Интуиция и математика

  

  

  

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ËÌË ÍÎÏÐÏÑÏÐÒ ÓÔÑÕÏÖÏ×Ï ØÏÕÙ {1To2?2 563718GH :137?F18B@?7oI13@o a n

1, 1, 0, −2, −4, −4, 0, 8, 16, 16, 0, −32, . . . ,

G3@01?IIGH :1 :0B867G

Ú-.-.-Û

an+2 = 2an+1 − 2an

31 3@B0@18426 G37186;26 a = a = 1 . S1:G3@62 I?1hk1F62B `102G7B n91 57?IB. m?106; 0?AG00?I@I4k :137?F18B@?7oI13@?a IB3 8 FBII4a 212?I@ I? 6I@?0?3G?@ 6 24 :013@1 813:17oTG?23; 91@1842 0? _?:@12Ü n4a 57?I 6b?@3; 8 86F? a = x  5@1 :137? :1F3@B I18A6 8 Ú-.-.-Û FB?@ 0

1

n

n

xn+2 = 2xn+1 − 2xn ,

B :137? 31A0Bb?I6; IB x :0681F6@ A A8BF0B@I12G G0B8I?I6H x − 2x + 2 = 0. Ú-.-.=Û t@1 G0B8I?I6? IBF1 0?J6@o :17G56@o F8B A10I; x 6 x  6 :171x6@o a =c x +c x . 5@1 6 hGF?@ 1hb62 0?J?I6?2 Ú-.-.-Û. XIB5?I6; A1I3@BI@ c  c 219G@ h4@o 1:0?F?7?I4 6T IB5B7oI4k G37186a (a = a = 1) . {1@ 31h3@8?II1 6 83? ?376 918106@o 1 TBFB5? ABA @BA181a. >1 IB3 6I@?0?3G?@ F0G91?. ‡10I6 G0B8I?I6; Ú-.-.=Û 1ABT48BH@3; A12:7?A3I426 n 2

1

n

n 1 1

2

n 2 2

1

0

x1 = 1 + i,

6 `102G7B F7; a 8 37G5B? a

x2 = 1 − i,

2

1

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0

= a1 = 1 n

n

n

n

n

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2 3 ln (1 + x) = x − x + x − . . . , 2 3

3k1F;@3; 76Jo :06 < 1  5@1 1h›;3I;?@3; IB7656?2 G `GIA _6a 3@1;b6k 37?8B|x|131h4k @15?A 3 21FG7?2 |x| = 1 . >1 A@1 3ABx?@ :15?2G 0;F 1 = 1 − x + x − .... Ú-.-.UÛ 1+x @1x? 3k1F6@3; @17oA1 :06 |x| < 1W >6ABA6k 131h4k @15?A I?@ TIB2?IB@?7o 8 IG7o I? 1h0BbB?@3;... c 2

2

4

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:1@12 F01h?a. nB3J60?I6? 63k1FI191 1h›?A@B 2BI6:G7;_6a 0, 1, 2, 3, . . . ,

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ax = b

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é øêììýêï
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í îïî çïøêïôõéçç
é ìèêé  îóí ëîõïù
ôï  èëð øì øêïôóõû øï  êïõõéõìéîèìê  êïòòïí öèì ô îììêùóçïèçì   ïøóëó ì çïöïéè ìý
îçìôéççìé øìîììêùóçïèçìé ëõì éçóé ïêóïçèìô ûòçì éçóð ùôï  !¶­½°¸±´¯ ³¸´º¾Ã¯¿¯±º¯í $ %& "# êïôçìé  ôéùéçó  ùõóç ôéîèìêìô çï îìëóçûë û õï òé ùû çóòóí ó ï¶À µ´¸±´¯øêìó ³¸´º¾Ã¯¿¯±º¯í ùé ôéîèìê øì ùõóçé êïôéç ' ( ùé  ëèôóèéõüç
òó ) éëèü òïõéçüîïð çïèð îï  * ïòí ôììý  é  ìôìêðí ôçéùêðéèëð îïî ý
øìëèìêìççðð óùéð çéøêéê
ôçìëèó çéïõ éýêïóöéëîó  ôóêûë & êé  ûõüèïèé  ïùçóò öóëõìò ôì  çóîï  è ôëé  èó" ùìøìõçóèéõüç
é êïìôìê
ìý ïõ éýêïóöéëîó ó èêïçë+ éçùéçèç
öóëõï  -x = {x1 , x2 , x3 },

y = {y1 , y2 , y3 }

x + y = {x1 + y1 , x2 + y2 , x3 + y3 }. xy = |x| |y| cos α,

z

z = x × y,

z y ϕ

x

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é çé ôéîèìêí ï ëîïõðê  éîèìêçìé "#  ôéùéçóé ùï  éè ôéîèìêí çì çéè îìòòûèïèóôçìëèóí í ó ïëëì + óïèóôçìëèóí èóôç
  ïîìç   ïøêóòéêí . êïôùïí ô ìýìó ëõûöïð ëøêïôéùõóô ùóëèêóýû  |z| = |x| |y| sin α,

x y

z

xy

(a × b) × c 6= a × (b × c)

x × y 6= y × x

(x + y) × u = x × u + y × u.

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é öóëõï øìõé  çì ô õðçûèü ë ýìõéé ìý  é  èìöîó  êé  çóð çó#.+ûëèüí îïî ó øêé ùéí êéöü óùéè ìý õéòéçèï ë èì  õóúüô êïèìò é  í öèì êïôçì çé íï õ  ýìòû çïøéêéù  ïùïççìòû öóëõû öóëõéí ôì  òì çìí
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é öóëõï ó ùûïõüç
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é & )  èìíîïî  ìôìêðèí ùôé ýìõüúóé êï çó +

é öóëõï çï ìùðèëð" ô ïêôïèéêé òïèéòïèóîóí èì ùû  ëõó îìòøõéîëç  ïõüç
é ó ùôì  ç
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i2

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: óøéêîìòøõéîëç
é öóëõï îôïèéêçóìç
& í ìèîê
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é :ïòóõüèìçìòí " ðôõð, èëð öéè
êé òéêç
òó ìý ; éîèïòó ôóùï ùé í í í ) ùé  ëèôóèéõüç
é öóëõïí ï í í í ) öéè
êé ýï  óëç
éùóçó +
í ûùìôõéèôìêð  ó ëììèçìúéçóðò Z = α + iβ + jγ + kδ,

α β γ δ

1 i j k

ìèîûùï "ë ûöéèìò èì ìí öèì ) ìý
öçïð éùóçó + ï & ëõéùûéè i2 = j 2 = k 2 = ijk = −1, 1

ij = −ji = k,

jk = −kj = i,

æçì ùïöéòîôïèéêçóìç
ëöóèï  è ìýìý  éçóéò îìòøõéîëç
öóëéõí çì  èì çéøìõðýìõéé øìø
èîï ô
ùïèü éõïéòìé  ï ùé  ëèôóèéõüçìé  < ïëúóêéçóé
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ò óëöóëõéçóéò çï øéêô
øìêï çéêéùîì øêóôìùóè î òïëõû÷  ìêóó  * êûùç
é  éìòéèêóöéëîóé  ïùïöó çïöóçï  è êéúïèü  ëð ñîïî
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çé îìòòûèóêû  è @  ki = −jk = j.

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é ëîìêìëèó  ó ëîõïù
ôï èëðí ó êéûõüèóêû   ïð ëîìêìëèü ìîï
ôïéèëð êïôçì  

v = ω × r,

ω

ω1

v1 = ω1 × r

ω2

v2 = ω2 × r

öèì ó ùïéè çû ç
 ô
ôìù  <éûõüèóêû   éé ùôó éçóé øêìóë ìùóè ë ôéî  èìêçì  û õìôì  ëîìêìëèü   v = ω1 × r + ω2 × r = (ω1 + ω2 ) × r, ω = ω1 + ω2

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P Q RS 3TU3TV W QTR3S 3Y 1Z[ V W 3U Y Z3TV W Q] Z[2 Q^ X X\ X X P X _W T3`a QTR3S 3Y 1ZW 1 U3V 3 [2 2 Q QUV 1 W] 1V WTb cbdb ef^ \ X X X X \ X g W 1 UQTQV 1S hZ[ 1 U UQY _34 RQX 1 3UX \ Y Z3TV 14 R 1X 1T1UQi V Tj S 1Y QV ZQ 3_Z34 ` V 3kUQ2 a a b l 3UQ] QV ha kV 3 a a A B C R jm 34 b A B C X nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nnn nnn nnn nnn nnn nnn nnn nnn nnn nnn nnn nnn nnn nnn nnn nnn nnn nnn nnn nnn nnn nnn nnn nn nn nn nn nn nn nn nn nnnn

nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn

C

B A

¸º®Æ A Æ Æ o´Ä¶º

A, B, C

½¯p ­µ ±­ ´¿±´ 9 ³¸°»´9

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ó ýéïéò ëóèûï+ ó  èóøï-èì  í îìèìêïð ó  ìýêï éçï çï êóë  qrsqtu uvw  < ïûòééèëðí òì çì ý
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êïëëîï ïèü ì ôëé ôïêóï+ óð í çì ò
ý
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¸º®Æ A ÆA Æ Ç¸°»­° ¶´¸´Ä¯ ½´»­±´ 9

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2 1

n n 2 n

2 1

2 n

(x1 − λy1 )2 + . . . + (xn − λyn )2 > 0.

E137? 0B3A04@6; 3A1h1A 62??2

λ2 (y12 + . . . + yn2 ) − 2λ(x1 y1 + . . . + xn yn ) + (x21 + . . . + x2n ) > 0

-K

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ôìùóèí øìëîìõüîû ô  èìò «  óîèìí ëõûöïé ñ  ìõìôçïð ýìõü÷ øéêéçìëóèëð ô ùêû ìé ) ôì çóîïéè çéìý ìùóòìëèü îïî èì óçïöé ùìîï
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1 1

2 1

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2 n

2 1

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< ï  ûòééèëðí óëèóççïð øêóêìùï òïèéòïèóöéëîó ïîèìô çé ôëé ùï  éì  òéèêóöéëîïð èìý
+ óïõüç øìçðèüí
ëîïûêïôçéçó

éòí øìöéòû ùõð ûëèì  öóôìëèó õóçé  çì  ëóëèéò
ùó  ƒéêéç  , , „& " èêéýûéèëð çéìèêó + ïèéõüçìëèü ùé  ëèôóèéõüç
öïëèé  ëìýëèôéçç
 çïöé  çóøóëó "èìöéî & òïèêó +
í éëèéëèôéççì ìýêïèóèüëð çïøêðòû  î  ï  êéúéçóðëøéîèêï , , „& " x˙ = Ax

A

ìçéöçìí ùõð èì ì èêéýûéèëð çéîïð ëèïùóð øêéùôïêóèéõüçì ì ìýûöé çóð ó øêóô
îïçóð î ëøé + óïõüç
ò îïèéìêóðò ò
úõéçóð   èìò ìèçìúé çóó éìòéèêóöéëîïð óçèéêøêéèï+ óð óòééè èì øêéóòû  éëèôìí öèì ìçï ùïçï îïî ý
ïøêóìêóí ó éìòéèêóöéëîó  ïîèí îïî øêïôóõìí õéîì ìý ; ðëçóèü ùï é çìôóöîû 

x(t) = c1 eλ1 t + . . . + cn eλn t .

ÞÌ … ã×ã  âã ÑÔÐÐàáÎÔÔ „:0?F?7?II4a 6I@?0?3 8 FBII12 A1I@?A3@? :0?F3@B87;?@ 81 :013 1 A1765?3@8? :0B867oI4k 2I19190BII6A18. S1ABx?2 :0?F8B06@?7oI1 `102G7G ta7?0B V − R + G = 2, Ú=.D.CÛ 9F? V – 56371 8?0J6I R – 0?h?0 G – 90BI?a 7Hh191 0B3 32B@068B?2191 2I19190BII6ABÝ.

å ì  òì çìí çéøêïôóõüçì ì 

¸º®Æ A Ɔ Æ =-

{1To2?2 s7B3@65IGH 21F?7o 2I19190BII6AB 0B3@;I?2 1F IG 6T ?91 90BI?a 6 IB I?? G71x62 13@B7oI4?. { 0?TG7o@B @? :17G56@3; :713A6a 90B` – `69G0B @6:B 6T1h0Bx?II1a IB 063. D.<. { :013@?aJ?2 37G5B? 2I191G917oI6AB Úh?T 8IG@0?I I6k 8?0J6I IB 90B`?ې V = R = n  G = 2  0B8?I3@81 Ú=.D.CÛ 8?0I1. SB7?? 0?JB?@ 6IFGA_6;. Ž376 Ú=.D.CÛ 8?0I1 F7; 7H h191 2I19190BII6AB 3 G 90BI;26 @1 F1hB87?I6? ?b? 1FI1a 90BI6 Ú063. D.UÛ FB?@ I184a 2I19190BII6A 3 563712 8?0J6I  – 0?h?0 R = R + k + 1  – 90BI?a G = G + 1  9F? V = V +k – 56371 I184k 8?0J6I. E1s@12G k 0

0

0

V 0 − R0 + G0 = V − R + G = 2,

5@1 TB8?0JB?@ F1ABTB@?7o3@81.

¸º®Æ A Ƈ Æ „h0B@623; @?:?0o A 37G5BH :0B867oI191 2I19190BII6AB A1@104a kB0BA@?06TG?@3; @?2 5@1 8 ABxF1a 8?0J6I? 3k1F6@3; 1FI1 6 @1 x? 56371 K 0?h?0 6 1FI1 6 @1 x? 56371 N 0?h?0 190BI6568B?@ ABxFGH 90BIo. „5?86FI1 :13A17oAG :?0?35?@ 0?h?0 2I19190BII6AB ABA 3 :121boH 6F?6  @BA 6 3 :121boH – KV  FB?@ 2R  6h1 8 @12 6 F0G912 NG 37G5B? ABxF1? 0?h01 356@B?@3; F8BxF4. mBA62 1h0BT12 I?1h k1F621   N · G = 2R,

K · V = 2R,

2 2 −1+ K N

==

R = 2.

