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1 Mathematica , Mathematica Mathematica 1 Mathematica 2 2 Mathematica

2 sin x x 2 y ( ) Mathematica Mathematica 1 Reduce TEX HTML Maple 2 Mathematica Maple MuPAD 3 MuPAD Maple 1 Mathematica 2 Maple 3 MuPAD 2

3 Mathematica WWW 2 Mathematica Mathematica Apple Macintosh Sun Solaris Linux Microsoft Windows Windows NT Mathematica UNIX X Mathematica X Mathematica Mathematica mkdir mathmath cd mathmath pwd mathmath mathematica & Mathematica & Mathematica Mathematica Mathematica 3

4 Mathematica GUI (Graphical User Interface) Microsoft Windows Macintosh Mathematica 3 Mathematica Mathematica 1+2/3 Shift Return ( Enter... ) Return Backspace Shift Return Return Shift Return Return Shift Mathematica Mathematica π In[1]:= Out[2]= Out[1]+1/3 Shift Return Out[1] %1 %1+1/3 4

5 Out[2] Out[3] % %% 250*6 200*12 %%+% X Windows macintosh Next Shift Return In[..] Out[..] a:=2/3+1 := : =?a a := b=3/7+1 = b := = := = c b+c ( c ) c b 2nd 2 nd the2nd number1 the cow 5

6 the cow thecow anapple anapple Sin Log?Num* Num?*ber* ber 5/ /3//N 0.0 //N N[5/99,30] 5/ N[1/3,1] 0.3 %* N[1/3,1]*2 5/3//N N[5/3] postfix full form Prefix N@5/3 N[5/99,30] 5/99~N~30 infix 1+2 Help 6

1 Log 1: 4 + ( ), - ( ), * ( ), / ( ) ^ 2 3 2^3 * 2 5 10 25 25

7 1 Log 1: 4 + ( ), - ( ), * ( ), / ( ) ^ 2 3 2^3 * (2+3) 1+2/5 2/5+1 2/5 1 (1+2)/5 2/(5+1) 2/5 3 (2/5) 3 2/(5 3) 7

8 *^ 3*^ *^ *^ 20 Sin[3.14] Mathematica Sqrt ( ) Abs ( ) Log( ) Log Log[10,100] Log[10,100] log π Pi e E Sin[Pi] 0 Sin[ ] Sin[Pi//N] 0 Log[E] Log[ ] Pi E Mathematica N 5 ( ) x y x y Clear[x] x Mathematica (Mathematica File Exit ) x^2 + x y + y^2-4 y x x y x y * xy yx //Simplify x^2 + 2x y + y^2 // Simplify 8

9 Sin[x+y]+Sin[x-y] // Simplify Sin[x]^2+Cos[x]^2 // Simplify Simplify Simplify = Sqrt[3 + 2Sqrt[2]] // Simplify (Sqrt ) Simplify ( ) FullSimplify Sqrt[3 + 2Sqrt[2]] // FullSimplify Factor Expand Factor Expand TrigFactor TrigExpand Sin[2x] + Sin[2y] // Factor 2Cos[x-y]Sin[x+y] Factor TrigFactor Factor[Sin[2x]+Sin[2y], Trig->True] TrigFactor 1/(1-x)+O[x]^3 O O 3 O[x]^n x n 1/(1-x)+O[x]^3 //Normal x=0 O[x]^3 Series[1/(1-x),{x,0,2}] 9

10 1/(1 x) x = 0 x 2 Normal O[x] x = 0 Series Infinity x Series[1/(1-x),{x,Infinity,2}]//Normal 6 Plot[Sin[x],{x,-Pi,Pi}] 2 sin x : sin x {x,-pi,pi} x π π Pi x Sin Cos Tan 1-x^2 x/(1-x) Tan x 10

11 Plot[1/x, {x,-2,2}, PlotRange->{-3,3}] PlotRange->{-3,3}??Plot Plot Log[x] (log e x) Sqrt[x] ( x) x (-1)^x (( 1) x ) Re Im Abs Conjugate Plot[{Re[(-1)^x],Im[(-1)^x]},{x,0,5}] ( 1) x f:=x^2 Plot[f,{x,-1,1}] f:=x^2 x^2 f Plot[1-f,{x,-1,1}] f x^2 1-f 1-x^2 graph1=plot[f,{x,-1,1}] graph1 Show[graph1] Show[graph1,PlotRange->{-1,3}] Show graph2=plot[f-0.5,{x,-1,2}] Show[graph1,graph2] graph1 graph2 x f1 f2 f3 f2 x = 0 x 2 f3 x = 1/x 2 11

