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;ukljh_ \\_^_gb_ \ kbkl_fm 0DWKHPDWLFD QZklv , F_lh^bq_kdb_ mdZaZgby ih ki_pbZevghfm dm...
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DZn_^jZ fZl_fZlbq_kdh]h ZgZebaZ
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Kh^_j`Zgb_ Ij_^bkeh\b_ KijZ\hqgbd ih kbkl_f_ 0DWKHPDWLFD ;ukljh_ agZdhfkl\h =jZnbdZ Kibkdb/ \_dlhju/ fZljbpu MjZ\g_gby b aZ^Zqb gZ wdklj_fmf :gZeba Ijh]jZffbjh\Zgb_ :gbfZpby b a\md Ebl_jZlmjZ
5 6 6 9 < 45 47 53 58 59
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Ij_^bkeh\b_ IZd_lu dhfivxl_jghc fZl_fZlbdb kms_kl\_gguf h[jZahf f_gyxl fbj h[jZah\Zgby b gZmdb1 Hgb ^_eZxl ^hklmiguf ©gZ [ulh\hf mjh\g_ª ijbf_g_gb_ fhsguo fZl_fZlbq_kdbo f_lh^h\ ijb j_r_gbb ijbdeZ^guo aZ^Zq/ kms_kl\_ggh ih\urZxl gZ]ey^ghklv b dhgdj_lghklv Z[kljZdlguo dhgp_ipbc dZd \ ijhp_kk_ h[mq_gby/ lZd b \ bkke_^h\Zgbyo1 GZaj_\Z_l ij_h[jZah\Zgb_ mq_[guo dmjkh\ 0 fZl_fZlbq_kdbo/ _kl_kl\_gghgZmqguo b ijbdeZ^guo gZ hkgh\_ bkihevah\Zgby kh\j_f_gguo iZd_lh\ dhfivxl_jghc fZl_fZlbdb/ lZdbo dZd ©0DWKHPDWLFDª/ ©0DSOH 9ª b ^j1 GZ AZiZ^_ lZdh_ ij_h[jZah\Zgb_ m`_ b^zl k gZqZeZ <30o ]h^h\1 Wlh \Z`g_cr__ ihke_ ,QWHUQHW gh\h\\_^_gb_ ihke_^gbo e_l \ h[jZah\Zl_evguo l_ogheh]byo1 < Jhkkbb fZeh^hklmigu dZd mq_[gu_ ihkh[by ih iZd_lZf/ lZd b kZfb iZd_lu/ qlh aZljm^gy_l ijbf_g_gb_ i_j_^h\uo h[jZah\Zl_evguo l_ogheh]bc1 K 4<<:2<; mq_[gh]h ]h^Z <=M jZkiheZ]Z_l ebp_gabhgguf iZd_lhf ©0DWKHPDWLFD 515ª/ ijbq_f ebp_gaby jZaj_rZ_l mklZgh\dm kbkl_fu gZ \k_ dhfivxl_ju \maZ1 GZklhys__ ihkh[b_ ij_^gZagZq_gh ^ey [ukljh]h agZdhfkl\Z k g_dhlhjufb ba \hafh`ghkl_c iZd_lh\ ©0DWKHPDWLFD 5ª b ©0DWKHPDWLFD 6ª1 QZklv , kh^_j`bl [hevrh_ dhebq_kl\h ijbf_jh\/ \ qZklb ,, [m^_l kh^_j`Zlvky \ hkgh\ghf kijZ\hqguc fZl_jbZe1 QblZl_ex fh`gh ihkh\_lh\Zlv ^Ze__ bkihevah\Zlv dgb]b >4/ 5/ 6@ b/ dhg_qgh/ ih[hevr_ jZ[hlZlv kZfhklhyl_evgh1
7
>ey ih\ur_gby wnn_dlb\ghklb kZfhklhyl_evghc ^_yl_evghklb mqZsboky kha^Zg wdki_jbf_glZevguc \ZjbZgl ]bi_jl_dklh\h]h we_dljhggh]h kijZ\hqgbdZ b mq_[gh]h ihkh[by gZ jmkkdhf yaud_ 0 jmdh\h^kl\Z ih iZd_lm %©0DWKHPDWLFD%ª gZ ieZlnhjf_ :LQGRZV1 KijZ\hqgbd ho\Zlu\Z_l h[sb_ ijbgpbiu jZ[hlu k kbkl_fhc %©0DWKHPDWLFD%ª b ijh]jZffbjh\Zgby \ g_c/ hibkZgb_ nmgdpbc kbkl_fu %©0DWKHPDWLFD%ª/ khijh\h`^Z_fh_ ijbf_jZfb/ ZenZ\blguc bg^_dk b l_fZlbq_kdb_ bg^_dku ih lZdbf l_fZf/ dZd %©:gZeba%ª/ %©:e]_[jZ%ª/ %©>bnn_j_gpbZevgu_ mjZ\g_gby%ª/ %©Qbke_ggu_ f_lh^u%ª b l1i1 =jZnbq_kdbc bgl_jn_ck ^_eZ_l bkihevah\Zgb_ kijZ\hqgbdZ ijhkluf b m^h[guf1 Fgh]hhdhgguc bgl_jn_ck :LQGRZV iha\hey_l mqZsbfky b ij_ih^Z\Zl_eyf bkihevah\Zlv iZjZee_evgh kijZ\hqgbd gZ jmkkdhf yaud_ b Zg]ehyauqguc iZd_l %©0DWKHPDWLFD%ª/ qlh j_adh m\_ebqb\Z_l \hafh`ghklb kZfhklhyl_evghc ^_yl_evghklb klm^_glh\ _kl_kl\_gghgZmqguo nZdmevl_lh\ +\dexqZy fZl_fZlbq_kdbc/ IFF b nbabq_kdbc, ih ijbf_g_gbx iZd_lZ %©0DWKHPDWLFD%ª ijb bamq_gbb dZd h[sbo/ lZd b ki_pbZevguo ^bkpbiebg/ Z lZd`_ \ GBJK1 Ijbf_g_gb_ iZd_lZ ©0DWKHPDWLFDª b ]bi_jl_dklh\h]h we_dljhggh]h kijZ\hqgbdZ [ueh [u ihe_aguf b \ rdheZo/ jZkiheZ]Zxsbo ^hklZlhqgh kh\j_f_ggufb ,%00kh\f_klbfufb dhfivxl_jZfb1 < gZklhys__ \j_fy wdki_jbf_glZevguc \ZjbZgl we_dljhggh]h kijZ\hqgbdZ ijhoh^bl ©h[dZldmª 0 bkihevam_lky \ mq_[ghf ijhp_kk_ gZ fZl_fZlbq_kdhf nZdmevl_l_ <=M ijb \_^_gbb aZgylbc ih ki_pdmjkZf dZn_^ju fZl_fZlbq_kdh]h ZgZebaZ/ ijb \uiheg_gbb dmjkh\uo b ^biehfguo jZ[hl1