{3? 81T21xI4? 0?J?I6; 7?9A1 1:0?F?7;H@3; :0;242 :?0?h1 012ä A1@104a FB?@Ü @?@0BsF0 , K = 3, N = 3 − AGh, K = 3, N = 4 − F1F?ABsF0, K = 3, N = 5 − 1A@BsF0, K = 4, N = 3 − 6A13BsF0. K = 5, N = 3 −

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 ëôéè çï  ïùïöû øêìõóôïéè ëìôëéò óçì  øìù ìù  . êïôóõüç
 òçì  ì  êïççóî ïêïîèéêó  ûéèëð é  é ó èéòí öèì çéîìèìê
é é  ì ùôó éçóðí øìôìêìè
ô øêìëèêïçëèôéí øêóôìùðè î ëïòìëìôòé  éçó   . êóöéò òçì é  ëèôì.èïîó øìôìêìèìô ëìëèïôõðéè  êûøøû  ìôìêìè ô )  èì ôëéùï ôêïìýìéçóé ôìîêû  çéîìèìêì  ìëó   ëõó êéöü óùéè ì  êûøøé øìôìêìèìôí èì çïöïéè øìëõéùìôïèéõüçìé ô
øìõçéçóé øìôìêìèï í øìèìò   ïî ùìîï  ïõ é  é  õéêí  èì ëçìôï øìôìêìè  ƒ èìý
çé çïêûúïèü ýïõïçë ïî + éçèìôí çé ýûùéò ô ìùóèü ô ùéèïõó  ˆ ï  õéé ë  éè êï  ôóôïéèëð øêóòéêçì øì èïîì  ë éòé  ( êéùó  êûøø øìôìêìèìô éëèéëèôéçç
ò ìýêï  ìò ô
ùéõð  è çéîóé  êûøø
ëóòòéèêó  í çïõóöóé îìèì  êëèïôóèëð
ôõéöéè  ï ëìýì  ëû  éëèôìôïçóé øêïôóõüç
òçì  ì  êïççóîìô  = ïèéò ôìøêìë  îïîóé ô ëóòòéèêóó êïëëòïèêóôïéòì ì èóøï ôì  òì  çøêé
 æçéé ìîïêéúéçóé 
ôïéèëð ôëé  ì øðèü  êûøøï èéèêï ùêïíîûýï & ) ó øìõûöïéèëð "  ëèôïÿ . éêéýìê ëôìùóèëð î ìøêéùéõéçó  + éõìöóëõéçç
êéúéçó  çéêïôéç  øêó*ìöéôóùçìò ì êïçóöéçóó í  ïî ó ë õ  ùüòó ý
ôïéè  ˆ ìîûòéçè
øêéù ; ðôóõíýóì  êï ó  êïëëîï  ïõí öèì  ï öéõìôéî ) çé ðëçì  =D R3

a◦b

a

b

R3

2 2 + >1 K N K N >2

( õéùûð øêóòéêû ìêìúé  ì ùïçèóëèïí îìèìê
 ôëé ôêéòð ìý ; ðëçðéèí öèì ùéõïéè )  ïòéèóòí ëõéùû   éé   ïúï + éõü ) çéîó  ìý  ó  ô õðùí ùõð îìèìêì ì òïèéòïèóöéëîóé øìùêìýçìëèó çéëû  éëèôéçç
 ‰ îï ïçç
 øûèü ìøêéùéõéçóð ó øìóëîï êûøø ëóòòéèêó  ìèç  ùü çé ûë
øïç êì  ïòó  - ç ùõóççéé øêéù
ùû  é  ì ó ûèìòóèéõüçéé  . ì  èìòû ô ùïççìò ëõûöïé çéè çóîïîì ì êéìçï éòû ëõéùìôïèü  Š ôìè ôçéúçóé ïèêóýûè
èì ì øûèó  ï ëõû óôï è ôçóòïçóðí øìëîìõüîû óòééèëð ìùóç êìòïùç
 øõ  ëí îìèì ê
 øéêéôéúóôïéè ôëé òóçûë
 - øóëïçç
 ëøìëìý êéúéçóð øìòé  ïéè  ï ùïöû ô èéìêéèóîìêûøøìôû  ëêéùû êìùëèôéçç
ïîèìô  = ïùïöï û é çé ëèìóè ìëìýçðîìò  ‰ çéé ìýçïêû óôï èëð òçì ìöóëõéçç
é ó øõìùìèôìê  ç
é ëôð ó  : êûøøï óîìëïùêïí çïøêóòéêí ìîï
ôïéèëð èéëçì ëôð  ïççì  ë øêìýõéòì  êïêéúóòìëèó ïõ éýêïóöéëîó ûêïôçéçó  ô êïùóîïõï  :ïõûï ìýçïêû óõí öèì èï êûøøï øêìëèï "çé óòééè ëìýëèôéçç
çìêòïõüç
øìù êûøø & í ) ó  èì øêóôìùóè î çéêïêéúóòìëèó ûêïôçéçó  ëèéøéçó ó ô
úé  5

[P(QN  !M! QQN 6  ‚  òéêçìò øêìëèêïçëèôé óòééèëð ‹ øêïôóõüç
òçì  ì  êïççóîìô  # øêìëèêïçëèôï ýìõüúé  êï  òéêçìëèó ) èìõüîì $  ®º»³½¯¶®  òéêç
 " ùêï & èéèêï ùê & í îûý ó òçì  ì  êïççóî ¿Ã´ 9 ®µÃ¯±±² 9 îûýû ïçïõì  ìîèï ,  æîìëï ùê ó ùìùéîï ùê èïî é îïî ìîèï ùê ó îûý" & ùôì  ëèôéçç
ùêû  ùêû  û ) + éçèê
øðèóû  ìõüç"
 êïçé  ùìùéîï ùêï ðôõð  èëð ôéêúóçïòó óîìëï ï + éçèê
èêéû ìõüç
êïçé óîìëïùêï ) ôéêúóçïòó ùìùé îïùêï$ ùêïí  *éèêï ùê ùôì  ëèôéç ëïò ëéýé  òé ïçóîé éëèü èéìêéòï ì èìòí öèì êéûõüèóêû  ó  òìòéçè ëóõí ìèçìëóèéõüçì í êïôéç êéûõüèóêû   éòû òìòéçèû øõ  ë òìòéç  èû êéûõüèóêû èìöîó   é  ëóõ
n

L0 = r1 × F1 + . . . + rn × Fn ,

0

LC

ìèçìëóèéõüçì èìöîó "óòé  é êïùóûëôéîèìê & í è é  , $ %& "  èì çé ûõìôóõ ùéèïõé  ) çé ëûèü ôï çìí øìëîìõüîû êéöü ì ùêû  ìò   ï èéêêóèìêóó ôéîèìêçì  ì óëöóëõéçóð ìêòûõï , $ %& ûëèïçïôõóôïéèëð " ô ìùçû ëèêìöîû ˆ é ëèôóèéõüçìí ñˆý
ìôéîèìêçìé÷ ùìîï  ïèéõüëèôì ) óòéõì ùõóçû ýìõéé ëèï ëèêïçó + Œ æ ýéùï õï çé ô èìòí öèì øóëïçóç
òçì ì  > éùï ý
õï ô èìòí öèì  ï ùéêé ôüðòó. êóòéê õéëï çéóõõôóùçì  ëé èûòïççìí êïëøõ
ôöïèìí çéìýì  êóòì   ëèêóêûéè ìý  û   ïîìçìòéêçìëèü   ï ù
 òïèéòïèó  öéëîó  ïîè îïî ý
øêóøóëïç î ìøêéùéõéççìòû øõïëèû  * ïò ôóùçï é  ì øêóêìùïí éì ëôð  ó óí ëïòìé õïôçìéí èïò ìç ùìëèûøéç ùõð óçèûóèóôçì ì =< F = F1 + . . . + Fn

C

rC

L0 = LC + rC × F.

LC = (r1 − rC ) × F1 + . . . + (rn − rC ) × Fn = L0 − rC × F.

ôìëøêóðèóðôóùé   ï öû ìò øìõé ìç èì é óòééè ì ùéçóéí çì ) ô òïëîéí ô ûëéöéççìò  æòéççì èïîóé òûõð ó øêìó  ôìùðè ôøéöïèõéçóé õûýó  ç
éí øìêì

ùïð ëòûèç
é øìùì  êéçóð ì ëû  éëèôìôïçóó èïóçëèôéçç
 èï  = çïçóð  ˆ ìîìøïèüëð ùì óëèóç
í ô  èó ëõûöïð í ì  çïöïéè øêìýóèü èûççéõü ô çéìýóèïéòû  = ìçûí ùé ó ûöïéò
 ïîè øìøïùïéè ô éëèéëèôéççû  ùõð çé ì ëêéùû

=U

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

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èì çéôéëéõþ è ùçì øêó  çïèüëð ) òìöûëü ôì ëçé  ìùó î óøçìèó éêû ) ìç øìòì éè éêé çéùéõ  èõóöçì ô
õðùóúü  : óøçì  øìòì  þ ìçéöçì  ìöûëü øìøêé çéòû  ì èéøéêü ð èóò ìê  ûëü ? i>Ž‡S„m { 3?2IBF_B@12 8?A? n?I? S?AB0@ 1@A047 s``?A@68I4a 2? @1F 0?J?I6; 9?12?@065?3A6k TBFB5 3 :121boH B79?h0B65?3A1 91 1:63BI6; 8 A110F6IB@Bk A1@104? @?:?0o IBT48BH@3; F? AB0@18426ª. ^ABx?2 :713A13@o 8 R 1:6348B?@3; G0B8I?I6?2 86FB D.|.-Û ax = a x + a x + a x = α, Ú íîïî øêïôóõìí ó ýéïéòó ûèìöçðèü óëèìêóöéëîóé ùéèïõóí øìëîìõüîû «?ìé
ëóõüçì òçì øéêéøûèïõìëüí çé ìèéõìëü ý
ûôéõóöóôïèü ïìë  ùïç  ëõûöïéí çïøêóòéêí óùé îììêùóçïèçì ì ìøóëïçóð òì çì ûëòìèêéèü ûçìò

é û ùêéôçó é  óøèðç  ( óëèéòïèóöéëîó ìçï çïöïõï êï  ôóôïèüëð .  ’ éê  òï ó < ˆ éîïêèìò

óõó èïî é æ   ü  èìçí : D é  ýçó + óí çïôéêçðîïí òçì  óé ùêû< ûîûóéíøêóõì îïçûôúóé ô D éèû ) ) ) ƒ ) )

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2

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ax = α,

bx = β,

3

2

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cx = γ

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√ r = n − 1,

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2

3

4

x(τ ) = {τ, τ 2 , τ 3 , τ 4 },

τ > 0.

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ϕ(τ ) = (τ − t)2 (τ − s)2 = α0 + α1 τ + α2 τ 2 + α3 τ 3 + α4 τ 4 ,

6 8 AB5?3@8? :713A13@6 H 81To2?2 2I1x?3@81 @15?A x  F7; A1@104k hx + α = 0, 9F? h = {α , α , α , α }  0

1

2

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4

hx = h1 x1 + h2 x2 + h3 x3 + h4 x4 .

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ϕ(t) = ϕ(s) = 0.

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p

Dp

TB2?I4 z = px . E1s@12G ϕ  p z = p ϕ(z)  1@AGFB q

ϕ

q

  p p = ϕ(1) , q q

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i i

i

i

DK

n

96k 563?7. >B:062?0 3 6 √2 76I?aI1 I?TB863624 :13A17oAG √ λ1 3 + λ2 2 = 0

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λ

λ

λ

λ

λ

λ

λ

λ

Ì

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¯  

3

2

3

eβ

ϕ(4) =?

<|

ϕ(4) = 0

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i

i

i

ϕ(λ) = λθ ,

i

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 k i

i

T i

i

ϕ(λ)

λ

<-

T i

6 37Gx6@ 0?h012 Ú676 5B3@oH 0?h0BÛ I184k 2I19190BII6 A18 @1

M

l{ϕ(λ1 ) + . . . + ϕ(λs )} = lϕ(λ1 + . . . + λs ) = lϕ(2π) = 0,

?376 x? l 7?x6@ IB 90BI6 M  @1 1:;@o :17G5B?@3; IG7o

l{ϕ(λ1 ) + . . . + ϕ(λs )} = lϕ(λ1 + . . . + λs ) = lϕ(π) = 0.

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T(x) x

F(x)

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<<

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Ý «*ìöçéé  ìôìêðí  øìõç
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èìöîï í ó ) ô óêêï + óìçïõüç
 NP

f (x) = 1

x

f (x) = 2


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zk3 − 1 3zk2

0

A

B

C

ΓA = ΓB = ΓC ,


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6T1h0Bx?II12 :GIA@6012 :06 1@1h0Bx?I66 6T 1@0?TAB [0, 1] f  84:BFB?@ 30?FI;; @0?@o 1 , 2 6 13@BH@3; F8? A0BaI6? Ú8 TBJ@06k18BII1a 1h7B3@6ې 3 3   1 6  2 , 1 . 0, ÚU.D.<Û 3 3 Š?9A1 86F?@o 5@1 :06 :18@10I1a 6@?0B_66 f 6T 1@0?TA18 U.D.<Û Ú 1:;@o 84:BFG@ 30?FI6? @0?@6 6 @. F. ˜@1 :17G5B?@3; 8 :0?F?7?W { :0?F?7? :17G5B?@3; TIB2?I6@1? ABI@10181 2I1x?3@81. S0G9626 3718B26 I65?91 I18191. ˜@1 x? @19FB FB?@ :1FA1 8B ^2?a7BW „ABT48B?@3; 15?Io 2I191. E1AB ABI@10181 2I1x?3@81 3@016713o 563@1 G21T06@?7o I42 84h0B348BI6?2 @0?@?a 6T 1@0?TAB B :1@12 6T 13@B8J6k 3; 5B3@?a – 0?TG7o@B@ :0?F3@B87;73; 21I3@012 ABTG63@6A6 62?Hb62 3G9Gh1 BABF?265?3A6a 6I@?0?3. m?:?0o x? 84;3I; ?@3; 5@1 ™21I3@0š x68?@ 0;F12 TB 3@?I1a. XBG0;FIB; `GIA _6; f s7?2?I@B0I1 13Gb?3@87;?@ :01_?FG0G ™84h0B348BI6; @0?@?aš. [P(QN  !M! QQN 6  - ý  ïð óùéìõì  óð êï  êéúóòìëèó ûêïôçéçó  ôîõ  öï  ïð èéìêé  " çï  éìòéèêóöéëîó òû >#êïöçì êïû êï îïêèóçé ô ôóùé òïõéçüîì  ì êï  òéçèï & ýï  óêûéèëð øêì  . ûëèü ) ì  êïçóöéççïð ìýõïëèü øõìëîìëèó óõó øêìëèêïçëèôï & ë  êï  " çóøìôéê + é  í ó øûëèü çéøêéê
ôçìé ìèìýêï éçóé øéêéôìùóè ô  ïòîçûèû  çìëèü   ëõó èìöîï õé óè ôçûèêó í èì ûêïôçéçóé óòééè êéúéçóé ô "øìèìòûí öèì ìýêï ìýð  ïç çïîê
ôïèü ôçûèêéç  çìëèü & . êìôéêóèüíõé óè õó ôçûèêó í òì çìí çïøêóòéêí ø
èïðëü ëçðèü ëùéêçûèü & ë ñ  ôì  ùóîï÷ êóë  ‹ $& ˆ êû  óòó ëõìôïòóí ùé ìêòóêûð " øìôéê çìëèü ó çé îïëïðëü "øêó èìò çûõðí ûôéëèó  ’ ìêòïõó øìùïõüúé ìè íèìï òìøìèìòí ëîï

éòí ë ïèü éé ô îïîû  èì èìöîû  ìôïèü

çì èïî  çé õé óè ôçûèêó í éëõó çéøêéê
ôçïð ùé ìêòï+ óð  ìòìèìøóð & èïîìôïí öèì " ó Ω

Γ

f

f (Γ)

