数学 - Python - 解析学 - 多変数の関数 - 高次偏導関数、テイラーの定理(指数関数、グラディエント(gradient)、内積(ドット積、スカラー積))

解析入門〈3〉

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解析入門〈3〉(松坂 和夫(著)、岩波書店)の第14章(多変数の関数)、14.2(高次偏導関数、テイラーの定理)、問題3-(c).を取り組んでみる。

3. 3. 
      
      $\begin{array}{}\frac{\partial \; <!--\; partial\; differential\; -->z}{\partial \; <!--\; partial\; differential\; -->u}\\ =gradf\left(x,y\right)·\; <!--\; middle\; dot\; -->\left(e^u,-e^{-u}\right)\\ =\left(\frac{\partial \; <!--\; partial\; differential\; -->z}{\partial \; <!--\; partial\; differential\; -->x},\frac{\partial \; <!--\; partial\; differential\; -->z}{\partial \; <!--\; partial\; differential\; -->y}\right).\left(e^u,-e^{-u}\right)\\ =\frac{\partial \; <!--\; partial\; differential\; -->z}{\partial \; <!--\; partial\; differential\; -->x}{e}^u-\frac{\partial \; <!--\; partial\; differential\; -->z}{\partial \; <!--\; partial\; differential\; -->y}{e}^{-u}\end{array}$
      $\begin{array}{}\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}z}{\partial \; <!--\; partial\; differential\; -->u^2}\\ =\left(\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}z}{\partial \; <!--\; partial\; differential\; -->x^2},\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}z}{\partial \; <!--\; partial\; differential\; -->y\partial \; <!--\; partial\; differential\; -->x}\right)·\; <!--\; middle\; dot\; -->\left(e^u,-e^{-u}\right)e^u+\frac{\partial \; <!--\; partial\; differential\; -->z}{\partial \; <!--\; partial\; differential\; -->x}{e}^u-\left(\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}z}{\partial \; <!--\; partial\; differential\; -->x\partial \; <!--\; partial\; differential\; -->y},\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}z}{\partial \; <!--\; partial\; differential\; -->y^2}\right)·\; <!--\; middle\; dot\; -->\left(e^u,-e^{-u}\right)e^{-u}-\frac{\partial \; <!--\; partial\; differential\; -->z}{\partial \; <!--\; partial\; differential\; -->y}{e}^{-u}\\ =\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}z}{\partial \; <!--\; partial\; differential\; -->x^2}{e}^{2u}-\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}z}{\partial \; <!--\; partial\; differential\; -->y\partial \; <!--\; partial\; differential\; -->x}+\frac{\partial \; <!--\; partial\; differential\; -->z}{\partial \; <!--\; partial\; differential\; -->x}{e}^u-\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}z}{\partial \; <!--\; partial\; differential\; -->x\partial \; <!--\; partial\; differential\; -->y}+\frac{{\partial \; <!--\; partial\; differential\; -->}^2}{\partial \; <!--\; partial\; differential\; -->y^2}{e}^{-2u}+\frac{\partial \; <!--\; partial\; differential\; -->z}{\partial \; <!--\; partial\; differential\; -->y}{e}^{-u}\\ =\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}z}{\partial \; <!--\; partial\; differential\; -->x^2}{e}^{2u}-2\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}z}{\partial \; <!--\; partial\; differential\; -->x\partial \; <!--\; partial\; differential\; -->y}+\frac{\partial \; <!--\; partial\; differential\; -->z}{\partial \; <!--\; partial\; differential\; -->x}{e}^u+\frac{\partial \; <!--\; partial\; differential\; -->z}{\partial \; <!--\; partial\; differential\; -->y}{e}^{-u}+\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}z}{\partial \; <!--\; partial\; differential\; -->y^2}{e}^{-2u}\end{array}$
      $\frac{\partial \; <!--\; partial\; differential\; -->z}{\partial \; <!--\; partial\; differential\; -->v}=\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->x}{e}^v-\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->y}{e}^{-v}$
      $\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}z}{\partial \; <!--\; partial\; differential\; -->v^2}=\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}z}{\partial \; <!--\; partial\; differential\; -->x^2}{e}^{2v}-2\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}z}{\partial \; <!--\; partial\; differential\; -->x\partial \; <!--\; partial\; differential\; -->y}+\frac{\partial \; <!--\; partial\; differential\; -->z}{\partial \; <!--\; partial\; differential\; -->x}{e}^v+\frac{\partial \; <!--\; partial\; differential\; -->z}{\partial \; <!--\; partial\; differential\; -->y}{e}^{-v}+\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}z}{\partial \; <!--\; partial\; differential\; -->y^2}{e}^{-2v}$