;ukljh_ agZdhfkl\h1 AZimklbl_ iZd_l 0DWKHPDWLFD1 GZ[_jbl_ 5.5 b gZ`fbl_ deZ\brb 6KLIW . (QWHU +l1_1 m^_j`b\Zy gZ`Zlhc deZ\brm 6KLIW/ gZ`fbl_ (QWHU,1 < hdg_ kbkl_fu \u m\b^bl_ ke_^mxs__= In[1] : = 2 + 2 Out[1] = 4 G_ h]hjqZcl_kv/ _keb \Zf ijb^_lky ih^h`^Zlv g_kdhevdh k_dmg^/ ihdZ ijhbkoh^bl aZ]jmadZ y^jZ kbkl_fu1 KdZ`_f/ lhqgh_ agZq_gb_ qbkeZ 10 2 [m^_l \uqbke_gh ^hklZlhqgh [ukljh/ _keb \u gZ[_j_l_ 5A43 b gZ`f_l_ 6KLIW . (QWHU/ qlh ^Zkl In[2] : = 2^10 Out[2] = 1024. AgZq_gb_ 100 2 jm]hc kihkh[ \ bkihevah\Zgbb nmgdpbb 1> @/ ^Zxs_c ijb[eb`_ggh_ qbke_ggh_ agZq_gb_ Zj]mf_glZ/ gZijbf_j/ lZd In[6] : = N[22/7]
Out[6]
= 3.14286
Kbkl_fZ 0DWKHPDWLFD iha\hey_l k gm`ghc lhqghklvx \uqbkeylv g_dhlhju_ fZl_fZlbq_kdb_ dhgklZglu b bkihevah\Zlv bo1 GZijbf_j/ In[7] : = N[Pi] Out[7] = 3.14159 3i/ 433@ beb 1 >3i, 1000]. B \u fh`_l_/ gZdhg_p/ magZlv/ gZkdhevdh HA3L [hevr_/ q_f 3LAH/ \uqbkeb\ N [E^Pi-Pi^E,10]. 0DWKHPDWLFD < kbkl_f_ fh`gh bkihevah\Zlv fgh]b_ h[s_mihlj_[bl_evgu_ fZl_fZlbq_kdb_ nmgdpbb/ h[hagZq_gby ^ey gbo \iheg_ _kl_kl\_ggu/ l_f g_ f_g__ bo ke_^m_l kljh]h ijb^_j`b\Zlvky1
In[8]: = Exp[3] 3 Out[8] = E In[9]: = N[%] Out[9] = 20.0855 In[10 ] := Log[%] Out[10] = 3. In[11] := Log[E ] Out[11] = 1 In[12] := Log[10,100] Out[12] = 2 In[13]: = Sin[Pi] Out[13] = 0 In[14]:= Tan[Pi/4] Out[ 14] = 1 In[15]:= ArcTan[-1] Out[ 15] = -Pi/4 In[16]: = Sqrt[0.04] Out[16] = 0.2 In[17]: = Sqrt[-1] Out[17] = I H[jZlbl_ \gbfZgb_= 4, Zj]mf_glu nmgdpbc aZdexqZxlky \ d\Z^jZlgu_ kdh[db> 5, bf_gZ nmgdpbc/ \kljh_gguo \ kbkl_fm 0DWKHPDWLFD/ gZqbgZxlky k aZ]eZ\guo [md\1
Qlh[u g_ aZ^_j`b\Zlvky ijb gZ[hj_ [he__ keh`guo \ujZ`_gbc beb dhibjh\Zgbb bo/ ^Z\Zcl_ l_i_jv \hkihevam_fky \hafh`ghklyfb bgl_jn_ckZ +h[hehqdb, kbkl_fu 0DWKHPDWLFD1 >ey gZqZeZ ba f_gx )LOH \u[_jbl_ imgdl 6DYH $V b aZibrbl_ ijhlhdhe ijh\_^_gguo \Zfb jZkq_lh\ \ nZce P\41PD +`_eZl_evgh \ k\hc dZlZeh],1 ?keb \u aZohlbl_ ih\lhjblv dZdh_0lh \uqbke_gb_/ lh \u fh`_l_ mklZgh\blv dmjkhj \klZ\db \ khhl\_lkl\mxs_c kljhd_ ^\hcguf s_eqdhf b gZ`Zlv 6KLIW . (QWHU1 Wlh `_ fh`gh k^_eZlv bgZq_= mklZgh\bl_ ,0 h[jZaguc dmjkhj gZ d\Z^jZlgmx kdh[dm kijZ\Z hl nhjfmeu +dmjkhj ijb wlhf baf_gbl k\hc \b^, b s_edgbl_ h^bg jZa1 Kdh[dZ ©ihq_jg__lª1 L_f kZfuf \u \u^_ebeb yq_cdm/ kh^_j`Zsmx gm`gmx nhjfmem1 L_i_jv \u[_jbl_ \ f_gx kbkl_fu 0DWKHPDWLFD imgdl $FWLRQ/ Z \ g_f imgdl (YDOXDWH 6HOHFWLRQ1 Gm`gh_ \uqbke_gb_ [m^_l \uiheg_gh1 Ijb `_eZgbb \u^_e_ggu_ yq_cdb fh`gh dhibjh\Zlv b jZafgh`Zlv h[uqgufb ^ey kbkl_f k ]jZnbq_kdbf bgl_jn_ckhf ijb_fZfb +dghidZfb beb f_gx,1
5
[ \ ij_^u^msbc j_amevlZl1
In[22]: = %/.{a->x, b -> x^2} 2XW>55@ +[ 0 [A5,A6
=jZnbdZ Kbkl_fZ 0DWKHPDWLFD [h]ZlZ ]jZnbq_kdbfb \hafh`ghklyfb1 Wlh ± ihkljh_gb_ ]jZnbdZ h^ghc nmgdpbb In[23]:= Plot[Sin[x], {x,0,Pi}] 4 31; 319 317 315 318
4
418
5
518
6
Out[23]=-Graphicsbeb g_kdhevdbo kjZam= In[24]:=Plot[{Sin[x],Sin[2x],Sin[4x]},{x,0,Pi}] 4 318
318
4
418
5
518
6
0318 04
Out[24] = -Graphicsihkljh_gb_ ]jZnbdh\ nmgdpbc ^\mo i_j_f_gguo In[25]:=