0

Γ

f (Γ)

f (x) = 0

Ω

f (Ω)

f (Γ)

0

f (Γ) f (Γ)

f (Γ)

0

0

0 h(x, τ )

h(x, 0) ≡ f (x),

f (Γ)

h(x, 1) ≡ w 6= 0,

h(x, τ ) 6= 0


f (Γ) w 6= 0

x∈Γ

f()

f()

O

O

¸º®Æ Á ÆA Æ Ã ¿¯ ñ·µ¸¯±±´®µ 7 Ä øêó í í öèì çï
ôï è ±¯Ã²¸´p ¿¯±±´®µ 7 ¼ ¿¯Ê´¸»­À ɺº õ ýý
ó  êéûõüèïè èéøéêü  ôûöóè èïî  ·¸­Ã±¯±º¯ Ň Æ Æ Æ ¸­¾¸¯ Ç º»´ à ¯®½º- ý ±¯çìëèü ®· B ¯®µÃ·¯µ ±¯Ã²¸´ p ¿¯±±´ 9 ¿¯Ê´¸»­ÉººÆ èûè øìîûøïéèëðí îìçéöçìí ùìêì ì  ñ é ëû  éëèôûéè÷ ó ñçé ôóçìôïè÷ ùìîï 
ôïèü ìùóçïîìôì èêûùçìí ï ñøêé  ûòø + óð çéô
êì ùéççì  ëèó÷í ðëçìé
òóùéõìí çé ìùóèëð  . ì  èìòû çï øêïîèóîé øìõü  û  èëð êï  õóöç
 òóëõóöïëèç ëõûöïðòó  *éìêéòï > êïû  êï ) ìùóç ó  çó  ˆ êû  ì  ôïêóïçè  í ï øìèìòûûùïéèëð çéôô
èìöîû êì ùéççì øêìùé

ìêòóêìôïèü ô í èì )  ôëé ô øìêðùîé öèì øìèìò çé ùé ìêòóêûéèëð & ˆ é ìêòï + óð òì" éè ý
èüí çïøêóòéêí õóçé  çì  x ∈ Γ τ ∈ [0, 1]

Ω

0∈Ω

f (Γ)

Γ

Γ

w 6= 0

õð éé çéô
êì ùéççìëèóí ìöéôóùçìí çéìý ìùóòï ó ùìëèïèìöçï çéøêìèóôì øìõì çïð çïøêïôõéççìëèü ôéîèìêìô ó çï  . êìëèìèï ìøóëïççì  îïêèóç
ô ìøêéùéõéççì  ëèéøéçó ìýòïçöóôï  . êó  òéêçì èïî éí îïî ëóèûï+ óð ô èéìêéòé  ìêùïçï "õïôï ‚ & ‰ëóõóð øì çïôéùéçó  øìêðùîï ùéëü ûôìùðè ôéëüòï ùïõéîì ìè çïõðùçìëèóí ó èìøì õì  óð ó  ñòóêï îïêèóçìî÷ øéêé ìùóè ô ïýëèêïîèçû  çóúûí ùé û öéõìôéîï çéè ìø
èï ó ì  û  éçó   èó ûëõìôóð øêì êïöç
éí ìèð ó çéëèêì óéí òìùéõó ìëìýéççì øìõéç
 ,  *éçùéç + óð øéêéêïýìèîó óëîûëëèôéçç
øìëèêìéçó  ô éëèéëèôéçç
é ùìôìõüçì ëóõüçïí ó ìçï øêìðôõðéèëð çï ôëé ûêìôçð  ’ ûçî + óð éëõó óêêï+ óìçïõüçì éëõó óêêï+ óìçïõüçì øìçïöïõû øêéùëèïôõðéèëð óëîûëëèôéççì   *éò çé òéçééí øêó ùé   é ëöóèïð öûôëèôï ùóëîìò ìêèï 
ˆ

h(x, τ ) = (1 − τ )x + τ f (x). x

f (x) =

(

f (x)

Γ

0,

x

,

1,

x

,

fk (x) → f (x)

k → ∞,

fk (x) = lim (cos k! πx)2n . n→∞

. ìõûöïéèëðí îìçéöçìí èï é ëïòïð ûçî + óðí çì ë ûîï  ïçóéò çï ôì  òì  çìé øêìóë ì ùéçóéí öèì ïêïîèéêó ûéè ëìôëéò ùêû ì  øìù ìù   ïî ô òéùó  + óçé  < é  ûõüèïè ýìõé  çó óîëóêûéè øïèìõì  ìïçïèìòí ó  ûöïéè øêóöóç
ó õéöóè ) + éõóèéõü  òïèéòïèóîé ôì  òì ç
èïîóé é çïøêïôõéçóð ùé ðèéõüçìëèó 

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a1 , a2 , . . . , an , . . . ,

ÚC.-.UÛ

b = 0, β1 , β2 . . . ,

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2

x = 0, α1 α2 . . . ,

y = 0, β1 β2 . . .

Ža 21xI1 31:13@B86@o @15AG 1@0?TAB

z = 0, α1 β1 α2 β2 . . .

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ôï è ùóï ìçïõüç
ò  U< b

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- èêé  ìî öóëéõí ô îï

ùìò óîëóêûéò øì ìùçìòû  õéòéçèû ïîëóìòï ó  îìèìê
ìýêïûéò òçì éëèôì í ó êïëëòìèêóò òçì "éëèôïå ô
ýìêï & í D é  îì ôóùéèüí öèì øéêé ìùóè ô ôøêïôì ó ùêûøêóì ëùôó  é ìùçì  öïëèó  õéòéçèìô çï êïëëèìðçóé öïëèó ) çï êïëëèì  ðçóé ôõéôì  ˆ êû  óòó ëõìôïòóí òì

çì êï  êé  ïèü çï ùôé öïëèó óí øéêéùôó ïçïõì ïð èéíóöçìÿ îïî èôéêù
é èéõïí ìýêï  ìôïèü ëõûöï " ®·»»¯ ¿­¼µ êïëëòïèêóôïéèëð &  ïîìçé + ìöéôóùçìí öèì à  ï®7´µ¸¯¾´¶ ì õ  ýïð øìùøìëõéùìôïèéõüçìëèü ô ëûòòé èì é ùïéèîïîôéëüèôéêùìèêé
  ìî øìëõé øêéùôïêóèéõüçì  øéêéëèêì  îó òçì éëèô èéõ   .
ì èìòûí éëõó ëéòé  ëèôì í èì ó êïîïýóèì ó  çïöïõüçì çï òóõõóìç ýéë îìçéöç ëìôìîûøçìëèé

ùì  èïîì  ëìôìîûøçìëèó òì çì ëõì óèü ìèêéóìî îûëîìôí ï øìèìò òóõõóìç ìèêé îìô øìëèïôóèü êðùìòí ìýêï  ìôïô < ï  ûòééèëðí  èì øìîï õóúü ®Ä¯µ±­° ¸­Ã±´®´®µ­Ã½¯±±´®µ 7 îûýìô çéìùó  " çïîìô
êï òéêìô & í øêéùëèïôõð  ïð ëìýì  øéêô
 úï ô çïøêïôõéçóó êé ûõüèïèï > ïçï ï Ï *ïêëîì ìí îìèìê
 ô çïöïõé øêìúõì ì ôéîï ìëû  é ëèôóõ  ïûëùìê  ˆ ïõéé òì çì øìîï ïèüí öèì ô öóëõì ñîûëîìô÷ òì çì ëùéõïèü îìçéöç
òí çì ùõð  èì ì èêéýûéèëð öûèü ýìõüúé èéêøéçóð  x

y ∈ [0, 1]

π

{x − y} = {kπ}

k Zα π

[0, 1] Zα

zα

Z

Z k = {Z + kπ}.

Zk

Z p (p > k) (p − k) π Zk

Zk 1 − (p − k) π

Zp Zk

p
[0, 1]

Z γk

Z γk

[0, 1]

[0, 1]

[0, 106 ]

{Z k } {Z γk }

[0, 1]

R3

í  ( ìëèìð  óé ó  ôëéôì  òì ç
 õéòéçèìô ëìéùóçóèüí ìýêï  ìôïô ìîêû çìëèüí èì ó ÿåýûùéè  ëõó îìç +
ìèêé  îï øìõûöïèüëð

éëèîóò øìôìêìèìò   < ï  êé  ïçóé ó ùôó éçóé öïëèé  øéêéôìùóè ô  êéûõüèïèé ëì ôìîûøçìëèü òçì éëèô øéêé ìùóè ô øìõçû  ëìôìîûøçìëèü UC [0, 1]

{z + kπ} z ∈ Z

Zp

Zk

Z γk

{Z γk }

Zk

{Z k }

ž Ÿ  Ÿ

Ð

“ ¦ £ • ’    ’ • H £  • ”£  ‘ Ñ

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Uc

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− 6

6

2 2

2 2.

B

− 5

5

5 1

1 1.

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Up

m?:?0o 81To2?2 ?b? F8? A13@6Ü C

− 4

4 4

4

0 0.

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− 3

3 3

3

3 3.

3

A > B,

B > C,

C > D.

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1 10 89 10 > , > . 90 10 10 90

UK

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2·

1 1 1 + 4 · + . . . + 2n · n + . . . = 1 + 1 + . . . , 2 4 2

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−100

200

100

−100

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1 . 4

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P (A) = P (B) = P (C) =

P (AB) = P (AC) = P (BC) =

P (AC) = P (A)P (C), P (BC) = P (B)P (C)

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Ln (x) =

x1 + . . . + xn . n

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n

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C=

n?A7B2I4a 8B06BI@ G@8?0xF?I6; 219 h4 T8G5B@o @BAÜ

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n

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x1 + . . . + xn n



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ëìîïð èìöçìëèü ìýéëøéöóôïéèëð û é øêó øìêðùîï çéëîìõüîó ùéëðèîìô  

x1 . . . xn = γ

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2

2

z = 2x4 − 3ax3 + a2 x2

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4

2

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z = (y − x2 )(y − 2x2 ),

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kx2 − 2mgx + const. 2

wT U (x) = 0 30BTG :17G5B?@3; kx = 2mg . 0

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ÌË  ÎÔÓÔ  Ò ÐàáãÐÏÎ • Ï O S6IB265?3A6? `1AG34 5B3@1 1:0?F?7;?@ 3@B@6AB. Ž376 IB:06 2?0 363@?2B 1:6348B?@3; F6``?0?I_6B7oI42 G0B8I?I6?2 Ú8 @12 5637? 8?A@10I42Û K.-.-Û x˙ = F (x, ε), Ú @1 F6IB26AB TB8636@ 1@ @191 5@1 :0163k1F6@ 3 0B8I18?36?2 @. ?. 3 0?J?I6?2 G0B8I?I6; ß

F (x, ε) = 0,

:06 6T2?I?I66 :B0B2?@0B ε . { :1:G7;0I1a 36@GB_66 A19FB F (x, ε) :0?F3@B87;?@ 31h1a 90BF6?I@ I?A1@101a `GIA_66 ϕ  0B8I18?36?2 37GxB@ A06@65? 3A6? @15A6 :1@?I_6B7B ϕ  8 A1@104k 90BF6?I@ ∇F (x, ε) 1hIG 7;?@3;. E06 2B712 81T2Gb?I66 εx `GIA_66 ϕ(x) = x  @1 ?3@o :06 :?0?k1F? A x + εx  A06@65?3AB; @15AB = 0 Ú@15AB :?0?96 hBÛ 811hb? :01:BFB?@ :06 ε > 0  B F7;x 1@06_B@?7oI4k 8 ε 1A0?3@I13@6 x = 0 :1;87;H@3; F8? I184? A06@65?3A6? @15A6. iIB71965I42 1h0BT12 8 0?TG7o@B@? 81T2Gb?I6; x + εx :06 3A17o G91FI1 2B712x :1 21FG7H ε < 0 FB?@ 8 1A0?3@I13@6 82?3@1 1FI191 – F8B 26I62G2B 6 1F6I 2BA362G2. t@1 x=0 :062?04 I?9B@68I191 kB0BA@?0B. E062?0 F0G9191 01FB 3@0GA@G0I1 G3@1a5681a A06@65?3A1a @15A6 FB?@ ϕ(x) = x . gB71? 81T2Gb?I6? x 21x?@ 76Jo I?TIB56@?7oI1 32?3@6@o A06@65?3AGH @15AG I1 1IB 13@B?@3; ?F6I3@8?II1a 6 31k0BI;?@ 381a @6: Ú26I62G2BÛ. { 37G5B? h17oJ?91 5637B :B0B2?@018 81TI6ABH@ 36@GB_66 I18191 @6:B6TG5?I6?2 A1@104k TBI62B?@3; @?106; AB@B3@01`. ^ABx?2 26I62G2 :1 x `GIA_66 1 1 ϕ (x) = x + λ x + λ x ÚK.-.=Û 4 2 1:0?F?7;?@3; G37186?2 3

3

4

4

2

λ

2

2

4

2

2

1

dϕλ = x3 + λ2 x + λ1 = 0. dx UQV QTV 3º [ Tf3 UW X X λ1 λ2

t@1 G0B8I?I6? @BA IBT48B?21a . E06 0BT 765I4k :B0B2?@0Bk  AGh65?3A1? G0B8I?I6? hGF?@ 62?@o c=

676 1F6I F?a3@86@?7oI4a A10?Io676 @06. E17G5B?@3; :18?0k I13@o 6T1h0Bx?IIB; IB 063. -|.-. g1xI1 76 s@G :18?0kI13@o 356@B@o 90B`6A12 I?A1@101a 2I191TIB5I1a `GIA_66 x(λ , λ )W {11hb? 91810; 21xI1 I1 IB 3B212 F?7? TF?3o :1F0BTG2?8B ?@3; I?5@1 h17oJ??. >B:062?0 ?376 363@?2B 8 IB5B7oI4a 21 2?I@ IBk1F6@3; 8 @15A? A  6 :B0B2?@0 λ :7B8I1 G8?76568B ?@3; @1 313@1;I6? 363@?24 2?I;?@3; 8F17o ACB – 8 @15A? :0163k1F6@ 3AB51A. E06 81T80Bb?I66 8 63k1FI1? :171 C x?I6? 313@1;I6? 363@?24 hGF?@ 2?I;@o3; λ:1 F0G912G :G@6 – I?5@1 801F? :?@76 963@?0?T63BÛ. { @1 x? 80?2; :06 3:? BEF Ú _6B7oI1a 0?9G76018A? :B0B2?@018 λ  λ 81T21x?I :7B8I4a :?0?k1F 6T A 8 B 8F17o ADB . >BA1I?_ TBJ@06k18BIIB; 5B3@o :18?0kI13@6 311@8?@3@8G?@ I?0?B76TG?242 ÚI?G3@1a56842 2? @B3@Bh67oI42Û 313@1;I6;2 363@?24. ‡Gh65?3A1? G0B8I?I6? TBFB?@ ABI1I65?3A6a 86F AB@B3@01 `4 3h10A6 62?Hb?a 8 @15A? S kB0BA@?0IGH 131h?II13@o. ‡ @BA12G 86FG 3 :121boH I?76I?aI4k TB2?I A110F6IB@ 381F;@ 3; 0BT765I4? F0G96? G0B8I?I6;. E1 :?0812G 8:?5B@7?I6H 0BT 765I4k @6:18 131h?II13@?a 21x?@ h4@o F1817oI1 2I191. „F IBA1 1ABT48B?@3; 5@1 8 @6:65I4k TVX \ UV \X Z3 \ TV 34kW`[2  37G5B;k :06 IB76566 F8Gk :B0B2?@018 219G@ 83@0?5B@o3; 76Jo F8B @6:B 131h?II13@?aÜAB@B3@01`B 3h10A6 6 AB@B3@01`B 3A7BF A6. E06 @0?k 6 h17?? :B0B2?@0Bk 0BT765I4k 131h?II13@?a I?3A17o A1 h17oJ? I1 6 @B2 FB?@3; 6k :17IB; A7B336`6AB_6; – 8 TIB 56@?7oI1a 3@?:?I6 h7B91FB0; i0I17oFG. àQm 1kQZW 1b ýêïèóò ôçóòïçóéí öèì òóçóòûò ûçî + óó øêóôì ùóõëð ô îïöéëèôé -øêóòéêï ëèêûîèûêçì çéûëèì  öóôì  ìëìýéççìëèó ó  ï ôì  òû  éçóð  "á # ,& îôïùêïèóöç
é öõéç
"ó õóçé ç
é & ñõéïõó ì ôïç
÷í øì èìòû ôì  òû  éçóé òéçðéè èìõüîì øïêïòéèê ûçî + óó  ,& í îìèìêïð "ô ìèõóöóé ìè ñìùóçìîì ì÷ & ëèêûîèûêçì ûëèì  öóôïí çì èì"á #û é çé øêìëèïð èéìêéòï  ß ÌÞ “ áÎ  âá  Î  ãÙ  ÑáÏ – ÓÔÖÏÑá ² 1