      $\begin{array}{}\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}z}{\partial \; <!--\; partial\; differential\; -->u\partial \; <!--\; partial\; differential\; -->v}=\left(\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->x}\left(\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->x}{e}^v-\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->y}{e}^{-v}\right),\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->y}\left(\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->x}{e}^v-\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->y}{e}^{-v}\right)\right)·\; <!--\; middle\; dot\; -->\left(e^u,-e^{-u}\right)\\ =\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}z}{\partial \; <!--\; partial\; differential\; -->x^2}{e}^{u+v}-\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}z}{\partial \; <!--\; partial\; differential\; -->x\partial \; <!--\; partial\; differential\; -->y}{e}^{u-v}-\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}z}{\partial \; <!--\; partial\; differential\; -->y\partial \; <!--\; partial\; differential\; -->x}{e}^{v-u}+\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}z}{\partial \; <!--\; partial\; differential\; -->y^2}{e}^{-v-u}\end{array}$
      $\begin{array}{}\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}z}{\partial \; <!--\; partial\; differential\; -->u^2}+2\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}z}{\partial \; <!--\; partial\; differential\; -->u\partial \; <!--\; partial\; differential\; -->v}+\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}z}{\partial \; <!--\; partial\; differential\; -->v^2}\\ =\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}z}{\partial \; <!--\; partial\; differential\; -->x^2}\left(e^{2u}+e^{2v}+2e^{u+v}\right)+\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}z}{\partial \; <!--\; partial\; differential\; -->y^2}\left(e^{-2u}+e^{-2v}+2e^{-\left(u+v\right)}\right)\\ -2\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}z}{\partial \; <!--\; partial\; differential\; -->x\mathrm{dy}}\left(e^{u-v}+e^{v-u}+2\right)+\frac{\partial \; <!--\; partial\; differential\; -->z}{\partial \; <!--\; partial\; differential\; -->x}\left(e^u+e^v\right)+\frac{\partial \; <!--\; partial\; differential\; -->z}{\partial \; <!--\; partial\; differential\; -->y}\left(e^{-u}+e^{-v}\right)\\ =\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}z}{\partial \; <!--\; partial\; differential\; -->x^2}{\left(e^u+e^v\right)}^2+\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}z}{\partial \; <!--\; partial\; differential\; -->y^2}{\left(e^{-u}+e^{-v}\right)}^2-2\left(e^n+e^v\right)\left(e^{-u}+e^{-v}\right)\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}z}{\partial \; <!--\; partial\; differential\; -->x\partial \; <!--\; partial\; differential\; -->y}+x\frac{\partial \; <!--\; partial\; differential\; -->z}{\partial \; <!--\; partial\; differential\; -->x}+y\frac{\partial \; <!--\; partial\; differential\; -->z}{\partial \; <!--\; partial\; differential\; -->y}\\ =x^2\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}z}{\partial \; <!--\; partial\; differential\; -->x^2}+y^2\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}z}{\partial \; <!--\; partial\; differential\; -->y^2}-2xy\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}z}{\partial \; <!--\; partial\; differential\; -->x\partial \; <!--\; partial\; differential\; -->y}+x\frac{\partial \; <!--\; partial\; differential\; -->z}{\partial \; <!--\; partial\; differential\; -->x}+y\frac{\partial \; <!--\; partial\; differential\; -->z}{\partial \; <!--\; partial\; differential\; -->y}\end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, exp, Function, Derivative

