Chapter 6 1 ParametricPlot 3 ParametricPlot ParametricPlot3D 6.1 [1] *1 In[1]:= v1 = {x, y}; v = {z, w}; [] In[]:= m1 = {{a, b}, {c, d}}; m = {{3, }, {1, }}; [3] 3 3.3 In[3]:= Out[3]//MatrixForm= [] m1 //MatrixForm a b c d In[]:= Out[]= v1.v wyxz *1 v1//matrixform ` x y 1
66 6 1 ParametricPlot [5] In[5]:= Out[5]= m1.v1 axby, cxdy [6] pr In[6]:= Out[6]= pr = m1.m 3ab, ab,3cd, cd [7] In[7]:= Out[7]//MatrixForm= pr //MatrixForm 3ab ab 3cd cd [8] MatrixPower (power) m 7 In[8]:= Out[8]//MatrixForm= MatrixPower[m, 7] //MatrixForm 617 51998 5999 516 [9] Transpose m1 (transposed matrix) In[9]:= Out[9]//MatrixForm= Transpose[m1] //MatrixForm a c b d [1] Inverse (inverse matrix) In[1]:= Out[1]//MatrixForm= Inverse[m1] //MatrixForm d b bcad bcad c a bcad bcad [11] Det (determinant) In[11]:= Out[11]= Det[m1] bcad [1] Tr (trace) In[1]:= Out[1]= Tr[m1] ad [13] (identity matrix) i In[13]:= i = IdentityMatrix[]
6. 67 Out[13]= 1,,, 1 n IdentityMatrix[n] 6.1 () [6][11][1][13] ( ) a b M = M tr(m)m + det(m)i = O c d I O! M = a b [] m1 I [13] i M tr(m)m + c d det(m)i Simplify In[ ]:= Out[ ]= Simplify[m1.m1 - Tr[m1]*m1 + Det[m1]*i],,, O 6. n M λ n v Mv = λv λ M (eigenvalue)v λ (eigenvector) Mathematica Eigensystem [1] [] ( ) 3 m 1 In[1]:= m = m; [15] Eigensystem es In[15]:= Out[15]= es = Eigensystem[m] 5,,1, 1,, 1 [16] MatrixForm In[16]:= Out[16]//MatrixForm= es // MatrixForm 5 1, 1, 1
6. 3 1 ParametricPlot3D 73 Out[ ]= 6. () disk(r, t) = (r cos t, r sin t) r 1, t π f ParametricPlot [3][31][3] In[ ]:= In[ ]:= disk[r_, t_] = {r Cos[t], r Sin[t]}; ParametricPlot[{disk[r, t], f[disk[r, t]]}, {r,, 1}, {t,, Pi}] Out[ ]= 3 1 1 3 6. 3 1 ParametricPlot3D xyz R 3 1 g : R 3 R 3 g x y = 1 x y = x + z y z 1 z x + z
7 6 1 ParametricPlot ParametricPlot3D [33] In[33]:= mm = {{1,, }, {,, }, {,, 1}}; Out[33]//MatrixForm= mm // MatrixForm 1 1 [3] In[3]:= Out[3]//MatrixForm= [35] 1 Eigensystem[mm] // MatrixForm 3 1 1,, 1, 1, 1,, 1 In[35]:= g[{x_, y_, z_}] = mm.{x, y, z} Out[35]= xz, y, xz [36] {(cos t, sin t, t/1) t 1π} (spiral helix) g In[36]:= spi[t_] = {Cos[t], Sin[t], t/1}; [37] ParametricPlot3D 3 In[37]:= ParametricPlot3D[spi[t], {t,, 1 Pi}] 1..5...51..5 1..5 1. 3 Out[37]= 1.5 [38] g In[38]:= ParametricPlot3D[g[spi[t]], {t,, 1 Pi}]
6. 3 1 ParametricPlot3D 75 1 1 6 Out[38]= [39] In[39]:= ParametricPlot3D[{spi[t], g[spi[t]]}, {t,, 1 Pi}] 1 1 Out[39]= 6 [] ParametricPlot3D sphere {(cos s sin t, sin s sin t, cos t) s π, t π} 6. 3 In[]:= sph[s_, t_] = {Cos[s] Sin[t], Sin[s] Sin[t], Cos[t]}; ParametricPlot3D[ {sph[s, t], g[sph[s, t]]}, {s,, Pi}, {t,, Pi}, PlotStyle -> {Opacity[.9, Green], Opacity[., Yellow]}]
76 6 1 ParametricPlot Out[]= sph[s, t], g[sph[s, t]] PlotStyle Opacity sph[s, t].9 g[sph[s, t]]. 6.5 () (torus) g {((3 + cos t) cos s, (3 + cos t) sin s, sin t) s π, t π} tor[s_, t_] = { Cos[s] (3 + Cos[t]), Sin[s] (3 + Cos[t]), Sin[t] } t {t,, Pi} [] 6.5 1 1 1 1 g mm Det[mm] 6 [1] xy, yz, zx In[1]:= xy[s_, t_] = {s, t, };
6.5 1 77 In[1]:= In[1]:= yz[s_, t_] = {, s, t}; zx[s_, t_] = {t,, s}; [] xy, yz, zx 3 (panels) In[]:= pan = ParametricPlot3D[ {xy[s, t], yz[s, t], zx[s, t]}, {s, -1, 3}, {t, -1, 3}, PlotStyle -> {Red, Green, Blue}] Out[]= s, t x, y, z [3] g In[3]:= pan = ParametricPlot3D[ {g[xy[s, t]], g[yz[s, t]], g[zx[s, t]]}, {s, -1, 3}, {t, -1, 3}, PlotStyle -> {Red, Green, Blue}] Out[3]= [] 3 g
78 6 1 ParametricPlot [] In[]:= Show[pan, pan, Boxed -> False, Axes -> False] Out[]= Boxed -> False Axes -> False Show[pan, pan,...] pan pan 6.6 (1) (rank) MatrixRank MatrixExp JordanDecomposition () ParametricPlot3D Hint. Mesh -> None (3) ParametricPlot3D Hint.