2

2

2

1

2

x4

εx2

εx2

x4

n1F1IB5B7oI6A12 @?1066 AB@B3@01` 356@B?@3; n?Is m12 :06 FG2B8J6a s21_61IB7oI1? IBT8BI6? – IB G018I?:1xB7Ga 369 IB7B âã â . „FIBA1 91810;@ ABA IBT18?Jo @BA 6 :1?F?Jo. { FBII12 37G5B? @BA 6 :17G56713o. ™E01IT6@?7oI1?š 62; IB cD

x λ1 λ2

ääB ää S Eää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää äää äää äää äää äää äää äää äää äää äää äää äää äää äää äää äää äää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää Cää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ä ä ä ä ä ä ä ä ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ää ääFää ää ää ää ää ää ää A

¸º®Æ å Æ Æ æ ­µ­®µ¸´Ê­ ®Å ´¸¶º ABA12@1 s@B:? 1h?3:?5671 @?1066 I?1h4AI18?IIGH :1:G7;0 I13@o 5@1 3 1FI1a 3@101I4 k101J1. ^ F0G91a – 1h0BT18B7B3o @BAB; ABJB 5@1 I? 0BTh?0?Jo 9F? 2B@?2B@6AB B 9F? :6B0. „31h1 8BxIGH 017o 8 @?1066 AB@B3@01` 690B?@ 3@0GA@G0 IB; G3@1a56813@o. S?71 8 @12 5@1 0BT765I4k 86F18 3T3f1ZZ3^ TV 14 :06I_6:6B7oI1 21x?@ h4@o 3A17oA1 G91FI1 – 6 :0?F2?@ 6TG5?I6; 0B3:17TB?@3;. { @1 x? 80?2; ;3I1 5@1 I?@ 32437B 6TG5B@o @? AB@B3@01`4 A1@104? 635?TBH@ :06 2B7?aJ?2 81T 2Gb?I66 363@?24. t@1 6 37Gx6@ 0BTG2I42 13I18BI6?2 F7; 71AB76TB_66 8I62BI6; IB 90Gh4k Ú3@0GA@G0I1 G3@1a5684kÛ 36@GB_6;k. V1810;@ 5@1 @?106; AB@B3@01` :1;867B3o 376JA12 :1TFI1. „IBF? TBI62B?@3; 81:013B26 A1@10426 h471 h4 ?3@?3@8?II1 TBFB@o3; ™TBF1791 F1š. >B:062?0.. . E18?F?I6? `GIA_66 f (x) 3AB7;0I1a :?0?2?II1a x 21xI1 c<

6TG5B@o 6337?FG; ?? 0;F msa710B a0 + a1 x + . . . + ak xk + . . . ,

9F? a = f (0) . „h45I1 k!190BI6568BH@3; 0B3321@0?I6?2 I?A1@10191 1@0?T AB s@191 0;FB A1@104a FB?@ F13@B@15I1 k101J?? :06h76x?I6? . ^ABx?2 ?376 f (0) = 0  @1 1 kB0BA@?0? A06@65?3A1a @15 f (x) A6 x = 0 21xI1 3GF6@o :1 :?0812G I?IG7?812G 57?IG 0;FB msa710B Ú:06 a > 0 – 26I62G2 :06 a < 0 – 2BA362G2 6 @BA FB7??Û. 601A1 0B3:013@0BI?I 26` 5@1 F7; `GIA_6a 2I196k :? 0?2?II4k F?71 1h3@16@ BIB71965I42 1h0BT12. >B 3B212 F?7? s@1 I? @BA 6 0BT›;3I?I6? 36@GB_66 – 1FIB 6T TBFB5 @?1066 AB@B3@01`. ƒGIA_6;ª (k)

k

0

2

2

f (x, y) = xy 2 + x101 + ϕ(x, y),

9F? 0BT71x?I6? 8 0;F msa710B ϕ(x, y) IB56IB?@3; 3 57?I18 :1 0;FAB I? 2?I?? -|=911ABT48B?@3; sA868B7?I@I1a 8h76T6 IG7; 2I19157?IG xy + x  I1 I? xy . >?@0686B7oI13@o :01h7?24 :1F5?0A68B?@ 37?FGHb6a TB 9BF15I4a :062?0. nB3321@062 `GIA_66 2

101

2

f (x, y) = 3x2 + 2x3 − 6xy 2 ,

g(x, y) = 3x2 + 2x3 − 6xy 2 + ϕ(x, y),

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cU

y

x

f(y)

Ì

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¸º®Æ å Æ  Ï  ÖÏ   ãÙ ÓãÑá ² 㠖 Ñ ž àÎ×ã

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ó  îçó  ó ñ*éìêóð îïèïëèêì ó éé øêóõì éçóð÷í . ìëèìçí ( è  ïêèí Eç  #á cC

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84@?AB?@ >6 2I191 I6 2B71 – h71A3k?2B :0?F 3ABT48B?@ hGFGb??. E1I;@I1 5@1h4 GF?0xB@o3; 8 0B2ABk F1T 817?II191 IBF1 :?0?k1F6@o IB h17?? @1IA1? 1:63BI6? F6IB26 A6 1:;@o G56@48B; 876;I6? :B0BT6@I4k :B0B2?@018 6 3B21? 97B8I1? – TB:G3A :01_?33B. y

x

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x mg

mg

¸º®Æ å Ɔ Æ Ç­¸­¿´¶® ǯ±½¯Ã¯ 8?3@6 3?h; 0?B7oIB; 363@?2B. E06 6T2?I?I66 ε 8F0G9 IB56IB?@ 269B@o ™369IB7oIB; 7B2:15ABš. t@1 21x?@ h4@o 5@1 G91FI1Ü 1 1ZWj ] Q U3Z1kZ31 ` 1m j 2 3_j V ` f1TU3Z1kZ3TV h ­WS W X \ • X R W2 3_j V W]® a X m 3_1S h Z QkWZ Q1V R 1_TUQ][ `QV h f _ g 11a X \ \ • ­UQU31 QV h ®a 1 1ZWj `1V `j V Tj `[ fW • X X Ô Q`Z1ZW1 R 1 1TV Q1V 1 QV hTj ­Z1V 1 1ZW4® b • \X X X X

{ 7Hh12 37G5B? 3 @?006@1066 ÚK.-.-Û 0?B7oIB; 363@?2B 84 97;F6@ TB9BFA1a 6 :06k1F6@3; 63AB@o F0G96? 21F?76. {4k1 F6@o TB 0B2A6 63k1FI1a :13@BI18A6 :?0?13243768B@o TBFB5G . . .‡ 3718G 6T 21F?76 :0?F3ABT48BHb?a hGFGb?? h17??2?I?? ;3I1 5@1 80?2?II 4?/ TBF?0xA6 369IB718 8 _?:6 1h0B@I1a 38; T6 219G@ :0681F6@o A h17oJ62 I?F10BTG2?I6;2. Ž376 @BA6? TBF?0xA6 `69G060GH@ 8 AB5?3@8? :B0B2?@018 @1 8 TBFB5? 1 h6`G0AB_6;k IBF1 1x6FB@o 3?0o?TI4k :06A7H5?I6a. ^B26 :1 3?h? I?1x6FBII13@6 :06 h6`G0AB_6;k :15@6 83? 9FB 38;TBI4 3 TBh4856813@oH. XBh48BH@3; ;8I4? 6 I?;8I4? F1:Gb?I6; :06 8481F? G0B8I?I6a 21F?76. S6``?0?I_6B7o I1? 1:63BI6? ABA :0B8671 :0?F:17B9B?@ 5@1 ™:01_?33 Gx? :1J?7š 8 @1 80?2; ABA s@B: ™8A7H5?I6;š 3:131h?I 63:10@6@o 1h?FIH. ‡ 3@B_61IB0I12G 0?x62G 6I19FB I?@ F10196. cp

[P(QN  !M! QQN 6 - ýï òçì ìöõéçï f1 (x, y) = x2 − 2xy + y 2 ,

ô
êì í èïî ùéç
ó øêó . éêô
 ìýêï  ïéèëð ô çûõü øêó í ôèìêì  ) îïî øêó  ? é ùû çóòó óòééèëð øêóç + óøóïõüçïð êï  çó + ï  ’ ûçî + óð ìøêéùé  õðéè ëéùõìí ó ëéùõìôìöõéçìô  ïêïîèéê îêóèóöéëîì  èìöîó çé òì éè ý
èü çï  êûúéç ùìýïôõéçóéò èêéèüéì øìêðùîï ó ô
úé  ˆ êû  óòó ëõìôïòóí õ  ýïð ûçî + óð ùé êðù * õìêï çïöóçïéèëð ë öõéçìô øìêðùîï çé òéçéé èêéèüéìí èïî

é óòééè ëéùõìôû  îêóèóöéëîû  èìöîû  . êóöéò ìýð  ïèéõüçì ëû  é  ëèôû  è çéõóçé ç
é îììêùóçïè
í èïîóéí öèì ô ìîêéëèçìëèó çûõð f2 (x, y) = x2 − y 2

x = y

x=y

x = −y

f2

f (x, y) = x2 − y 2 + ϕ(x, y),

ϕ(x, y)

u v

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é îììêùóçïè
í ô îìèìê
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ôï  èëð »´¸®´Ã®¶º»º ®¯¿½­»ºíîìèìê
é óëöéêø
ôï  è ëèêûîèûêçì ûëèì  öóô
é îêóèóöéëîóé èìöîó  

f [x(u, v), y(u, v)] = u2 − v2 .

f1

f (x1 , . . . , xn )

f

2 2 f = z12 + . . . + zk2 − zk+1 . . . − zn ,

f (0) = 0

cK

í

ž Ÿ  Ÿ ¡ è  •“   G ’ • ’ ¤   ¥ ”¦•  J  • ”£ I éê

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676 x=2

∞ X

x x3 x5 − + − ... 1! 3! 5!

(−1)k+1

sin kx , k

x ∈ (−π, π),

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x = x1 e1 + . . . + xn en ,

9F? – A110F6IB@4 e – hBT63I4? 8?A@104. Œ `GIA_61IB7o I191x 0;FB i

i

f (x) = c1 ϕ1 (x) + . . . + cn ϕn (x) + . . .

p|

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i

3

n

n

n

max f (x)

I6A19FB I? G:60B?@3; 8 :01h7?2G 3Gb?3@818BI6; 0?J?I6;. { h?3A1I?5I12?0I4k TBFB5Bk – s@1 3?0o?TIB; 91718IB; h17o. x∈X

>B:062?08B06B_61IIB; TBFB5B >oH@1IB 26I626TB_66 `GIA _61IB7B J=

Z1

xdx , 1 + (y 0 )2

y(0) = 0,

y(1) = 1

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n

yn (x) = x + sin2 nπx.

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y

f1(x)

f(x)

x

x O

a)

O

1

1 b)

¸º®Æ  Æ Æ í ¯Ê´¸»­Éº° Å ¯®¶´±¯Ä±´»¯¸±´ 9 ®Ê¯¸² max |f (x)| = 1.

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1

1

2

2

1

p=

2

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2

0

0

H(x, λ) = λf0 (x) + (1 − λ) f2 (x),

λ ∈ [0, 1].

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@6:B an =

1 1 + ... + n, 2 2

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à Òà •   ⠔ ÔÔ

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712 F?a3@8GHb62 :1 :0B867G Z∞

δ(x) ϕ(x) dx = ϕ(0)

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é øìõûøêðò
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1

 

`1S WkWZ[

xi

Z1] Q`WTWm [a V b 1b

p(x) = p1 (x1 ) . . . pn (xn );

KC

n

Ú-=.-.-Û

  

p(x)

Z1 ] Q`WTWV 3V ZQR Q`S 1ZWj b x X

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p(x) = p(x21 + . . . + x2n ),

@. ?. `GIA_6; p(x)  B TIB56@ 6

Ú-=.-.=Û

ln p(x) = ln p1 (x) + ln pn (x),

:13@1;IIB IB 3`?0Bk

Ú-=.-.DÛ S0G9626 3718B26 `GIA_66 -=.-.=Û 6 Ú-=.-.DÛ 62?H@ 1F I6 6 @? x? :18?0kI13@6 G018I; ÚB s@1 81T21xI1 76Jo A19FB 6k I102B76 Ú90BF6?I@4Û A1776I?B0I4 Ú1F6IBA181 676 :01@6 81:171xI1 IB:0B87?I4ې @. ?. ∇ ln p(x) = λ∇x . t@1 FB?@ n 0B8?I3@8 x2 = x21 + . . . + x2n = const.

2

p0i (xi ) + 2λxi = 0, pi (x)

6I@?906018BI6? A1@104k :0681F6@ A

2

pi (xi ) = ki e−λxi .

‡1I3@BI@4 1:0?F?7;H@3; I1026018A1a 6 TBFBI6?2 IB:06 2?0 8@10191 212?I@B Z∞

ki e

−λx2i

dx = 1,

„A1I5B@?7oI1 −∞

p(x) =

Z∞

2

x2i ki e−λxi dxi = σ 2 .

−∞

n (2πσ 2 )− 2

Kc

e

−

x21 +...+x2n √ 2πσ2 .