12 : f1=1/(x^2+1) f2=series[f1,{x,0,2}]//normal f3=series[f1,{x,infinity,-2}]//normal Plot[{f1,f2,f3},{x,-1,6},PlotRange->{0,1},GridLines->Automatic] Plot f2 f3 f1 Plot GridLines->Automatic 3 x PlotRange Plot[{f1,f2},{x,-0.5,0.5}] Plot[{f1,f3},{x,3,10}] {x,-1,6} PlotRange->{0,1} Plot aaa = Sequence[{x,-1,6}, PlotRange->{0,1}, GridLines->Automatic] Plot[{f1,f2,f3},Evaluate[aaa]] Sequence Plot Plot[{f1,f2,f3},aaa] aaa Plot Evaluate Plot rrr={x,-0.5,0.5} rrr Plot 12

13 Evaluate[bbb] z x y 4: x 2 y 2 Plot3D g:=x^2-y^2 graph3=plot3d[g,{x,-1,1},{y,-1,1}] x y 3 AxesLabel LightSources Plot3D[g,{x,-1,1},{y,-1,1}, AxesLabel->{"x","y","z"}, LightSources -> {{{1, 0.5, 0.3}, RGBColor[1,1,1]}}] 4 x, y, z {1, 0.5, 0.3} RGBColor[1,1,1] RGBColor

14 2 LightSources LightSources -> {{{1, 0.5, 0.3}, RGBColor[1,0,1]}, {{1, -0.5, 0.3}, RGBColor[0,1,0]}} Mathematica Mathematica PlotPoints Plot3D[g,{x,-1,1},{y,-1,1},PlotPoints->25] x 50 y 10 PlotPoints->{50,10} x x x2 + y 2 3 x ( ) x y Clear[x,y] r = Sqrt[x^2 + y^2] foutside = x/r^3 - x finside = 0 f = If[r>1, foutside, finside] Clear 3 If 2 3 f x y Plot3D ViewPoint 14

15 2 V Y X : Plot3D[f,{x,-2,2},{y,-2,2},PlotPoints->35, ViewPoint->{2.750, , 0.020}] Input 3D ViewPoint Selector Past ViewPoint Z ViewPoint ( See also: ) 6 ContourPlot Plot3D Contours->30 PlotPoints->50 ContourLines->False PlotRange 7 v=1/sqrt[(x-1)^2+y^2]-1/sqrt[(x+1)^2+y^2] v2=arctan[v] <<Graphics PlotField PlotGradientField[v2,{x,-2,2},{y,-2,2}] 15

16 : ( ) ( ) v2 x y V2 v2 7: 2 Sin Plot (Built-in Functions) Mathematica (Add-ons) <<Graphics PlotField 16

17 Graphics PlotField PlotGradientField Help Browser 8 Mathematica (Notebook) (cell) (bracket) <Shift>+<Enter> <Alt>+2 File Print Print Selection 17

18 Windows Machintosh File Save UNIX Windows.nb Explorer Macintosh Next UNIX.nb Mathematica File Load Open Mathematica Mathematica Edit Save selection as EPS EPS 9 = := == 2 1/Sqrt[2]==Sqrt[2]/2 True (Sqrt[2] 2 1/ 2 = 2/2 ) x==1 1==2 False 18

19 Solve[3x-2==1,x] {{x -> 1}} 3x 2 = 1 x 3x-2==1 /. x->1 /. x->1 x 1 /. x->1 x 1 x 1 3x-2 /. x->1 3x-2==1 /. x->0 {{x -> 1}} Solve[{x-y==2,x+3y==6}, {x,y}] x y = 2, x + 3y = 6 {{x -> 3, y -> 1}} {x-y==2,x+3y==6} /. {x -> 3, y -> 1} {x-y,x+3y} /. {x -> 3, y -> 1} /. %... Solve eq1={...,...} Solve[{x+y==3, x y==2}, {x,y}] {{x -> 1, y -> 2}, {x -> 2, y -> 1}} 19

20 x = 1, y = 2 x = 2, y = 1 Solve[x+1==x, x] NSolve NSolve[1-10x-9x^2+x^5==0, x] NSolve Solve Solve[Tan[x]==1/3, x] NSolve[Tan[x]==1/3, x] NSolve[Tan[x]-2x==0, x] FindRoot FindRoot tan x 2x = 0 f=tan[x]-2x Plot[f,{x,-3,3}, PlotRange->{-1,1}] x tan x PlotRange FindRoot[f==0, {x,1.2}] 0 10 {x+y, x y} == {3, 2} 20