Plot3D[Sin[x*y],{x,0,Pi}, {y,0,Pi}]
Out[25] = -SurfaceGraphicsiZjZf_ljbq_kdbo ]jZnbdh\ In[26]:=ParametricPlot[{Sin[t],Cos[3*t]},{t,0,2Pi}] 1 0.5
-1
-0.5
0.5 -0.5 -1
1
Out[26] = --GraphicsIn[27]:= ParametricPlot3D[{(4+Cos[t])Cos[u],(4+Cos[t])Sin[u],Sin[t ]},{t,0,2 Pi}, {u,0,2 Pi}, Ticks->None]
Jbk1519 Out [27 ] = -Graphics3DIn[28]:= ParametricPlot3D[{4+(4+u*Cos[t/2])Cos[t],Sin[t/2]*u, (4+u*Cos[t/2])Sin[t]}, { t,0,2 Pi}, {u,-1,1}, Ticks->None]
Jbk151: - Out [28 ] = Graphics3Db kh\f_klguc ihdZa g_kdhevdbo ]jZnbdh\/ ihkljh_gguo ih hl^_evghklb In [29 ]: =Show[%,%%]
Jbk151; Out [29 ] = Graphics3D
Kibkdb/ \_dlhju/ fZljbpu <_dlhju b fZljbpu ij_^klZ\e_gu \ kbkl_f_ 0DWKHPDWLFD F ihfhsvx kibkdh\/ aZdexqZ_fuo \ nb]mjgu_ kdh[db/ b kibkdh\ kibkdh\1 6
In[ ]:= e1={1, 0, 0} Out[ ] = {1, 0, 0} In[ ]:= e2={0, 1, 0} Out[ ] = {{0, 1, 0} In[ ]:= e3={0, 0, 1} Out[ ] = {{0, 0, 1} Imklv In[ ]:= u={a,b,c} Out[ ] = { a, b, c} ?kl_kl\_gguf h[jZahf \\h^ylky hi_jZpbb gZ^ \_dlhjZfb= In[ ]:= v=a*e1+b*e2+ c*e3 Out[ ] = {a, b, c}
4
>_ckl\by k fZljbpZfb b \_dlhjZfb aZ^Zxlky _kl_kl\_gguf h[jZahf1 ey d\Z^jZlghc fZljbpu 0DWKHPDWLFD iha\hey_l gZclb hij_^_ebl_ev/ h[jZlgmx fZljbpm +ijb g_gme_\hf hij_^_ebl_e_,/ kh[kl\_ggu_ agZq_gby b \_dlhju1 Imklv In[ ]:= m2= {{1, 1, 1}, {2, 3, 4}, {4, 9, 16}} Out[ ] = {{1, 1, 1}, {2, 3, 4}, {4, 9, 16}} In[ ]:= MatrixForm [%] Out[ ] = 1 1 1 2 3 4 4 9 16. GZc^_f hij_^_ebl_ev= In[ ]:= Det [m2] Out[ ] =2 GZc^_f h[jZlgmx fZljbpm= In[ ]:=m3= Inverse [m2] Out[ ] = {{6, -(7/2), 1/2}, {-8, 6, -1}, {3, -(5/2),1/2}} In[ ]:= MatrixForm [%] Out[ ] = 6 -(7/2) 1/2 -8 6 -1 3 -(5/2) 1/2 Ijhba\_^_gb_ fZljbpu gZ h[jZlgmx d g_c ^he`gh ^Zlv _^bgbqgmx fZljbpm:
In[ ]:= m3 . m2 Out[ ] ={{1, 0, 0}, {0, 1, 0}, {0, 0, 1}} In[ ]:= MatrixForm [%] Out[ ] = 1 0 0 0 1 0 0 0 1
5
Ijhba\_^_gb_ P6 - P5 lh`_ bf__l kfuke/ gh wlh ± ihwe_f_glgh_ ijhba\_^_gb_ kibkdh\/ Z g_ ijhba\_^_gb_ fZljbp$
In[ ]:= m3 * m2 Out[ ] = {{6, -(7/2),1/2}, {-16, 18, -4}, {12, -(45/2), 8}} In[ ]:= MatrixForm [%] Out[ ] = 6 -(7/2) 1/2 -16 18 -4 12 -(45/2) 8 Kh[kl\_ggu_ agZq_gby b \_dlhju gZoh^ylky [_a hkh[uo keh`ghkl_c +fu g_ ijb\h^bf ]jhfha^db_ hl\_lu,= In[ ]:= Eigenvalues [m2] In[ ]:= Eigenvectors [m2] Ijb g_h[oh^bfhklb bo fh`gh bkdZlv ijb[eb`_ggh +hl\_l [m^_l ^Zg \ \_s_kl\_gguo qbkeZo k fZrbgghc lhqghklvx,= In[ ]:= Eigenvalues [N[m2]] In[ ]:= Eigenvectors [N[m2]] Ihke_ lh]h dZd \u j_rbeb aZ^Zqm k i_j_f_ggufb [/ \/ W b l1i1/ b i_j_^ i_j_oh^hf d gh\hc aZ^Zq_/ ihe_agh ihqbklblv [u\rb_ \ mihlj_[e_gbb kbf\heu dhfZg^hc &OHDU$OO>[/ \/ W b l1i1@
6
MjZ\g_gby b aZ^Zqb gZ wdklj_fmf Kbkl_fZ 0DWKHPDWLFD mki_rgh j_rZ_l jZaghh[jZagu_ mjZ\g_gby b bo kbkl_fu1 MjZ\g_gby aZibku\Zxlky \ kbkl_f_ 0DWKHPDWLFD ihkj_^kl\hf ^\hcgh]h agZdZ jZ\_gkl\Z 1
6
+ x - 2 = 0