Ú-=.-.<Û

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1

2

2

3

n

n+1

1

n+1

k

arctg u − arctg v = arctg

7?9A1 :0681F6@ A

u−v 1 + uv

Ú-=.D.CÛ . êó  èìò ? ïîëôéõõ ý õ çïòéçóè èéòí öèì îï ùïð ëèêìîï ìêòûõ éì «êûîìøóëé  ëìùéê ïõï ìúóýîû n X

k=1

1 n = arctg . 1 + k + k2 n+2




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y = y 0 − 2u,

A1@104? I? 2?I;H@ @BAx? `GIA_66 z = 2x−y . E1s@12G TB2?IB 30BTG G:01bB?@ Ú-=.< .cې :0681F; A G0B8I?I6H z = 2x − y dz = dx, 2−z

A1@101? 7?9A1 6I@?9060G?@3;. {1@ h17?? 371xI4a :062?0Ü G0B8I?I6? ih?7; 8@10191 01FB 3 dy 2 = − y, y Ú-=.<.pÛ dx

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A1@101? ABA 7?9A1 Gh?F6@o3; 6I8B06BI@I1 1@I136@?7oI1 :0? 1h0BT18BI6a x = kx0 ,

y=

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t@6 x? :0?1h0BT18BI6; I? 876;H@ IB `GIA_6H z = xy . E1 s@12G :?0?k1F 1@ :?0?2?II4k (x, y) A (x, z) G:01bB?@ Ú-=.<.pې FB8B; 7?9A1 6I@?9060G?21? G0B8I?I6? dx zdz = . z 2 − 3z + 2 x

Œ0B8I?I6? @?:71:0181FI13@6

∂T ∂2T =a ∂t ∂x2

I? 2?I;?@3; :1F F?a3@86?2 :0?1h0BT18BI6a t = √kt  x = √kx  A A1@1042 @BAx? 6I8B06BI@IB `GIA_6; τ = x/ t . w3:17oT1 8BI6? τ 381F6@ 63k1FI1? G0B8I?I6? 8 5B3@I4k :016T81FI4k A 1h4AI18?II12G 0

0

d2 T aτ dT + = 0. dτ 2 2 dτ

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@1 6T s@191 `BA@B 21xI1 848?3@6 2B33G :17?TI4k 37?F3@86a IB56IB; 3 TB2?I4 G0B8I?I6; f (x) = x sA868B7?I@I42 f (x) =  A1@101? 21x?@ 1ABTB@o3; GF1hI??. Žb? 7G5J? ?376 :0? H(x) 1h0BT18BI6a @6:B H 6T8?3@I1 I?3A17oA1. ^ABx?2 `GIA_6; f (x) =

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6I8B06BI@IB A :0?1h0BT18BI6;2 H (x) = 1/x  H (x) = 1 − x  B @BAx? 1

H3 (x) =

1 , 1−x

H4 (x) =

x , x−1

2

H5 (x) =

x−1 . x

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E137? 719B06`26018BI6; I?0B8?I3@81 :?0?k1F6@ 8 lg A < 2 lg k − p < lg(A + 1).

E17B9B; k = 2  p = 2q  :17G5B?2 m

lg A < 2m lg 2 − 2q < lg(A + 1).

SB7?? 13@B?@3; 3137B@o3; IB @?10?2G ‡01I?A?0B. >B s@12 `1I? 21x?@ 81TI6AIG@o 8:?5B@7?I6? 5@1 6 3 36@GB_6; @BA 676 6IB5? 0BT0?JB?@3;. >1 s@1 s21_66. f (n) ƒ102B7oI1? F1ABTB@?7o3@81 3Gb?3@818BI6; I?845637624k `GIA_6a F1817oI1 :013@1. w3k1FI42 :GIA@12 ;87;?@3; IG2? 0B_6; 8456376@?7oI4k :0190B22 5@1 7?9A1 3F?7B@o. ^IB5B7B 21xI1 :?0?356@B@o :0190B224 6T 1FI1a A12BIF4 :1@12 6T --D

F8Gk :1@12 6T @0?k 6 @BA FB7??. { 0?TG7o@B@? ABxFB; :01 90B22B :17G56@ 381a I12?0 k . E0190B22G 1h1TIB562 5?0?T  B 311@8?@3@8GHbGH `GIA_6H 5?0?T ϕ (n) . P nB3321@062 @?:?0o `GIA_6H  ?376 ϕ (n) 1:0?F?7?I1 ϕ (n) + 1, ?376 I? 1:0?F?7?I1. h(n) = Ú-<.-.-Û 0, ϕ (n) „IB I? 84563762B. S7; F1ABTB@?7o3@8B :0?F:171x62 :01@68 I1? @1 ?3@o h(n) = ϕ (n) :06 I?A1@1012 p . >1 s@191 I? 21x?@ h4@o @BA ABA h(p) 6= ϕ (p)  ?376 TIB5?I6? ϕ (p) 1:0?F?7?I1 6 1:0?F?7?I1 ?376 ϕ (p) I? 1:0?F?7?I1. h(p) ˜ ÌÞ ð  Ô ž âÔ SBIIB; 1h7B3@o 6T1h67G?@ :B0BF1A3B26 I1 TF?3o 1I6 8 h17o J6I3@8? 381?2 6T 1h7B3@6 1J6h15I4k G21TBA7H5?I6a. m?2 I? 2?I?? s@6 1J6hA6 :17?TI1 ™:?0?x6@oš 5@1h4 TIB@o ™AGFB k1 F6@o I? IBF1š. {1@ 1F6I 6T :062?018.  ï  ìôéò ûçî + ó  ô
öóëõóòì  í éëõó ùõð õ  ýì ì ëû  éëèôûéè øêéùøóëïçóéí øì îìèìêìòû  ï îìçéöçìé öóëõì úï  ìô ô
öóëõðéèëð . êìçûòéêûéò èïîóé ûçî + óó ó ìøêéùéõóò k

k

n

n

n

p

p

p

p

f (n)

n

f (n)

èï )ûçîìè+ óð óçéèïîô
ùïõéé öóëõóòïí èïî îïî øêó ìèõóöïéèëð ìè í øêó  ( ùêû  ì  ëèìêìç
í ìçï ô
öóëõóòïí øìëîìõüîû î ô
öóëõðéòìòû  çïöéçó  ôëéùï òì çì øêóýïôóèü  ì  çóîïéè ôìøêìëí  ïöéò ý
õì òûùêóèü ë ûçî + óé  ‚#   & . êóöó  "# ## ûçî + ó  çï ôéëüòï ëéêüé
ìçï çïð ý
Dõï ýïðô
öóëõóòï øìø
èîï øêó ìøêéùéõóèü ô
öóëõóòû èïîí öèìý õ  ýìò í ) øêóç + óøóïõüçì ìý  êéöéçï çï øêìôïõ  æõó øìõûöï  èëð ëõóúîìò û  îóé îõïëë
ûçî + ó  ùóë  " îêéùóèóêû   óé øìçðèóé ô
öóëõóòìëèó & í óõó îïîóé  èì ìýð ëóõìô ïèéõüçì óòé  è ìýõïëèü çéìøêéùéõéççìëèó  . ìø
èîï ùììøêéùéõóèü
ò òéèìùìò èì

é çé øêì ìùóè çéêï  êéúóòìëèü øêìýõéò
ìëèïçìôï &  ìêéçü çéøêóðèçìëèé  "ùìôìõüçì ìöéôóùéç  - øêéùéõéçóé ûçî + ó  ë øìòì  ü   ïôéùìòì ô
øìõçóò
øêðò
ùé  ëèôó  èóøï ëõì óèüí ûòçì  "

óèü & øêóôìùóè ô óèì  é î ûêïôçéçóðò g(n) = fn (n) + 1.

n = 1

n=2

f1

f2

fn (n)

f (n)

n

f (n) f (n)

‚ , ,& "#
+      

îìèìê
éëèéëèôéççì ìèçìëóèü î ô
öóëõóò ò ûçî ó ðòêéúéçóé  øêìèóôçìò ëõûöïé èéìêóð øêéôêï  ïéèëð ô òïõìøìõé çû ó êûúîû èì éé ôêéòð ‚ , ,& øêóç + óøóïõüçì çé ôëé ùï êï  êéúóòì "#
 øêóòéê   ìùóç øìøûõðêç u(a, n) = 0,

n = f (a)

--<

< ïëëòìèêóò ìëò
ëõéçç
é èéîëè
í îï ù
 ó  îìèìê
ëìùéê óè çé ýìõéé ëèï ëõìô ó ìøêéùéõðéè îïîìéèì öóëõì  - öéôóùçìí îìõóöéëèôì èï îóøìùìýç èéîëèìô ì  êïçóöéçìí øì  èìòû ì  êïçóöéçì òçì éëèôì ìøêéùéõéçç

ò ìýêï  ìò öóëéõ  ( õéùìôïèéõüçìí ô  èìò òçì éëèôé ëû  éëèôûéè òïîëóòïõüçìé öóëõì  . êóýïôóò î èìòû öóëõû éùóçó + û . êéù
ùû  ó  ïý  ï + ëìùéê óè òéçéé ëèï ëõìôí çì ìøêéùéõðéè öóëõì çï éùóçó + û ýìõüúéé òïîëóòïõüçì ì Œ ( ì óëèóîï  ùéëü òçì  ìøõïçìôï   ì ùï  ìôìêðè ìý ì  êïçóöéççìëèó òçì

éëèôï öóëéõí ìøêéùéõðéò
êï  ïòóí øìùêï  ûòéôï  è öèì  èì ôêìùé îìùóêìôîó îïî ý
èüçïë ùêû

êï  ïòóí îìèìê
é ìøêéùéõð  è öóëõï çé ëï  òóîìùóêìôîïí øì ëéýéí ïïêïììøóêïðëü  óé êï 
þ . ìõûöïéèëð û é çé éëèîïð  ôóôï   ïð ëóëèéòï êï   ˆ ïõéé  ƒ óëõì ïèìòìô ôì ëéõéççì  í óõó òïîëóòïõüçìé öóëõìí îìèì  êìé îèìçóýûùü îì ùïçóýûùü  ïøóë
ôïõ çï ýûòïé  ƒ èì èïîóé êï
ìøêéùéõð  èþ  óöé  ì  . õ  ë ôì  òì çïð ìøìêï çï ýìõüúû  ëìôìîûøçìëèü öéõìôéöéëîó  çïçó   . êìëèé  úó  øêóòéê ) ñöóëõì òóîêìç ô ù   òé÷ ƒ éè
êé ëõìôï  ïùï  è ôøìõçé ìøêéùéõéççìé öóëõìí çì û  çïèü é  ì òì çìí õóúü ìêìöé êïëøìõï ìôìêðí  ïð óç ìêòï + óé  í îìèìêïð ô ëïòì  êï  é çé ëìùéê óèëð  çé ìëèïéèëð  øìëõé ìöóëèîó ìè çéëûêïçìëèé ìè øïêïùìîëï çóöé ì ˜ Ì ¯ àâ  ÎÑÔÖ  Òà ÐÏáÔÖÒ gI1x?3@81 IBT48B?@3; X 1U \X TW`Z[m  ?376 3Gb?3@8G?@ s``?A @68IB; :01_?FG0B F7; 84;3I?I6; @191 :06IBF7?x6@ 676 I?@ :016T817oI4a s7?2?I@ s@12G 2I1x?3@8G. gI1x?3@81 IBT48B?@3; X 1U \X TW `Z3 R 1X 1kWTS Wm [m  ?376 3G b?3@8G?@ s``?A@68IB; :01_?FG0B F7; :137?F18B@?7oI191 :1 01xF?I6; ?91 s7?2?I@18 A1@10B; 8 :0?F?7? 635?0:48B?@ 83? 2I1x?3@81. w2??@ 2?3@1 37?FGHb6a `GIFB2?I@B7oI4a 0?TG7o@B@. 1TV ` 1V 1U TW `Z3 R 1 1kWTS Wm 31a Z3 Z1 1U TW`Z31 ¦\ g \ X \X X X \X m Z3Y 1TV `3 R3S 3Y WV 1S hZ[2 ˜ 1S [2 kW T1S . ^k?2B F1ABTB@?7o3@8B :013@B. ^IB5B7B G3@BIB8768B?@3; 3:0B 8?F76813@o 37?FGHb?91 `BA@BÜ m Z3Y 1TV `3 R3S 3Y WV 1S hZ[2 1U TW`Z3 V 3 y _Q W V 3S hU3 V 3 y _Qa U3 y _Q 1^ ˜1S [2 kWT1S S X \X X U TW `Z3 R 1 1kWTS Wm 3 Z1 V 3S hU3 a Z3 W ¯ a 9F? ¯ 1h1TIB5B?@ S S S \X X F1:17I?I6? S  @1 ?3@o 2I1x?3@81 :171x6@?7oI4k _?74k 56 3?7 A1@104? I? :06IBF7?xB@ S . S1ABTB@?7o3@81 s@191 :0?F3@B87;?@ 31h1a I?371xI1? G:0Bx I?I6?.SB7oI?aJ6a k1F 0B33GxF?I6a @BA18.EG3@o S , S , S , . . . – s``?A@68I1? :?0?5637?I6? 83?k 0?AG0368I1 :?0?5637?II4k --U 1

2

3

2I1x?3@8. „h0BTG?2 2I1x?3@81 6T @?k 563?7 x  A1@104? :06 IBF7?xB@ S . mBA62 1h0BT12 R¯R313@16@ 6T s7?2?I@18  GF1 x 87?@810;Hb6k G37186H x ∈/ S . E1s@12G R¯ I? 21x?@ 318:BFB@o I6 3 1FI62 6T S ÚF1ABT48B?@3; 1@ :01@68I191 – 8 F8? 3@015A6 I1 I? :013@1Û – 37?F18B@?7oI1 R¯ 0?AG0368I1 I?:?0?5637621 B R – I?0?AG0368I1. { :06I_6:? 3Gb?3@818BI6? I?845637621a `GIA_66 6 0? AG0368I1 :?0?563762191 I1 I?0?AG0368I191 2I1x?3@8B – s@1 1@0Bx?I6; 1FI191 6 @191 x? ;87?I6; 8 F8Gk 0BTI4k T?0AB 7Bk. Ž3@o ?b? 2I191 F0G96k T?0AB7. >?A1@104? 6T I6k 3@1;@ :11FB7o 6 8 I6k 6T1h0Bx?I6? @191 x? ;87?I6; F1:17I;?@3; :13@101II626 F?@B7;26 Ú@?10?24 V?F?7; IB:062?0Û. wT8?3@I4 2I1915637?II4? :1:4@A6 1:0?F?7?I6; 8456376 21a `GIA_66. { -KDC 91FG moH06I9 88?7 `GIA_66 8456376 24? A1I?5I1a 2BJ6I1a 6T8?3@I1a I4I? ABA m Q WZ Q hiX WZ^ y Qb { @12 x? 91FG V?F?7o t0h0BI ‡76I6 88?76 X 1U \X TW`Z[ 1 º ZU˜WW b XB@?2 ˜?05 E13@ gB0A18 – 84F?7;76 6I4? A7B334 \ `GIA_6a. Œ3676;26 2I196k 2B@?2B@6A18 h471 G3@BI187?I1 5@1 83? :G@6 ABA 918106@3; 8?FG@ 8 n62 – :0681F;@ A 1FI12G 6 @12G x? A7B33G 845637624k `GIA_6a. ˜ ̜ N ÔÏ • ã  áÏÖÒ Ð  ÏÍ àÑáÖ㠌0B8I?I6? -<.<.DÛ p(z , . . . , z ) = 0, Ú 9F? p – :176I12 3 _?7426 A1s``6_6?I@B26 IBT48B?@3; _W3^ ºQZV 3`[m 6 :1F0BTG2?8B?@ 0?J?I6? 8 _?74k 5637Bk. mBA6? G0B8I?I6; 3 F0?8I6k 80?2?I :06@;968B76 2B@?2B@6 A18 ABA 0G7?@AB 6901A18. ^:131h4 0?J?I6; 1@765B763o `0B9 2?I@B0I13@oH. { -K|| 91FG V67oh?0@ 8 381?2 TIB2?I6@12 3:63 A? 2B@?2B@65?3A6k :01h7?2 :1F I12?012 -| :13@B867 81:013Ü ™3Gb?3@8G?@ 76 ?F6I4a B79106@2 F7; 84;3I?I6; 0BT0?J621 3@6 7Hh191 G0B8I?I6; Ú-<.<.DÛWš. E01h7?2B 1ABTB7B3o @0GFI1a 6 :71F1@810I1a. E137?FIHH @15AG 8 ?? 1@06_B@?7oI12 0?J? I66 :13@B867 gB@6;3?865 8 -Kc| 91FG. ™X06@?76š :17G5676 9012BFI1? GF1817o3@86? :13A17oAG 2B@?2B@6A6 ÚV?F?7o Ss 863 EG@IB2 n1h6I31I gB@6;3?865 6 F0.Û F?a3@818B76 3 6T; b?3@812 6 8F1kI18?I6?2. ‡ @12G x? I?1h4AI18?II1 A0B3681? x

x

p

;