21 {{11,12},{21,22}}. {x,y} == {1,2} Mathematica MatrixForm[{1,2}] MatrixForm[{{11,12},{21,22}}] Det[{{1, -1}, {1, 3}}] m {{1,-1},{-1,2}} MatrixPower[m,2] m m.m m m m*m m.m MatrixPower[m,-1] 1/m m^-1 MatrixExp[m] Exp[m] MatrixExp[ t {{0, 1}, {-1, 0}} ]//FullSimplify FullSimplify 11 v e Clear[v,e] Mathematica H = {{v,1},{1,0}} J = {{1,0},{0,1}} H - e J MatrixForm[H - e J] 21

22 f=det[h - e J] f Solve[f==0,e] f==0 e ee=e/.% ee H ee = Eigenvalues[H] vv=eigenvectors[h] ee ee[[1]] ee[[2]] Part[ee,1] Part[ee,2] vv H.vv[[1]] //Expand ee[[1]] vv[[1]] //Expand (. ) vv[[2]] ee[[2]] Expand Expand : v g1 = Plot[ee[[1]], {v,-10,10}] g2 = Plot[ee[[2]], {v,-10,10}] Show[g1, g2] 22

23 g12 = Plot[Evaluate[ee], {v,-10,10}] ee Evaluate g3 = Plot[v, {v,-10,10}, PlotStyle->Dashing[{0.01}]] e1 e2 v PlotStyle->... Dashing[...] 0.01 {0.01} {0.01,0.01,0.03,0.01} 12 f=x^2+x+1 f = x 2 + x + 1 f1=d[f,x] x df/dx f2=d[f,{x,2}] x d 2 f/dx 2 Plot[{f,f1,f2},{x,-1,1}] g=integrate[x^-1,x] x 1 x Plot[{x^-1,g},{x,-1,10}] Integrate[x^2,{x,-1,1}] 1 1 x2 dx Integrate[1/(x^2+1),{x,-Infinity,Infinity}] (x2 + 1) 1 dx NIntegrate[E^(-x^2),{x,-1,1}] DSolve[y [t]==-y[t],y[t],t] y(t) y (t) = y(t) DSolve[{y [t]==-y[t],y[0]==0,y [0]==1},y[t],t] 13 Mathematica 23

24 13.1 e ɛ 0 a F elec = e 2 /(4πɛ 0 a 2 ) Mathematica soden yuden a B = m 13.2 x exp( x 2 ) x Series Normal x X x 1 + x 2 x Series Normal x << Graphics Arrow Arrow Show[Graphics[Arrow[{0,0},{1,2}]], Axes->True] Graphics Show Arrow Graphics Axes AspectRatio->Automatic << Graphics Arrow := GVector GVector[vect_] := Graphics[Arrow[{0,0}, vect], Axes->True, AspectRatio->Automatic] 24

25 GVector[begin_, vect_] := Graphics[Arrow[begin, begin+vect], Axes->True, AspectRatio->Automatic] Show[GVector[{1,2}], GVector[{2,1}]] Show[GVector[{1,2}], GVector[{2,1}], GVector[{1,2}+{2,1}]] Show[GVector[{1,2}], GVector[{1,2},{2,1}], GVector[{1,2} + {2,1}]] (1,2) (2,1) v1 v2 v12 { } 120 (2π/3 ) M MatrixForm M θ ( ) cos θ sin θ sin θ cos θ M v1. ( ) 25

GraphicsWithPlotFull.nb Plot[{( 1), ( ),...}, {( ), ( ), ( )}] Plot Plot Cos x Sin x, x, 5 Π, 5 Π, AxesLabel x, y x 1 Plot AxesLabel

GraphicsWithPlotFull.nb Plot[{( 1), ( ),...}, {( ), ( ), ( )}] Plot Plot Cos x Sin x, x, 5 Π, 5 Π, AxesLabel x, y x 1 Plot AxesLabel http://yktlab.cis.k.hosei.ac.jp/wiki/ 1(Plot) f x x x 1 1 x x ( )[( 1)_, ( )_, ( 3)_,...]=( ) Plot Plot f x, x, 5, 3 15 10 5 Plot[( ), {( ), ( ), ( )}] D g x x 3 x 3 Plot f x, g x, x, 10, 8 00 100 10 5