In[ ]:= x ∧ 3 + x - 2 == 0 Out[ ]= - 2 + x + x ∧ 3==0 In[ ]:= Solve [%] Out[ ]= -1 - I Sqrt[7] -1 + I Sqrt[7] {{x -> 1}, {x -> -------------- }, {x -> --------------}} 2 2 beb kjZam In[ ]:= Solve [x ∧ 3 + x - 2 == 0] Out[ ]= -1 - I Sqrt[7] -1 + I Sqrt[7] {{x -> 1}, {x -> -------------- }, {x ->-------------- }} 2 2 H[jZlbl_ \gbfZgb_ gZ fgbfmx _^bgbpm , \ aZibkb ^\mo dhjg_c1 Kbkl_fZ 0DWKHPDWLFD gZoh^bl k ihfhsvx dhfZg^u 6ROYH \k_ +\_s_kl\_ggu_ b dhfie_dkgu_, j_r_gby Ze]_[jZbq_kdbo mjZ\g_gbc kl_i_gb g_ \ur_ 7/ ijblhf b ^ey mjZ\g_gbc k iZjZf_ljhf = In[ ]:= Solve [x ∧ 4 + a∗ ∗ x - 2 == 0, x] Out[ ]= … hl\_l kebrdhf ]jhfha^hd/ \u [_a ljm^Z gZc^_l_ _]h kZfb1 J_r_gby Ze]_[jZbq_kdh]h mjZ\g_gby kl_i_gb 8 beb \ur_ fh]ml [ulv gZc^_gu/ \hh[s_ ]h\hjy/ qbke_ggh1 KdZ`_f/ mjZ\g_gb_ [∧8.[05 3 kbkl_fZ 0DWKHPDWLFD j_rbl lhqgh= In[ ]:= Solve [x ∧ 5 + x - 2 == 0] Out[ ] = … 3 0 hldZ`_lky= Z mjZ\g_gb_ [ ∧ 8 . [ 0 : In[ ]:= Solve [x ∧ 5 + x - 7 == 0] Out[ ]= {ToRules[Roots[x^5 + x == 7, x]]} Ijb^_lky j_rZlv mjZ\g_gb_ qbke_ggh= In[ ]:= NSolve [x^5 + x - 7
== 0]
Out[ ]= {{x -> -1.21388 - 0.924188 I}, {x -> -1.21388 + 0.924188 I}, {x->0.508469 -1.36862 I}, {x->0.508469 +1.36862 I}, {x ->1.41081}} LjZgkp_g^_glgu_ mjZ\g_gby j_rZxl/ aZ^Z\Zy gZqZevgh_ ijb[eb`_gb_ d dhjgx1 In[ ]:= FindRoot [Cos[x] == 2∗ x; {x,0}] Out[ ]= {x -> 0.450184} Fh`gh j_rZlv kbkl_fu mjZ\g_gbc= In[ ]:= Solve [{a*x +b*y == g, c*x + d*y == h} ,{x,y}] Out[ ]=
{{x ->
g a
+
b (c g - a h) c g - a h --------------, y -> -( ------------ ) } a (-(b c) + a d) -(b c) + a d
b In[ ]:= FindRoot [{ x + y == Sin[x], x - y == Cos[x], {x,0}, {y,0}] Out[ ]= {x -> 0.704812, y -> -0.0569214} Kbkl_fZ 0DWKHPDWLFD iha\hey_l j_rZlv ^bnn_j_gpbZevgu_ mjZ\g_gby b bo kbkl_fu= In[ ]:= DSolve [y''[x] - 2y'[x] + 1 == 0, y[x], x] x 2 x Out[ ] = {{y[x] -> - + C[1] + E C[2]}} 2 J_r_gb_ kh^_j`bl ijha\hevgu_ ihklhyggu_ K>4@ b &>5@1 Fh`gh gZclb qbke_ggh_ j_r_gb_ aZ^Zqb Dhrb ^ey ^bnn_j_gpbZevguo mjZ\g_gbc beb bo kbkl_f= In[ ]:= NDSolve[{x'[t]==-y[t]-x[t],y'[t]==2x[t]y[t],x[0]==1,y[0]==-1},{x[t],y[t]},{t,0,10}] Out[ ]={{x[t] -> InterpolatingFunction[{0., 10.},<>][t], y[t] -> InterpolatingFunction[{0., 10.}, <>][t]}} In[ ]:= Plot[Evaluate[{x[t],y[t]}/.%],{t,0,10}, PlotRange->All]
4 318
5
7
9
;
43
0318 04
Out[ ]=-GraphicsFh`gh bkdZlv wdklj_fmfu nmgdpbc h^ghc b g_kdhevdbo i_j_f_gguo1
In[ ]:=FindMinimum[Exp[x]*Cos[x] ,{x,0}] Out[ ]= {-0.0670197, {x -> -2.35619}} In[ ]:=FindMinimum[Sin[x]*Cos[y] ,{x,0},{y,0}] Out[ ]= {-1., {x -> -1.5708, y -> 0.}} AZ^Zqb ebg_cgh]h ijh]jZffbjh\Zgby g_[hevrhc jZaf_jghklb fh`gh j_rZlv/ bkihevamy kbkl_fm 0DWKHPDWLFD In[ ]:= ConstrainedMax [19x - 47y + 28z, {x + y+ z > 0, x + y + z < 1, x > 0, y > 0, z > 0}, {x, y, z}] Out[ ]= {28, {x -> 0, y -> 0, z -> 1}}
:gZeba Ohly kbkl_fm ³0DWKHPDWLFD´ b _c ih^h[gu_ ^h\hevgh qZklh gZau\Zxl kbkl_fZfb dhfivxl_jghc Ze]_[ju/ \ gbo lZd beb bgZq_ ij_^klZ\e_gu \k_ nmg^Zf_glZevgu_ jZa^_eu fZl_fZlbdb1 ^Ze__ ± ^_jaZcl_$ >bnn_j_gpbjh\Zlv \ kbkl_f_ ³0DWKHPDWLFD´ g_ ijhklh/ Z hq_gv ijhklh$ < dZq_kl\_ Zj]mf_glh\ dhfZg^u ^bnn_j_gpbjh\Zgby '>1/1@ gm`gh mdZaZlv lm nmgdpbx/ dhlhjmx fu gZf_j_gu ijh^bnn_j_gpbjh\Zlv/ b lm i_j_f_ggmx +beb i_j_f_ggu_,/ ih dhlhjhc +dhlhjuf, ke_^m_l ^bnn_j_gpbjh\Zlv1
[Q
: \hl lZd \uqbkey_lky qZklgZy ijhba\h^gZy
∂VLQ+[\], ∂]
nmgdpbb VLQ+[\], ih i_j_f_gghc ] = In[ ]:=D[Sin[x y z],z] Out[ ]=x y Cos[x y z] Kf_rZggu_ qZklgu_ ijhba\h^gu_ lZd`_ \uqbkeyxlky [_a ijh[e_f ± \hl ∂ 7VLQ+[\], = ∂[ 5 ∂\∂]