1

n

--C

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k

1

m

1

k

1

n

m

n

< Z3Y 1TV `3 ˜1S [2 R3S 3Y WV 1S hZ[2 kWT1S _W3ºQZV 3`3 ` A V 3m W V 3S hU3 V 3m TS kQ1a U3 y _Q 3Z3 j `Sj 1V Tj m Z3Y 1TV `3m \ R3S 3Y WV 1S hZ[2 ] ZQk1ZW4 Z1U3V 3 3 y 3 R3S WZ3m Q b X P (x1 , . . . , xk )

ˆ ìîï  ïèéõüëèôì  õéòéçèïêçì   ëõó ) òçì éëèôì øìõì óèéõüç
çïöéçó  í èì ìøêéùéõðéè í îïî ùóì ïçèìôì  - ýêïèçì  . ûëèü ) òçì éëèôì èé í øêó îìèìê




A

P (x1 , . . . , xk )

p(a, x1 , . . . , xk ) = a − P (x1 , . . . , xk ) = 0

A

A

a

« . ìöèó ôé ùé ùïõéé òì çì ëöóèïèü + éõ
öóëéõ 

í è é 

Q(a, x1 , . . . , xl ) = 0

--c

k=1

A

)

øêìëèì òçì éëèôì

êï êéúóòì  *ì ùï

A

) 

òçì éëèôì øìõì óèéõüç
çïöéçó  øìõóçìòï

x(1 − Q2 (a, x1 , . . . , xl ))

mBA62 1h0BT12 3Gb?3@8G?@ :176I12 2I1x?3@81 :171x6 @?7oI4k TIB5?I6a A1@10191 8 @15I13@6 318:BFB?@ 3 2I1x? 3@812 :013@4k 563?7. j17?? @191 ?91 21xI1 A1IA0?@I1 GAB TB@o k1@; s@1 2B71 5@1 FB?@ 6TTB :71k1a 1h1T06213@6. ‡BxF4a :176I12 -<.<.<Û kB0BA@?06TG?@3; 3@?:?IoH n 6 563712 m :?0?2?II4k Úx . S7; 7Hh191 F61`BI@18B 2I1x?3@8B 21xI1 GABTB@o :176I12 Ú-<.<.< Û 3 n 6 4 ÚI1 81T21xI1 h17o A J62 mÛ 76h1 m 6 9 ÚI1 21x?@ h4@o h17oJ62 nÛ. ˜63?7 n 6 :10;FAB F8Gk@0?k F?3;@A18 ABA :0B8671 F13@B@15I1 F7; m 3B24k 371xI4k 37G5B?8. mBA191 :10;FAB n 6 m F13@B@15I1 6 F7; TB:636 GI68?03B7oI191 :176I12B ÚBIB719 GI68?03B7oI1a 2BJ6I4 moH06I9Bې 9?I?060GHb?91 7Hh1? F61`BI@181 2I1 x?3@81 :1 ?91 I12?0G. >B `1I? 1@A048BHb?913; :?aTBxB -|; :01h7?2B V67oh?0 @B ABA@1 1@k1F6@ IB 8@101a :7BI. Ž? 1@06_B@?7oI1? 0?J?I6? FB?@ :013@1a :?0?81F IB F0G91a ;T4A `BA@B 3Gb?3@818BI6; 0?AG0368I1 :?0?563762191 I1 I? 0?AG0368I191 2I1x?3@8B. ˜@1 AB3B?@3; 845637624k `GIA_6a y = f (x , . . . , x )  @1 s@1 `GIA_66 90B`6A A1@104k Ú2I1x?3@81Û G = {x , . . . , x , y = f (x , . . . , x )}, F61`BI@18. mBA62 1h0BT12 F61`BI@184 G0B8I?I6; 1ABT48BH@3; ?b? 1FI62 8B06BI@12 Ú;T4A12Û 6TG5?I6; 8456376213@6. { ABA12 @1 32437? – sA868B7?I@I42 8 ABA12@1 – h17?? s``?A@68 I42 8 ABA12@1 – 2?I??. j17oJ6I3@81 6337?F18BI6a 8 0B3 32B@068B?21a 1h7B3@6 @0BF6_61II1 1:60B?@3; IB 2BJ6I4 moH 06I9B 676 IB 0?AG0368I4? `GIA_66. { @12 6 F0G912 37G5B? :01_?33 845637?I6a ABA h4 :1A04@ 8GB7oH ÚG3A17oTB?@ 6T:1F A1I@017; k1@; :0B867B 1:63BI4Û. XF?3o x? 845637;H@3; TIB 5?I6; 1h4AI18?II191 :176I12B A1@104a 21xI1 ™:1@019B@oš. S7; I?A1@104k TBFB5 s@1 21x?@ h4@o 3Gb?3@8?II1. ˜ ̱ «àÏÎàÐÒ |à  àÕÙ { 38;T6 3 ™F61`BI@181a s:1:??aš 9?F?7?83AB; @?2B@6AB 3Gb? 3@8?II1 :01;3I67B3o. { 381? 80?2; V?F?7H :06J713o 2I191 --p 1

1

n

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4t , (1 − β 2 )

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0

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@17oA1 63@6IB. *éøéêü ùôï ëõìôï ì ëïòìò ûêïôçéçóó 

éççìëèü  õéîèêóöéëîì  ì øìõð ôùïõó ìè  ïêòìçóöéëîì  ì ó  õû  öïèéõð ïøêðòéçðéèëð øì  ïîìçû

‹  & "# # #  

ùé ) îêû  ìôïð öïëèìèï í ) èïî çï
ôïéò
 ôìõçìôì ôéî èìêí ) êïùóûë  ôéîèìê  . ìëîìõüîû ùé > êì  õü øêóøóëïõ öïëèó + ïò ùõóç
ôìõç øì ïçïõì  óó ë

ìèìçïòóíèì ô îïöéëèôé ôìõçìôì  ûçî + óó ý
õì éëèéëèôéççì óëøêìýìôïèü îìøó  ‹   & ë  ïòéçì  êï  ôé öèì ýûîô
çï öèì  çóýûùü ùêû  ìé  * ïî "# ôìõçìôïð ## ôì çóîõï

ûçî + óð E = E0 ei(ωt−kr) ,

ω r = {x, y, z}

(ω = 2πν) k

E

‹  ,&

"# # ìèõóöï  ïðëð ìè ‹   & õóúü øéêé  ïøóëü  øïêïòéèêìô ë ûöéèìò ìêìúì ó  ôéëèç
ëôð  é "# # # ‹  $& "# # ùé ) øìëèìðççïð . õïçîïí ùéõéççïð çï  ˆ ó éêéç + óïõüçìé ûêïôçéçóéí îìèìêìòû ûùìôõéèôìêðéè ‹  ,& í õé  "# # îì øìùýóêïéèëð  ˆ ó éêéç + óêìôïçóé ‹  ,& ùïéè "# # i

ψ(r, t) = ψ0 e ~ (Et−pr) ,

E = ~ω,

~

p = k~, 2π

ï èïî é *éøéêü ó  ëõéùûéè ‹  ‚ & "# # öèì çóð ó øêéùëèïôõðéè êéùóç éêï  ëìýì  ìùóç ó  ôì  òì ç
ôïêóïçèìô  ïøóëó ûêïôçé D  ýì  úï  ô çìôìò çïøêïôõéçóóí îïî øêïôóõìí ìîï 
ôïéèëð øéêô
òí èêéýûð ùïõüçé úé ì ùôó éçóð  èìò ëò
ëõé ðëçìíöèì çï  ïøóëó "#‹ # ‚ & îôïçèìôïð òé ïçóîï çé  ïîïçöóôïéèëð  Ë Ë ÌÞ ¯ ãÑØÎÏÑáÎã  à  Ôà ÖÏÕ  ^:G3@623; 3 I?h?3 IB X?27H. „:0?F?762 3A1013@o 0B3:01 3@0BI?I6; 62:G7o3B 3xB@6; 8F17o 3@?0xI; 3 3?5?I6?2c S Ú063. -<.-Û. EG3@o 8 62:G7o3? FB87?I6? G8?765?I1 IB 4P  B :71@ I13@o – IB 4ρ . XB 80?2; 4t 5?0?T S :01aF?@ 2B33B 8?b?3@8B . 4m = 4ρSc4t -DD i ∂ψ = Eψ, ∂t ~

p2x + p2y + p2z ∂2ψ ∂2ψ ∂2ψ + + = ψ. 2 2 2 ∂x ∂y ∂z ~2 p2x + p2y + p2z 2m

∂ψ i~ = ∂t 2m

H



=E

∂2ψ ∂2ψ ∂2ψ + + ∂x2 ∂y 2 ∂z 2



,

S

c

¸º®Æ † Æ Æ ^67B F?a3@8GHbB; IB :71bBFAG S  15?86FI1 F = 4P · S . wT2?I?I6? A1765?3@8B F86x?I6; 0B8I1 62:G7o3G 3674 @. ?.  1@AGFB 4mc = F 4t 4ρSc2 4t = 4P S4t,

5@1 :137? :?0?k1FB A :0?F?7G (4P, 4ρ → 0) FB?@

Ú-C.=.UÛ  ìèéõìëü ý
ìýêïèóèü ôçóòïçóé çï ôï ç
  ûòïçóèïêç
 ïëøéîèí õéëìëèïôõéçóé

ï  ó   ùéëü çï  ïùçéò øõïçé  - ý  éó  ôéëèçìí öèì úîìõüç
é  ïùïöó çï ûêïôçéçó  ô

ôï  è çïóýìõüúóé èêûùçìëèó  = ïùçóò öóë  õìò ôëé øêìëèìí çì ó  çïöïõüçì ô
õðùóè çéøêóëèûøçì  ƒ èì øêóçðèü  ï çéó  ï îïîóé îìç +
û ôïèóèüëðí öèì öéòû øêóêïôçðèü  :õï  ï êï ýé’ôéëèçìéí  ï  èëðí êûîó ìøûëîï  èëðí îõûýìî çé êïëøûè
ôïéèëð  ó óîó  èéìêéèóîó ôóêèûì  çì ôõïùé  è óëîûëëèôìò êéúéçóð èïîì  ì ëìê  èï ïêð ïùïö  æ çïùì øêó  çïèüí öèì øì ôïèîìëèóí ûòéçó  ìêóéçèóêìôïèüëð ó éççìëèó çï êéûõüèïèí ) ìçó ðôçì  ïçóòï è õóùóêû   éé øìõì é çóé ’ ìêòûõï ‹ , „& ) ìùçï ó  ó  óöéëîó òóçóïè  ê  ˆ õð èôéêùì  ì "# í ùé ) òìùûõü ç ï  ëèéê çðëõûöïé ìçï ùïéè  ï  ï ôìøêìë ûøóêïéèëð ô ûêïôçéçóé ëìëèìðçóð   ëõó øìõì 

óèüëð çï ‹ , ‹& "# è é  í èì í öèì øêóôìùóè î  ïçó éççìòû êéûõü èïèû s

c=

c=

p

E/ρ

I

E

PV =

P = ρ RT µ

dP . dρ

m RT, µ

dP = RT = P dρ µ ρ

s

P . ρ

"#

‹ , %&

- úóýìöçìëèü ‹ , %& ëìëèìóè ô èìòí öèì øêó  ôûîìô
îìõéýïçóð "# èéòøéêïèûêï çé ûëøéôïéè ô
êïôçóôïèüëðí øì  èìòû ôòéëèì "#‹ , ‹& çïùì c=

-D<

êïëëòïèêóôïèü ïùóïýïèóöéëîó  øêì + éëë íìèîûùï í öèì û é øêïôóõüçì ìøêéùéõðéè ëîìêìëèü ôûîï  g?xFG :01562 A 8481FG `102G74 -C.=.UÛ 21xI1 h471 h4 :0?F›;86@o 2I191 :0?@?IT6a IB56IB;Ú 3 I?1:0?F?7?II13@6 3B2191 :1I;@6; 3A1013@6 c . >B:062?0 `102B 62:G7o3B 21x?@ TB863?@o 1@ t  6 @19FB I?:1I;@I1 5@1 @BA1? . >1 24 83? s@1 13@B87;?2 TB ABF012 @?2 h17?? 5@1 8 :?0812c :06h76x?I66 0?TG7o@B@4 8?0I4. æòééè øêïôì çï ëû  éëèôìôïçóé ó ùêû ïð èìöîï  êéçóðí èêéýû   ïð ñùì óòïçóð÷  ïùïöó ùì îìç + ï   ì èì ùï çïùì ûöóè
ôïèüí öèì # çïëé õéçóð ô
øóôïéè áá ïõîì  ìõð  *ìöçì èïî é # ûëóõó  êéúïéè  ïùïöû çï áá  Š îìòû ìöéèëð ôëé ëèìí èì çï ìùóç çéùìëèï  ó  øêì + éçè û ìùóè ìöéçü òçì ì ëóõ ó ôêéòéçó  „h0B@62 @?:?0o 8I62BI6? IB F0G9GH 3@101IG F?7B. Ž376 3@?0x?Io 0B3:171x?I 8F17o 136 x  @1 3@?:?Io 3xB@6; u(x, t) 8 1hb?2 37G5B? hGF?@ `GIA_6?a F8Gk :?0?2?II4k. >BaF?2 F6` `?0?I_6B7oI1? G0B8I?I6?A1@1012G GF187?@810;?@ u(x, t) .nB3 321@062 F8B 3?5?I6; x 6 x + 4x Ú063. -<.=ې B @BAx? TBA7H 5?II4a 2?xFG I626 1h›?2 . ^67B F?a3@8GHbB; 4V = S4x P/ργ = const

c=

p

γP/ρ

J

J

J

J

x

¸º®Æ † Æ Æ ¬­®³¸´®µ¸­±¯±º¯ ®p ­µº° x

x+ x

IB 3?5?I6? x  0B8IB kS

∂u , ∂x x

9F? k 1:0?F?7;?@3; `6T6A1a TBFB56 Ú9BT @8?0F4a 3@?0x?IoÛ. n?TG7o@60GHb?a 3671a F?a3@8GHb?a IB 1h›?2 4V  hGF?@ ! ∂u ∂u ∂ u − 4x, F = kS = kS Ú-C.=.pÛ 2

∂x

x+4x

∂x

x

-DU

∂x2

6 @19FB TBA1I >oH@1IB FB?@ 817I181? G0B8I?I6?

