∓ tan x tan y

;

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