In[ ]:=D[Sin[x y z],z,y,x,x] Out[ ]= -5*x*y^2*z^2*Cos[x*y*z] - 4*y*z*Sin[x*y*z] + x^2*y^3*z^3*Sin[x*y*z] Wlh `_ fh`gh aZibkZlv bgZq_= In[ ]:=D[Sin[x y z],{x,2},y,z] Out[ ]= -5*x*y^2*z^2*Cos[x*y*z] - 4*y*z*Sin[x*y*z] + x^2*y^3*z^3*Sin[x*y*z] Iheguc ^bnn_j_gpbZe \uqbkey_lky ihkj_^kl\hf dhfZg^u 'W= In[ ]:=Dt[Sin[x y z]] Out[ ]= Cos[x y z] (y z Dt[x] + x z Dt[y] + x y Dt[z])
In[ ]:=Dt[f[Sin[x y z]]] Out[ ]= Cos[x y z] (y z Dt[x] + x z Dt[y] + x y Dt[z]) f'[Sin[x y z]] A^_kv 'W>[@/ 'W>\@ b 'W>]@ 0 ^bnn_j_gpbZeu i_j_f_gguo [/ \ b ]1
DhfZg^Z 'W ijbf_gy_lky b ^ey \uqbke_gby iheguo ijhba\h^guo nmgdpbc fgh]bo i_j_f_gguo= In[ ]:=Dt[f[Sin[x y z]],x] Out[ ]= Cos[x y z] (y z + x z Dt[y, x] + x y Dt[z, x]) f'[Sin[x y z]] A^_kv 'W>\/[@ b 'W>]/[@ 0 ihegu_ ijhba\h^gu_ i_j_f_gguo \ b ] ih i_j_f_gghc o1 Gh fh`gh b kbkl_fm ³0DWKHPDWLFD´ ©ihkZ^blv \ dZehrmª1 HgZ ©agZ_lª/ qlh _[_ 0 g_^bnn_j_gpbjm_fZy nmgdpby/ b hldZau\Z_lky \uqbkeylv __ ijhba\h^gmx ^Z`_ \ l_o lhqdZo/ ]^_ nmgdpby ^bnn_j_gpbjm_fZ1 LZd kihdhcg__$ In[ ]:=D[Abs[x],x]/.x->1 Out[ ]= Abs'[1] Bgl_]jbjh\Zgb_ \ kbkl_f_ ³0DWKHPDWLFD´ +dZd b \ `bagb, keh`g__ ^bnn_j_gpbjh\Zgby1 NhjfZevgh \k_ ijhklh= g_hij_^_e_gguc bgl_]jZe \uqbkeyxl ihkj_^kl\hf dhfZg^u ,QWHJUDWH= ^ey ∫ [ Q G[ ihemqZ_f
In[ ]:=Integrate[x^n,x]
Out[ ]=
1 + n x -----1 + n
Gh In[ ]:=Integrate[Sin[Sin[x]],x] ^Zkl Out[ ]= Integrate[Sin[Sin[x]],x] LZdbf h[jZahf/ kbkl_fZ ³0DWKHPDWLFD´ hldZaZeZkv [jZlv g_[_jmsbcky bgl_]jZe1 Gh b g_ \k_ [_jmsb_ky bgl_]jZeu kbkl_fZ ³0DWKHPDWLFD´ ]hlh\Z [jZlv= \ hl\_l gZ aZ^Zgb_ In[ ]:= Integrate[1/(1+x^4+x^9+x^11),x] kbkl_fZ ³0DWKHPDWLFD´ ihdZ`_l \k_/ qlh hgZ fh`_l \ ^Zgghf kemqZ_ +kh\k_f dZd q_eh\_d,= Out[ ]= Integrate[(15/16 - (7*x)/8 + (13*x^2)/16 (3*x^3)/4 + (5*x^4)/8 - x^5/2 + (3*x^6)/8 - x^7/4 + x^8/8 - x^9/16)/ (1 - x + x^2 - x^3 + 2*x^4 - 2*x^5 + 2*x^6 - 2*x^7 + 2*x^8 - x^9 + x^10) , x] + Log[1 + x]/16 Dhg_qgh/ \ kemqZ_ hij_^_e_gguo bgl_]jZeh\ m^Z_lky k^_eZlv ih[hevr_ 0 dhfZg^Z ,QWHJUDWH iha\hey_l \uqbkeylv b hij_^_e_ggu_ bgl_]jZeu/ Z ^ey l_o/
dhlhju_ g_ [_jmlky k ihfhsvx dhfZg^u ,QWHJUDWH/ bf__lky dhfZg^Z 1,QWHJUDWH/ iha\heyxsZy \uqbkeylv hij_^_e_ggu_ bgl_]jZeu ijb[eb`_ggh1 E
BlZd1 \uqbkebf
∫ OQ[G[ D
In[ ]:=Integrate[Log[x],{x,a,b}] Integrate::gener: Unable to check convergence. Out[ ]= -(a (-1 + Log[a])) + b (-1 + Log[b]) Gh g_[_jmsb_ky bgl_]jZeu kbkl_fZ \havf_l lhevdh qbke_ggh= In[ ]:=Integrate[Sin[Sin[x]],{x,0,Pi}] General::intinit: Loading integration packages -- please wait. Out[ ]= Integrate[Sin[Sin[x]], {x,0,Pi }] In[ ]:=NIntegrate[Sin[Sin[x]], {x,0,Pi }] Out[ ]= 1.78649 Fh`gh \uqbkeylv ih\lhjgu_ bgl_]jZeu/ gZijbf_j/ 4
4− [ 5
∫ G[ ∫ +[ 3
5
+ \ 5 ,G\
− 4− [ 5
In[ ]:=Integrate[x^2+y^2,{x,0,1},{y,-Sqrt[1-x^2],Sqrt[1x^2]}] Pi Out[ ]= — 4 b 4
4− [ 5
∫ G[ ∫ 4 G\
−4
− 4− [ 5
In[ ]:=Integrate[1,{x,-1,1},{y,-Sqrt[1-x^2],Sqrt[1-x^2]}] Out[ ]= Pi Ijb bgl_]jbjh\Zgbb kbkl_fZ ³0DWKHPDWLFD´ fh`_l aZ[em`^Zlvky ± kh\k_f dZd q_eh\_d1 LZd i_j\hh[jZagZy nmgdpby ^ey [_a mdZaZgby ijhf_`mldh\/ gZ dhlhjuo wlh \_jgh1
4
[5
[m^_l gZc^_gZ dZd
: ^ey
4
∫ 7 + VLQ[G[
4 [
fu
ihemqbf ihkj_^kl\hf dhfZg^u ,QWHJUDWH>42+7.6LQ>[@,/[@ j_amevlZl/ g_ y\eyxsbcky i_j\hh[jZaghc ih^ugl_]jZevghc nmgdpbb1 LZdh\Zy i_j\hh[jZagZy h[yaZgZ [ulv g_ij_ju\ghc/ ihemq_ggZy `_ nmgdpby bf__l jZaju\u \ lhqdZo \b^Z 5+N + 4,Π/ \ q_f e_]dh m[_^blvky/ ihkljhb\ ]jZnbd ihkj_^kl\hf dhfZg^u 3ORW>(/^[/053L/53L`@ b ijhZgZebabjh\Z\ ih\_^_gb_ ihemq_gghc nmgdpbb \ mdZaZgguo lhqdZo1