Ú-C.=.KÛ 9F? c = pk/ρ – 3A1013@o 817I4 ?376 918106@o 3 IB@;xA1a – G3A10?I6?. ∂ u/∂t . êó øéêé ìùé ô øêìëèêïçëèôì ∂2u ∂2u = c2 2 , 2 ∂t ∂x

2

2

∂2u = c2 ∂t2



∂2u ∂2u ∂2u + + 2 2 ∂x ∂y ∂z 2



.

Ë Ë Ì ÎãÖ  à  Ôà áàØÕÏØÎÏÖÏ  ÏÑáÔ S7; 8481FB G0B8I?I6; 0B3:013@0BI?I6; @?:7B h17oJGH 5B3@o 0B33GxF?I6a :0?F4FGb?91 0BTF?7B 21xI1 13@B86@o h?T 6T2? I?I6a. {2?3@1 Ú-C.=.KÛ :17G5B?@3; ∂T ∂ T =a , Ú-C.D.-|Û ∂t ∂x 9F? T (x, t) – @?2:?0B@G0B ABA `GIA_6; x 6 t . iIB71912 3674 Ú-C.=.pÛ 8 FBII12 37G5B? hGF?@ A1765?3@81 @?:7B F

2

2

Q = kS

∂2T 4x, ∂x2

A1@101? 8@?AB?@ 8 1h›?2 4V Ú@?:?0o k – A1s``6_6?I@ @?:71 :0181FI13@6Û. XB@?2 @1 x? 3B21? Q :1F356@48B?@3; F0G962 3:131h12 Q = ρλS4x

∂T 4t, ∂t

9F? λ – @?:71?2A13@o ∂T 4t – :184J?I6? @?2:?0B@G04 TB ∂t . 80?2; 4t  5@1 6 FB?@ Ú-C.D.-|ې a = k/ρλ mBA62 1h0BT12 :06I_6:6B7oIB; 0BTI6_B 3 :0?F4FGb62 37G 5B?2 TBA7H5B?@3; 8 @12 5@1 8 FBII1a 36@GB_66 I? 0Bh1@B?@ BIB719 TBA1IB >oH@1IB. { 0?TG7o@B@? 8@10B; :016T81FIB; :1 80?2?I6 TB2?I;?@3; IB :?08GH. -DC

Ë Ë Ìœ ð ÐÔÑáÔÓàÑâÏÐ ØÏ Ï ž ÔÔ ( éò
êïëëû ùéçó  øêéù
ùû  ó ùôû êï  ùéõìô ô ìöó  éççìò ìè ùéèïõé  ôóùé óëöéêø
ôï  è ýìõüúû  öïëèü ó  óöéëîó  ïùïö  ëé ùï éëèü çéöèì ôêìùé ëóõ
íôõóð  é  çï ùôó éçóéí ó çéöèì ôêìùé îììêùóçïè
íìøêé  ùéõð   é  ìîûë ôçóòïçóð  æ éëèü ôëé  ì ùôï ôïêóïçèï éëõó çé  ìôìêóèü " ìý óëîõ  öéçóð & øêó ùôó

éçóé ìëèïçïôõó  ( óõï øìêì ùïéè ùôó éçóéí " ôïéèëðí & ˆ ôó éçóé øêìóë ìùóè øì óçéê + óóí ëóõï ìøêéùéõðéè ûëîìêéçóé  æ ë ëóõì  í ó ë îììêùóçïèì  øêìóë ìùðè òéèïòìê ì 
øêó øéêé ìùé ô ùêû  û  ìýõïëèü ó  òé ïçóîóí ëîï éòí ô ìøèóîû &  ì øì ëûèó ) çóöé  ì " çéîêûèóõüç òéçðéèëð
é  îìõéýïçóð çïûêïôçéçóéò øêû óçé ìøóë
ôïéèëð ûêïôçéçóéò í ? ïðèçóî ) û  ìõí ) òìòéçè " ìý
öçì  ëóõ
& . êó óçéê + óó ñèì  ìí öèì îêûèóèëð÷í ñëóõï÷ ) òìòéçè øéêé ìùé î  õéîèêóöéëîìòû îìõéýïèéõüçìòû îìçèûêû ùéîìêï + óó òéçð  èëð  çïöóèéõüçéé )  ïêðùí ) óçùûîèóôçìëèüí ) éòîìëèü & í ï ûêïôçéçóé ) èì é ëïòìé" ñëóõï÷ )  õéîèêóöéëîìé çïøêð éçóé  . ì  èìòû êï  ìôìê
ì õûýóçç
ëôð  ð êï  çìêìùç
øêóîõïùç
ìý  õïëèé  ô ìøêéùéõéççì  ëèéøéçó øêìóëèéîï  è ó  õûîïôëèôï óõó çéìëôéùìò  õéççìëèó  ’ ó  óîï ôì ôëé ëôìó óøìëèïëð ùì ëó øìê ìøóêïõïëü ô ìëçìô  çìò çï õóçé  ç
é ùó éêéç + óïõüç
é ûêïôçéçóð  . ì  èìòû óëèìîó ñòó  ëèóöéëîì  ì øìùìýóð÷ êé  ûõüèïèìô ùìëèïèìöçì ìöéôóùç
 ‰ êïôçéçóð ìùçó ó èé øðèü éí êïèïîó  çó + ï ) ô çï  ôïçóð îì  ó + óéçèìôå  çïùì ëîï ïèü ì úóêìîìò óëøìõü ìôïçóó øêó èìò îìò  øõéîëç
öóëéõ  øêóç + óøéí ôëé ëîï  ïçì ý
õì ô õïôé  ? ïèéòïèóîï ëèì  # óè çï ïêó òéèóöéëîó ìøéêï+ óð í ùõð îìèìê
îìòøõéîëçïð øõìëîìëèü ) ó  êìôìé øìõé   ì  èì öéêéëöûê ìý  ì   éîìèìê
é ë éê
 çïçóð ôëé  èïîó ûëøéúçì

ûçî + óìçóêû  è çï îêï   èì  ì øìõðí ìý ìùðëü ýé  òçóòì  éùóçó +
ýû  ïõèéêóðí çïøêóòéê & öéò îìçîêéèçì  ïîõ  öïéèëð øêóöó  " ùó éêéç + óïõüç
é ûêïôçéçóð ñó  êï  è øì ôëéòû øìõ  ÷þ ˆ ï çïé éèìôòéëèì  ìí öèì ëêïû çïöóçï è ë øêóýéýéïð ôóùóòì î ûëõû ïòçéìý ï ìùð  = ïöéò î êéïõüçìòû îìõéýïçó  ùìýïô õðèü ìùóòìëèó

óîèóôçìé þ :õïôçïð øêóöóçï ) ô ùó

éêéç + óïõüç
ëôì  ëèôï í ‹ ‚  & "# ## èôïçóé  é  ô
öóëõéçóé øêìó  ôìùçì  ëôìùóèëð î ûòçì éçó  çï  æçèé  êóêì  ) î ùéõéçó   . ìõûöïéèëð ëû  éëèôéçç
 ô
ó  ê
ú ô øêìëèìèé  ? ì éè øìîï  ïèüëðí öèì  èì ôëé é ôìøêìë ûùìýëèôïí çé îïëï  ó  ëð ëûô îïöéëèôì  éëèôï ùéõï  èì çé èïî   ìõóöéëèôì ùé  ëèôóèéõüçì óçì ùï øéêé ìùóè  ˆ ï é çé  çïöóèéõüç
é èé çóöéëîóé ûøêì  éçóð òì  ûè úï   ï å < ïûòééèëðí ò
 ïìëèêðéò ôçóòïçóé õóúü çï ìùçì  ëèìêìçé ùéõï  -Dc F

z

KL M

F = k z˙

F =0

z˙ = 0

KL LM

F =

k¨ z

Iϕ ¨ = −kϕ ϕ kϕ

q

L

I

m¨ x = −kx

C

L¨ q=−

q , C

q/C

cos ωt

eiωt = cos ωt + i sin ωt,

cos ωt

i sin ωt

eiωt

(e

iωt 0

) = iωe

iωt

,

iω

úïêì ìòùïøìùçðèü èû óõó óçû  çïûöçû  ùóë+ óøõóçû çï ùêû ì   èï  *ïò èëð çìô
é îïèé  ìêóó ò
úõéçóðí ó êï  ôóèóé óùéè ëìôëéò ùêû  ì  ùìêì  ì  í ô
ìùð çï  ïîìçìòéêçìëèóí ùõð îìèìê
ëèïê
 ð 
î ôììý  é çé  ìùóèëð  * ïî óõó óçïöéí çì ëôì  ëèôì ‹ ‚  & ô èéìêóó õóçé  ç
ùó éêéç + ó  ## ôêìùé øì  ó + óìççì  ëóëèéò
ëöóë  ïõüç
ûêïôçéçó  ìîï 
ôïéèëð "#öéò  èì õéçóð ùõð ïêó

òéèóîó   ìçéöçìí òì éè øìîï  ïèüëð ëèêïçç
òí öèì óôììý  êïéè çéëìêï  òéêçì ýìõüúû  êìõü ô ùó ûêï í îìèìê
é óòé  è ùéõìí  é  ìôìêðí ë êï  çììýêï  ç
òó ûçî + óðòó  . êóöóç èìòû ùôé  ì  øéêô
í  ïêòìçóöéëîóò îìõéýïçóðò ô õóçé  ç
ùó ûêï ùé  ëèôóèéõüçì øêóçïùõé

óè + éçèêïõüçïð êìõü  ì  ôèìê
í êï  õì éçóé ûçî + ó  ô êð  ùóëøìõü
’ ûêüé øêéùëèïôõéçóé ô ôóùé ëûòò
 ïêòìçóî & øì  ôìõðéè  éîèóôçì " ëôì  ëèôì ‹ ‚  & ô ìý  éò ëõûöïé   ìôïèü  ìçéöçìí öèìý
ôëé "#  èì ì +##éçóèü ô øìõçì  òéêéí çïùì øêì  èó ëìýëèôéç  çîïî
òîìòøïëí øûèéò øìçìëììèôéèëèôû   é  èéêêóèìêóó  - ý  óé ëììýêï éçóð ìêìúóí ìçó çé  ïòéçð  è ìø
èïí ìëìýéççì ìèêó + ïèéõüçì  ì  ? ì çì ëêïûí çïøêóòéêí ëîï ïèüí öèì ùõð ô
öóëõéçóð ëûòò
ÿ eiωt

øìõé çì êïëëòìèêéèü öèì õéîìí ô
öóëõðéèëð îïî ëûòòï éìòéèêóöéëîì  øêì êéëëóó ë øìîï ïèé õéò S = cos ωt + cos(ωt + ϕ) + . . . + cos(ωt + nϕ)

Ω = eiωt + ei(ωt+ϕ) + . . . + ei(ωt+nϕ) ,

eiϕ

ei(n+1)ϕ − 1 iωt e . iϕ e −  1 

  ì ìêï ùì øìõé çééí éëõó îìòû  èì øìôé  õì - îìçöïèéõüçì øêéùôïêóèéõüçì øìòûöóèüëð ë ô
öóëõéçóéò  *ì ùï ìîûë ùìýïôõéçóð

óîèóôç
òçóò
öïëèé  øìõûöóè ëìôëéò ùêû  ìé ìëôé  éçóéí ï øêéìý   éìòéèêóöéëîì  øêì  êéëëóó ýûùéè ôêï
 ìôïçóé õðùéèü çéûùìýçì òïõéçüîóò ëûòò ìèîê

èóéòô ëûòòû  Ω=

S = Re(Ω)

S

S

Ë Ë Ì± — à Ï âÖã  áÏÖÏ – Ðà © ã  Ôâà XB:6J?2 G0B8I?I6? 0?F6I9?0B Ú-C.-.<Û F7; 1FI12?0I191 37G 5B; i~ ∂ ψ ∂ψ = . Ú-C.U.-=Û ∂t 2m ∂x g1xI1 3ABTB@o 5@1 -C.U.-=Û 8 ABA12@1 32437? 848?F?I1 6T :0?F:171x?I6; 1 86F?Ú 817I181a `GIA_66 Ú-C.-.=Û.m?:?0o ?3@? 3@8?II1 :1321@0?@o ABA6? `GIA_66 GF187?@810;H@ Ú-C.U.-=Û. „ABT48B?@3; 6k 2I191 Ú@1 ?3@o h17oJ? 5?2 TBFG248B713oÛ. ÿ ì çóîï  é ô  ïùïöé ì ùó êïî + óìççì  êéúéèîé  N

2

2

-Dp

SB7?? IB56IB?@3; ™`G@h17š :0?F3@B87;Hb6a 31h1a G87?AB @?7oI1? TBI;@6? 8 1h7B3@6 96:1@?@65?3A6k G21TBA7H5?I6a. ^0?F6 0?J?I6a Ú-C.U.-=Û ?3@o ψ(x, t) = ψ(x) e . Ú-C.U.-DÛ E1F3@B87;; Ú-C.U.-DÛ 8 Ú-C.U.-=Û 6 31A0BbB; IB e  :17G 5B?2 G0B8I?I6? F7; B2:76@GF4 ψ(x) ∂ ψ 2mE + ψ(x) = 0. Ú-C.U.-<Û ∂x ~ S63A0?@I4? 3:?A@04 81TI6ABH@ :062?0I1 @BA. n?J?I6?2 -C.U.-<Û 37Gx6@ Ú −iEt/~

−iEt/~

2

2

2

ψ = a sin(kx + b),

√ 2mE k= . ~

Ž376 5B3@6_B TBxB@B IB 1@0?TA?  [0, l] A0B?8426 G37186;26  @1 s@62 G37186;2 21xI1 GF187?@8106@o @0? ψ(0) = ψ(l) = 0 hG; kl = nπ; n = 1, 2, . . . , 5@1 FB?@ F63A0?@I4a IBh10 81T21xI4k sI?096a π ~ -C.U.-UÛ E =n , Ú 2ml B TIB56@ 6 5B3@1@ 8 367G Ú-C.-.DÛ. >?IB2I191 371xI?? 1:0?F?7;?@3; F63A0?@I4a 3:?A@0 B@1 2B 81F101FB. ^11@8?@3@86? sA3:?062?I@B7oI42 FBII42 TB24 AB?@ A0G9 6 @?106; :17G5B?@ 1`6_6B7oI4a 3@B@G3 ABA 0Bh1 @13:131hI4a 6I3@0G2?I@. ŠHh6@?76 6I@?0:0?@B_6a :06FG24 8BH@ 26`4 6 :6JG@ G5?hI6A6. t@1 3ABTBI1 1@IHFo I? 8 GA10 `6T6AB2 A1@104? IB 3B212 F?7? TB37Gx68BH@ 813k6b?I6;. ‡7B3365?3A6a 2B@?2B@6A IB 6k 2?3@? :101kG h4 I? 84FG2B7 hGFG56 :18;TBI 6I42 G3@B 812. Ž376 :1321@0?@o IB s3A6T A101@A1a _?:15A6 8?FGb?a 1@ Ú-C.-.=Û A Ú-C.U.-Uې @1 7196AG TF?3o 3:131hI4 0B3321@0?@o  ìôìêðè ô  èìò ëõûöïéí öèì öïëèó + ï çï ìùóèëð ô øìèéç + óïõü  çì  ’ðòéó óîó  n

2

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@17oA1 `6T6A6. ^ABx?2 8 s7?A@01F6IB26A? 0?J?I6?2 817 I18191 G0B8I?I6; 37Gx6@ A17?hBI6? F?a3@86@?7oI1a 6T2?0;? 21a 8?7656I4 ÚIB:0;x?II13@6 :17;ې B 63:17oT18BI6? A12 :7?A3I4k 0?J?I6a – 81:013 GF1h3@8B. XF?3o IB1h101@. ƒ6 T65?3AB; 6I@?0:0?@B_6; A17?hBI6a 1@3G@3@8G?@ 6 817I18B; `GIA_6; 6TIB5B7oI1 1:0?F?7;?@3; ABA A12:7?A3IB; 5@1h4 I? h471 76JI6k 81:01318.Ž376 8FG2B@o3; – s@1 9?I6B7oI4a k1F. E1 GABTBII1a :0656I? G0B8I?I6? 0?F6I9?0B Gx? 3B21 31F?0x6@ 2I62GH ?F6I6_G 6 :06I_6:6B7oI1 1@765B?@3; 1@ G0B8I?I6a @6:B Ú-C.=.KÛ 676 Ú-C.D.-|ې I1 :?0?A76AB?@3; 3 :1 37?FI626. Ž376 h4 0?5o J7B @17oA1 1 :163A? F63A0?@I4k 3:?A @018 @6:B Ú-C.U.-Uې @1 83? 21xI1 h471 h4 13@B86@o 8 @BA12 :1 71x?I66 356@B; ψ`GIA_6H GF1hI1a `6A_6?a F7; :012?xG @15I4k F?a3@86a. >1 8 A8BI@181a 2?kBI6A? ?3@o 2B33B TBFB5 A1@104k 24 I? TB@0B968B76. E1s@12G ™`6A_6Hš :06k1F6@3; 6T87?AB@o IB 38?@ 6 :063:13Bh768B@o IB:062?0 F7; 6TG5?I6; 6I@?0`?0?I_66 @?k x? s7?A@01I18. {1:013 1h 6I@?0:0?@B_66 `GIA_66 3@BI186@3; 13@0?? I1 8 0BI?? :06I;@12 32437? 1I ψ @BA 6 I? 0?JB?@3;. E163A6 7196A6 13@BIB8768BH@3; IB 817 IBk 8?01;@I13@6 5@1 IBF1 :06TIB@o ;87;?@3; 1:@62B7oI42 84k1F12 6T :171x?I6;. ‡17?h7?@3; :1:0?xI?2G ™I?:1I;@I1 5@1š I1 8I?JI;; 3@101IB F?7B 62?II1 8?01;@I13@IB – 6 ;T4A BF?A8B@?I. { _?712 A8BI@18B; 2?kBI6AB – 318?0J?II1 GI6AB7oIB; IBG5IB; F63_6:76IB IBG568JB;3; 3:0B87;@o3; 3 1hJ60I42 A0G912 ;87?I6a 8 1@3G@3@86? 6k :1I62BI6;. ƒ6T6A6 IB35?@ :1I62BI6; 219G@ 3:106@o I1 s@1 81:013 @?026I171966. Ž3@? 3@8?II1? 0BTF0Bx?I6? BGF6@1066 :1 :181FG 1@3G@3@86; 6I@?0 :0?@B_6a 8 1h45I4k :1I;@6;k – BF0?318B@o IBF1 3A10?? 83? 91 I? `6T6AB2 B ^1TFB@?7H. ˜@1 x? AB3B?@3; 97Gh6I4 IBJ?91 º \ Z_Qm 1ZV QS hZ3 y3 I?:1 I62BI6; @1 1IB 8?TF? 1F6IBA18B. ‡8BI@18B; 2?kBI6AB k101JB @?2 5@1 @B2 s@B 97Gh6IB T062B.

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v

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81:0134 TF?3o 21xI1 :17G56@o :17B9B;3o IB A0BaI? :013@4? 21F?76. E171x62 2B33B 83?a 2B@?066 0B8I12?0I1 TB:17 I;?@ JB0 0BF6G3B R  B 3A1013@o ABxF1aM2B@?06B7oI1a @15A6 1:0?F?7;?@3; TBA1I12 Ú-c.=.-Û. nB3321@062 I?A1@10GH 5B3@6 _G 2B334 4m IB A0BH 1h7B3@6. Ž? 3A1013@o hGF?@ 0B8IB v = HR, Ú-c.=.=Û A6I?@65?3AB; sI?096; A6I = 4mv , E 2 :1@?I_6B7oIB; – :1@ ≈ −γ M 4m , E 9F? γ – 90B86@B_61IIB; :13@1;IIB;.R wT TBA1IB 31k0BI?I6; sI?0966   v M 4m = const, −γ Ú-c.=.DÛ 2 R 84@?AB?@ 37?FGHb??. Ž376 A1I3@BI@B 8 Ú-c.=.DÛ h17oJ? | @1 0B3J60?I6? hGF?@ :01F17xB@o3; h?3A1I?5I1 R

2 R

2 R

2 vR = 2γ

M + k, R

k=

const , 4m

?376 2?IoJ? – @1 :1 F13@6x?I6H R = − 2γM 0B3J60?I6? k 13@BI186@3; 6 32?I6@3; 3xB@6?2. mBA62 1h0BT12 0B8?I3@81 M -c.=.<Û v − 2γ =0 Ú R 1:0?F?7;?@ :190BI65IGH 36@GB_6H.E1F3@B87;; 8 Ú-c.=.<Û v = 6 = HR 2 R

R

:17G5B?2

4 3 πR ρ, 3   8π 2 2 R H − γ ρ = 0, 3 M=

1@AGFB A06@65?3AB; :71@I13@o 8?b?3@8B 81 {3?7?II1a ρ∗ =

3H 2 , 8πγ

-
Ú-c.=.UÛ

5@1 8 @15I13@6 318:BFB?@ 3 1_?IA1a A1@10GH :17G567 ƒ06F 2BI ÚB8@10 @?1066 0B3J60;Hb?a3; {3?7?II1aÛ IB 13I18? 1h b?a @?1066 1@I136@?7oI13@6. XF?3o 21xI1 0?J6@o 2B71a A018oH ?b? 1F6I ™h17oJ1a 81 :013š. „_?IA6 30?FI?a :71@I13@6 8?b?3@8B 81 {3?7?II1a h76TA6 A A06@65?3A12G TIB5?I6H Ú-c.=.Uې :1s@12G :06h76x?II1 62??@ 2?3@1 Ú-c.=.<ې @.? . r dR µ v = = 2γ , Ú-c.=.CÛ dt R 676 √RdR = √2γµdt  5@1 :137? 6I@?906018BI6; Ú1@ 0 F1 R 6 1@ 0 F1 tÛ FB?@ p 2 R = 2γµ t. Ú-c.=.cÛ 3 n?JB; @?:?0o 3182?3@I1 Ú-c.=.Cې Ú-c.=.cÛ 6 dR = HR  :17G5B dt ?2 81T0B3@ {3?7?II1a R

3 2

7?@ . { 6T71x?II12 GF686@?7?I A1I@0B3@ 2?xFG 371xI13@oH TB FB56 6 :013@1@1a 21F?76. t7?2?I@B0I4a F1 IB68I13@6 :1Fk1F :1T817;?@ ABA I6 3@0BII1 918106@o 1 31@810?I66 260B I? 8:BFB; 8 367oI4? :01@6810?56; 3 3180?2?II1a A13217196?a. j6h76; 0BTG2??@3; 13@B?@3; 8I? A1IAG0?I_66. t=

2 ≈ 1010 3H

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A1 − 50◦ , A2 − 25◦

B1 − 75◦ , B2 − 50◦ .

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ïéèëð ùì ïýëìõ  èçì  ì çûõð çï  êïçóöçì  ë éêé  > ìõéé èìöçìí èéò  øéêïèûêï øêìøìê + óìçïõüçï í ùé ) êïëëèìðçóé ùì + éçèêï ìè êïë ëòïèêóôïéòì  èìöîó  . êéùøìõì óò ùïõééí öèì ôëé èéõï óòé  è ìùóçïîìô
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é êï òéê
õ  ýì  ì èéõï ëèêì  ì øêìøìê + óì çïõüç
ïýëìõ  èçì  èéòøéêïèûêé  . ûëèüí çïîìçé + í ëôì  ëèôï òìùéõó èïîì  ôèéòøéêïèûêì
í öèì õ  ýì  øêéùòéèí øéêéçéëéçç
 ó  ìùçì  èìöîó ô ùêû  û  í ë óçì   í èìèöïë øêó ìùóè ô ëìëèìðçóé èéøõìôì  ì êïôçìôéëóð ë çìôì  ìîêûD ïýìø  é  ëêéùì  
èçìí öèì  èï ëéõéççïð ýûùéè îï  ïèüëð é  ì ìýóèïèéõðò ýéë  îìçéöçì   ëïòìò ùéõéí éëõó ìçó øì éõï  è øêóýõó  óèüëð î  êïçóöçì  ëòéêï

éêéí èì ýûùûè ì õï ùïèüëð øì òéêé øêóýõó éçóð ó ûòéçüúïèüëð ô êï   ï  ó ó í ëììèôéèëèôéççìí ýûùûè ûîìêïöóôïèüëðí ó ìçó çóîì ùï çé ëòì  ûè ùìëèóöü  êïçó +
ëôìé  ì òóêï  = ùéëü éëèü çïù öéò øìêï  ò
ëõóèü   ïøêóòéêí çïù èéòí òì  ûè õó ìýóèïèéõó  èì  ì òóêï ìýçïêû óèü ëû  é  ëèôìôïçóé ìùéõüíèéòøéêïèûê îìçéöçìí
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í êïëîêéøì   ïéè  ,  ˆ õð øìçóòïçóð òé ïçó  òï õïèèéê  éîèï çïùì ô
øóë
ôïèü ûêïô  çéçóð ùóçïòóîóí êéúïèü ó í ïçïõó  óêìôïèüí ìúóýïèüëð ó òçì ì êï çï öóçïèü ëçïöïõï  æçèéêéëçìí öèì éëèü ùêû  ì  øûèü ô
ðëçéçóð  ïîìçìòéê  çìëèé ðôõéçóðí çé èêéýû  ó  ó ì  êéççì  îôïõó óîï+ óó  æòééèëð ô ôóùû òéèìù îìçèêìõð êï òéêçìëèé  . êóýõó óèéõüçì èïîì  í îïî ô  ïùïöé ìøêéùéõéçóð øéêóìùï îìõéýïçó  òïðèçóîï   ëõó  ïêïçéé ðëçìíöèì ìè í í íèì øêìýõéòï ëôìùóèëð î øìóëîû îìòýóçï+ óóòìó  éèí  ïôóëéèü í íóòéõóúü  é  êï  òéêçìëèü ôêéòéçó  * ïîïð îìòýóçï + óð éùóçëèôéççï í òïëëï ìîï
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ëóèûï+ óð í ó ìç ôøìõçé ? ëèêì  í îïî ó  óöéëîóí èïî ó òïèéòïèóöéëîó  ’ óõìëì óð  ùéëü ýï  óêû  éèëð çï øêìëèìò ëììýêï éçóó  êéúéçóé  ïùïöó çé òì éè  ïôóëéèü ìè óë øìõü  ûéòì  ëóëèéò
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ôï  èí öèì  ïùïöï ùìõ çï ïêïîèéêó  ìôïèüëð çïýìêìò ýé  êï  òéêç
øïêïòéè  êìôíæèïîí ëôð  ïççêïëëòìèêóò
îïîì  èì ûçî + óìçïõüçì   ïôóëóòìëèü  å  ûøêû ì  ïîêéøõéççìé îê
õì "øõïëèóçûôòéëèì & ô øìèìîé óùéïõüçì  óùîìëèó ìùçìêìùçïðí çéë óòïéòïð óùîìëèü ï ï " çéëîìõüîì ûøêì  ïéè  ïùïöû & ( óèûï + óð ìøêéùéõðéèëð øïêïòéèêïòó %‚  ç& "# # ùé ó ) òìùûõó ç  ï ó ëùôó  ïí ) òïëëï îê
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ò úï  ìò ô øìçóòïçóó  ïîìçìòéêçì ëèéûôéõóöéçóé  õïèèéêï  ( êï  û ðëçìí çïøêóòéêí öèì ûôéõóöéçóé éëèîìëèó ô êï í ô êï& ûôéõóöóôïéè îêóèóöéëîû  ëîìêìëèü èìõüîì ô " êï   å ˆ õð òïðèçóîï ýéêï òéêç
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é òé ïçó  ò
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çóîïî çé ûîõïù
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é ýéõîó ó øêìö & ó îõìçìô í ëøìëìýç
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ïèü  ƒ èì "èïîìé îõìç ) çéëû  éëèôéççìÿ  :õïôçìéí öèì ûçóöèì 

ïð øóèïðëü & í îõìç îïî ý
ûëóõóôïéèëðí øêóìýêéèïð ëøìëìýçìëèü " ëôì  êïýìèû ýìõéé 

éîèóôçì   ï  òýêóìçïõüçì  ëèïùóó ôëõïý
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í îìèìê
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ò "øì ôêéòéçïò øðèóùéëðèóõéèçé ùïôçìëèó & ëõéùëèôóðò   ïøêóòéêí éëõó ô  òýêóìçïõüçì  ëèïùóó øìýé ùïéè í õì  óöçì øêéùõì óèü ùõð øêéìùìõéçóð çéëìôòéëèóòìëèó èîïçé ìêó ó  çïõüç
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ùêû  ìè ùêû ï  ?
ëõü ëûòïëúéùúïðí îïî óøìèé ïí çì ëììèôéèëèôû   óé îëøéêóòéçè
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