>ey \uqbke_gby kmff \ kbkl_f_ ³0DWKHPDWLFD´ bf__lky dhfZg^Z 6XP= In[ ]:=Sum[x^i/i,{i,1,9}] 2
Out[ ]= x +
x - + 2
3 x - + 3
4 x - + 4
5 x - + 5
6 x - + 6
7 x - + 7
8 9 x x - + 8 9
In[ ]:=Sum[x^i y^j,{i,1,2},{j,1,3}] 2 2 2 2 3 2 3 Out[ ]=x y + x y + x y + x y + x y + x y Kbkl_fZ ³0DWKHPDWLFD´ mki_rgh jZaeZ]Z_l nmgdpbb \ jy^ L_cehjZ= In[ ]:= Series[Sin[x],{x,0,3}] 3 x 4 Out[ ]= x - - + O[x ] 6 Hl[jZku\Zlv hklZlhqguc qe_g iha\hey_l dhfZg^Z 1RUPDO= In[ ]:= Normal[%] 3 x Out[ ]= x - 6 In[ ]:= Normal[Series[Sin[x],{x,0,5}]]
3
5
x x Out[ ]= x - - + 6 120 In[ ]:= Normal[Series[Sin[x],{x,0,7}]] 3 5 x x Out[ ]=x - - + 6 120
-
7 x 5040
7
5
4
5
6
7
8
9
05
07
0*UDSKLFV0
Kbkl_fZ ³0DWKHPDWLFD´ fh`_l \uqbkeylv ij_^_eu 0 aZf_qZl_evgu_ b g_ hq_gv= In[ ]:= Limit[Sin[x]/x,x->0] Out[ ]=1 GZ ihiuldm ih^klZ\blv 3 \ \ujZ`_gb_ kbkl_fZ ³aZjm]Z_lky´= In[ ]:= Sin[x]/x /.x->0 1 Power::infy: Infinite expression - encountered. 0 Infinity::indet: Indeterminate expression 0 ComplexInfinity encountered. Out[ ]=Indeterminate ohly ijb ^jm]bo agZq_gbyo [ agZq_gb_ \ujZ`_gby [m^_l \uqbke_gh= In[ ]:= Sin[x]/x /.x->1 Out[ ]=Sin[1] In[ ]:= Sin[x]/x /.x->2 Pi Out[ ]=0 Infinity] Out[ ]=E In[ ]:= Limit[(1+1/x)^x,x->-Infinity] Out[ ]=E : \hl ijbf_j ihkeh`g__=
H 42+[ − D, gZ ij_^eh`_gb_ \uqbkeblv OLP [→D kbkl_fZ ³0DWKHPDWLFD´ ^Zkl qZklbqgh \_jguc hl\_l= In[ ]:= Limit[E^(1/(x-a)),x->a] Out[ ]=Infinity HgZ mki_rgh \uqbkebl e_\uc b ijZ\uc ij_^_eu/ In[ ]:= Limit[E^(1/(x-a)),x->a,Direction->1]
Out[ ]=0 In[ ]:= Limit[E^(1/(x-a)),x->a,Direction->-1] Out[ ]=Infinity gh lZd b g_ kfh`_l ihgylv/ qlh ^\mklhjhgg_]h ij_^_eZ g_ kms_kl\m_l In[ ]:= Limit[E^(1/(x-a)),x->a] Out[ ]=Infinity Dhg_qgh/ \ dgb]_ K10] Out[ ]=Limit[Abs[x], x -> 0] ohly ³0DWKHPDWLFD 6131´ wlh m`_ agZ_l ´ In[ ]:= Limit[Abs[x],x->0] Out[ ]= 0 b ^Z`_ fh`_l \uqbkeblv ij_^_e In[2]:=Limit[(Abs[a+dx]-Abs[a])/dx,dx->0]/.a->1 Out[2]=1 Gh ^bnn_j_gpbjh\Zlv nmgdpbx $EV k ihfhsvx dhfZg^u ' hldZ`_lky b ³0DWKHPDWLFD 6131´ M\u/ g_l \ `bagb kh\_jr_gkl\Z$ @@D D
/LPLW $EV[ /[ ® 4
4
Ijh]jZffbjh\Zgb_ ³0DWKHPDWLFD´ < kbkl_f_ \hafh`gu g_kdhevdh klbe_c ijh]jZffbjh\Zgby= nmgdpbhgZevgh_ ijh]jZffbjh\Zgb_/ ijh]jZffbjh\Zgb_/ hkgh\Zggh_ gZ ijZ\beZo ij_h[jZah\Zgbc/ ijhp_^mjgh_ ijh]jZffbjh\Zgb_/ ijh]jZffbjh\Zgb_/ hkgh\Zggh_ gZ h[jZ[hld_ kibkdh\/ ijh]jZffbjh\Zgb_/ hkgh\Zggh_ gZ h[jZ[hld_ kljhd/ h[t_dlgh0hjb_glbjh\Zggh_ ijh]jZffbjh\Zgb_ b/ dhg_qgh `_/ \hevguc +kf_rZgguc, klbev1 HklZgh\bfky gZ g_dhlhjuo ba gbo1 Kmlv nmgdpbhgZevgh]h ijh]jZffbjh\Zgby aZdexqZ_lky \ lhf/ qlh[u \uibkZlv nmgdpbx +\ rbjhdhf kfuke_ keh\Z,/ \uiheg_gb_ dhlhjhc b ^Zkl lj_[m_fuc j_amevlZl1 >Zlv hij_^_e_gb_ gh\hc nmgdpbb \_kvfZ ijhklh= In[1]:= dlina[x_,y_,z_]:=Sqrt[x^2+y^2+z^2] H[jZlbl_ \gbfZgb_ gZ ih^q_jdb\Zgby ijb Zj]mf_glZo \ e_\hc qZklb hij_^_e_gby nmgdpbb GOLQD b gZ hlkmlkl\b_ ih^q_jdb\Zgbc \ ijZ\hc qZklb1 >Z\ hij_^_e_gb_/ fh`gh \uqbkeylv agZq_gby nmgdpbb/ In[2]:= dlina[1,3.0,4] Out[2]= 5.09902 In[3]:= dlina[a,b,a-b] 2 2 2 Out[3]= Sqrt[a + (a - b) + b ] ^bnn_j_gpbjh\Zlv __ In[4]:= D[dlina[x,y,z],x] x Out[4]=-----------------2 2 2 Sqrt[x + y + z ] bgl_]jbjh\Zlv __ b l1i1 In[5]:= Integrate[dlina[x,y,z],x] Out>8@ gZc^bl_ kZfb1 Fh`gh lj_[h\Zlv/ qlh[u Zj]mf_glu nmgdpbb [ueb hij_^_e_ggh]h lbiZ= In[6]:=dlinaR[x_Real,y_Real,z_Real]:=Sqrt[x^2+y^2+z^2] Nmgdpby/ ihemq_ggZy \ j_amevlZl_ lZdh]h [he__ kljh]h]h hij_^_e_gby/ khhl\_lkl\_ggh/ [he__ ©dZijbagZª= In[7]:= dlinaR[1,2.,3.] Out[7]= dlinaR[1, 2., 3.] In[8]:= dlinaR[1.,2.,3.] Out[8]= 3.74166
Ke_^mxsZy kljhdZ ij_^klZ\ey_l j_ZebaZpbx f_lh^Z GvxlhgZ qbke_ggh]h hlukdZgby dhjgy mjZ\g_gby/ \uiheg_ggmx \ klbe_ nmgdpbhgZevgh]h ijh]jZffbjh\Zgby= In[9]:=newton[f_,x0_]:=FixedPoint[(#-f[#]/f'[#])&,N[x0]] A^_kv & ij_^klZ\ey_l kh[hc nhjfZevguc Zj]mf_gl \uibkZgghc \ kdh[dZo Zghgbfghc +[_aufygghc, nmgdpbb/ hkms_kl\eyxs_c rZ] bl_jZpbb/ ) hlf_qZ_l dhg_p aZibkb wlhc Zghgbfghc nmgdpbb/ nmgdpby )L[HG3RLQW>K/H[SU@ hkms_kl\ey_l ihke_^h\Zl_evgh_ ijbf_g_gb_ +kmi_jihabpbx, nmgdpbb K/ gZqbgZy k \ujZ`_gby H[SU/ ihdZ ihemqZ_fu_ agZq_gby g_ i_j_klZgml baf_gylvky/ nmgdpby 1 h[_ki_qb\Z_l \uiheg_gb_ qbke_gguo +g_ kbf\hevguo, \uqbke_gbc1 >Zggh_ hij_^_e_gb_ iha\hey_l bkdZlv dhjgb mjZ\g_gbc= In[10]:=g[x_]:=x^3-2x In[11]:= newton[g,{0,2,-2}] Out[11]= {0, 1.41421, -1.41421} Fu \b^bf/ qlh \ klbe_ nmgdpbhgZevgh]h ijh]jZffbjh\Zgby \Z`g_crmx jhev b]jZxl kihkh[u hj]ZgbaZpbb ihke_^h\Zl_evgh]h ijbf_g_gby +kmi_jihabpbc, nmgdpbc/ ^ey q_]h \ kbkl_f_ ³0DWKHPDWLFD´ bf__lky [h]Zluc gZ[hj kj_^kl\1 Hq_gv \Z`gZ jZagbpZ f_`^m g_f_^e_gguf ijbk\Zb\Zgb_f OHIW ULJKW/ dhlhjh_ \uihegy_lky \ fhf_gl \uiheg_gby ijbk\Zb\Zgby/ b hleh`_gguf ijbk\Zb\Zgb_f OHIW= ULJKW/ dhlhjh_ \uihegy_lky dZ`^uc jZa/ dh]^Z bkihevam_lky agZq_gb_ OHIW1 Wlh ohjhrh \b^gh \ ke_^mxs_f ijbf_j_/ ]^_ 5DQGRP>@ ± dhfZg^Z ± ^Zlqbd kemqZcguo qbk_e1 In[12]:= x:=Random[] In[13]:= y=Random[] Out[13]= 0.199496 In[14]:= {x,x,y,y} Out[14]= {0.784326, 0.453175, 0.199496, 0.199496}
;hevrb_ \hafh`ghklb ij_^hklZ\ey_l ijh]jZffbjh\Zgb_/ hkgh\Zggh_ gZ ijZ\beZo ij_h[jZah\Zgbc1 Ijbf_g_gb_ ijZ\beZ ij_h[jZah\Zgby +beb ijZ\be, UXOH d \ujZ`_gbx H[SU h[hagZqZ_lky dZd H[SU21UXOH1 Ijbf_jhf ijZ\beZ ij_h[jZah\Zgby fh`_l kem`blv h[uqgZy ih^klZgh\dZ= In[1]:= x^4-a^4 4 4 Out[1]= -a + x In[2]:= %/.x->a Out[2]= 0 ?s_ h^gh ihe_agh_ kj_^kl\h ± mkeh\b_/ h[hagZqZ_fh_ dZd H[SU2>FRQG In[2]:= delta[i_Integer,j_Integer]:=0/;i!=j delta[i_Integer,j_Integer]:=1/;i==j In[3]:= {delta[2,3], delta[1,1], delta[2.5,3]}
Out[3]= {0,1,delta[2.5, 3]} 1,{_,_}->0}] qlh ^Z_l In[5]:={delta2[2,3], delta2[2,2], delta2[2.5, 3]} Out[5]= {0,1, delta2[2.5, 3]} < hij_^_e_gbb GHOWD5 \Z`gu jZagbpZ f_`^m iZjZfb ^[B/[B` b ^B/B`/ Z lZd`_ ihjy^hd/ \ dhlhjhf aZ^Zgu ijZ\beZ ih^klZgh\db ^^[B/[B`0!4/^B/B`0!3` ± ijhlb\hiheh`guc ihjy^hd ^Zkl ^jm]hc j_amevlZl1 GZdhg_p/ fu fh`_f \hafmlblvky ©g_ohjhrbfª ih\_^_gb_f \kljh_gghc nmgdpbb /RJ/ ih^jZamf_\Zxs_c/ \hh[s_ ]h\hjy/ dhfie_dkguc Zj]mf_gl/ b ihkljhblv kh[kl\_gguc/ [he__ ]b[dbc eh]Zjbnf OQ g_ dhfie_dkgh]h Zj]mf_glZ= In[6]:={Log[E^z] z Out[6]= Log[E ] In[7]:= Log[a b c d E] Out[7]= Log[a b c d E] In[8]:= ClearAll[x,y,ln,f] +- qbkldZ jZg__ \\_^_gguo hij_^_e_gbc b ijZ\be -, In[9]:= ln[E]=1;ln[1]=0;ln[1.0]=0; ln [y_^x_]=x*ln[y]; ln[x_*y_]=ln[x]+ln[y];ln[x_Real]=Log[x] In[10]:= Out[10]= In[11]:= Out[11]= In[12]:= Out[12]= In[13]:= Out[13]= In[14]:=
ln[a b c d E^f] f + ln[a] + ln[b] + ln[c] + ln[d] ln[10.0] 2.30259 Log[10.0] 2.30259 ln[I] ln[I] Log[I] I Out[14]= - Pi 2 >ey wlh]h eh]ZjbnfZ OQ fh`gh lZd`_ aZ^Zlv ijhba\h^gmx b g_hij_^_e_gguc bgl_]jZe= In[15]:= Derivative[1][ln]:=(1/(#))& In[16]:= D[ln[x],x] 1
Out[16]= X In[17]:= Unprotect[Integrate] Out[17]= {Integrate} In[18]:= Integrate[ln[x_],x_]:=x ln[x]-x In[19]:= Protect[Integrate] Out[19]= {Integrate} In[20]:= Integrate[ t+2ln[t],t] 2 t Out[20]=- + 2 (-t + t ln[t]) 2 ohly b ihke_ wlh]h g_dhlhju_ ijh[e_fu hklZgmlky/ gZijbf_j/ bgl_]jbjh\Zgb_f= In[20]:= Integrate[2t*ln[t],t] Out[20]= 2 Integrate[t ln[t], t] +bo fh`gh meZ^blv/ hij_^_eb\ k\hx nmgdpbx bgl_]jbjh\Zgby,1
k
>ey l_o/ dlh ijb\ud d ijhp_^mjghfm ijh]jZffbjh\Zgbx b g_ ohq_l hl g_]h hldZau\Zlvky/ \ kbkl_f_ ³0DWKHPDWLFD´ ij_^hklZ\e_gu rbjhdb_ \hafh`ghklb1 Wlh ± dhfZg^u ,I/ :KLFK/ 'R/ :KLOH/ )RU/ 0RGXOH/ %ORFN b ^jm]b_1
: \hl \uqbke_gb_ nZdlhjbZeZ= In[3]:= fact[n_Integer]:= Module[{s},If[n>=0,s:=1;Do[s=s*i,{i,n}];s, Print["n<0"]]] In[4]:= {fact[3],fact[4],fact[0],fact[3.0],fact[-2]} n<0 Out[5]= {6, 24, 1, fact[3.], Null} Dhg_qgh `_/ \k_ klbeb ijh]jZffbjh\Zgby \aZbfgh ijhgbdZxl b \aZbfh^_ckl\mxl> khhl\_lkl\_ggh/ gZb[he__ wnn_dlb\guf ^ey \Zk y\ey_lky
\hevguc klbev/ dhlhjuc \u kZfb fh`_l_ \ujZ[hlZlv ^ey k_[y/ kf_rZ\ qbklu_ klbeb \ gm`ghc ijhihjpbb1 Gh agZlv ^ey wlh]h gm`gh fgh]h1 @_eZ_f mki_oZ$
:gbfZpby b a\md Djhf_ klZlbq_kdbo ]jZnbq_kdbo bah[jZ`_gbc ³0DWKHPDWLFD´ fh`_l lZd`_ ijhba\h^blv ZgbfZpbx1 :gbfZpby ihemqZ_lky ba ihke_^h\Zl_evghklb dZ^jh\ iml_f bo [ukljhc ijhdjmldb1 DZ`^uc dZ^j ± wlh ]jZnbq_kdh_ bah[jZ`_gb_ / dhlhjh_ fh`_l [ulv ihemq_gh ijb ihfhsb ]jZnbq_kdbo nmgdpbc iZd_lZ ³0DWKHPDWLFD´1 < bgl_jn_ck_ ³0DWKHPDWLFD´ ^ey :LQGRZV ZgbfZpby ^_eZ_lky lZd= jZaf_klbl_ dZ^ju \ ihke_^h\Zl_evghklb yq__d/ aZl_f \u^_ebl_ yq_cdb b aZimklbl_ ZgbfZpbx k ihfhsvx dhf[bgZpbb deZ\br &WUO.<1 Dh]^Z \u \\h^bl_ ihke_^h\Zl_evghklv dZ^jh\ ^ey ZgbfZpbb/ \Z`gh/ qlh[u dZ^ju [ueb kh\f_klbfufb1 >ey wlh]h/ gZijbf_j/ ke_^m_l mdZau\Zlv y\gu_ agZq_gby ^ey hipbb 3ORW5DQJH \f_klh agZq_gby ih mfheqZgbx $XWRPDWLF/ qlh[u fZkrlZ[ \ jZaguo dZ^jZo [ue h^bgZdh\uf1 Ijbf_j ihke_^h\Zl_evghklb dZ^jh\ ^ey ZgbfZpbb In[]:= Do [Plot3D[Sin[c*2Pi+Sqrt[x^2+y^2]], {x,-Pi,Pi},{y,Pi,Pi},PlotPoints->30, PlotRange-> {Automatic,Automatic, {-1,1}}],{c,0,1,0.1}]
i_j\uc dZ^j ihke_^h\Zl_evghklb
Qlh[u ihemqblv \ ³0DWKHPDWLFD´ a\md/ h[uqgh bkihevamxl dhfZg^m 3OD\1 DhfZg^Z 3OD\>I>W@/ ^W/ WPLQ/ WPD[`@ \u^Z_l a\md/ Zfieblm^Z dhlhjh]h hij_^_ey_lky dZd nmgdpby I \j_f_gb W \ k_dmg^Zo hl WPLQ ^h WPD[1 Ijhba\_^_f ]Zjfhgbdm qZklhlu 4333 =p ^ebl_evghklvx \ h^gm k_dmg^m1 In[]:= Play[Sin[2 Pi 1000 t], {t, 0, 1}] Out[]= -Sound-
Ebl_jZlmjZ 41 :ROIUDP 61 ³0DWKHPDWLFD´ $ V\VWHP IRU 'RLQJ 0DWKHPDWLFV E\ &RPSXWHU10 $GGLVRQ0:HVOH\ 3XEOLVKLQJ &RPSDQ\/ 4<<41 51 vydhgh\ <1I1 Kbkl_fu kbf\hevghc fZl_fZlbdb 0DWKHPDWLFD 5 b 0DWKHPDWLFD 610 F1= KD IJ?KK/ 4<<;1
KhklZ\bl_eb= BajZbe_\bq Ydh\ :jhgh\bq Kdey^g_\ :gZlhebc K_j]__\bq J_^Zdlhj ;mgbgZ L1 >1