u, v = symbols('u, v')
x = exp(u) + exp(v)
y = exp(-u) + exp(-v)
f = Function('f')(x, y)
expr = Derivative(f, u, 2) + 2 * Derivative(Derivative(f, v, 1),
                                            u, 1) + Derivative(f, v, 2)
for t in [expr, expr.doit(), expr.doit().factor()]:
    pprint(t)
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample3.py
  ⎛   ⎛  2                   ⎞│                 ⎛⎛    2             ⎞│        
  ⎜ v ⎜ ∂  ⎛ ⎛     -v    -u⎞⎞⎟│              -v ⎜⎜   ∂              ⎟│        
2⋅⎜ℯ ⋅⎜────⎝f⎝ξ₁, ℯ   + ℯ  ⎠⎠⎟│           - ℯ  ⋅⎜⎜───────(f(ξ₁, ξ₂))⎟│    u   
  ⎜   ⎜   2                  ⎟│    u    v       ⎝⎝∂ξ₂ ∂ξ₁           ⎠│ξ₁=ℯ  + 
  ⎝   ⎝∂ξ₁                   ⎠│ξ₁=ℯ  + ℯ                                      

  ⎞│            ⎞        ⎛   ⎛⎛    2             ⎞│            ⎞│             
  ⎟│            ⎟  u     ⎜ v ⎜⎜   ∂              ⎟│            ⎟│             
 v⎟│    -v    -u⎟⋅ℯ  - 2⋅⎜ℯ ⋅⎜⎜───────(f(ξ₁, ξ₂))⎟│    -v    -u⎟│    u    v - 
ℯ ⎠│ξ₂=ℯ   + ℯ  ⎟        ⎜   ⎝⎝∂ξ₂ ∂ξ₁           ⎠│ξ₂=ℯ   + ℯ  ⎠│ξ₁=ℯ  + ℯ    
                ⎠        ⎝                                                    

    ⎛  2                 ⎞│            ⎞         2                            
 -v ⎜ ∂  ⎛ ⎛ u    v    ⎞⎞⎟│            ⎟  -u    ∂ ⎛ ⎛ u    v   -v    -u⎞⎞    ∂
ℯ  ⋅⎜────⎝f⎝ℯ  + ℯ , ξ₂⎠⎠⎟│            ⎟⋅ℯ   + ───⎝f⎝ℯ  + ℯ , ℯ   + ℯ  ⎠⎠ + ──
    ⎜   2                ⎟│    -v    -u⎟         2                            
    ⎝∂ξ₂                 ⎠│ξ₂=ℯ   + ℯ  ⎠       ∂u                           ∂v

2                       
 ⎛ ⎛ u    v   -v    -u⎞⎞
─⎝f⎝ℯ  + ℯ , ℯ   + ℯ  ⎠⎠
2                       
                        

⎛   ⎛  2                   ⎞│                 ⎛⎛    2             ⎞│          
⎜ u ⎜ ∂  ⎛ ⎛     -v    -u⎞⎞⎟│              -u ⎜⎜   ∂              ⎟│          
⎜ℯ ⋅⎜────⎝f⎝ξ₁, ℯ   + ℯ  ⎠⎠⎟│           - ℯ  ⋅⎜⎜───────(f(ξ₁, ξ₂))⎟│    u    v
⎜   ⎜   2                  ⎟│    u    v       ⎝⎝∂ξ₂ ∂ξ₁           ⎠│ξ₁=ℯ  + ℯ 
⎝   ⎝∂ξ₁                   ⎠│ξ₁=ℯ  + ℯ                                        

⎞│            ⎞      ⎛   ⎛⎛    2             ⎞│            ⎞│                 
⎟│            ⎟  u   ⎜ u ⎜⎜   ∂              ⎟│            ⎟│              -u 
⎟│    -v    -u⎟⋅ℯ  - ⎜ℯ ⋅⎜⎜───────(f(ξ₁, ξ₂))⎟│    -v    -u⎟│    u    v - ℯ  ⋅
⎠│ξ₂=ℯ   + ℯ  ⎟      ⎜   ⎝⎝∂ξ₂ ∂ξ₁           ⎠│ξ₂=ℯ   + ℯ  ⎠│ξ₁=ℯ  + ℯ        
              ⎠      ⎝                                                        

⎛  2                 ⎞│            ⎞         ⎛   ⎛  2                   ⎞│    
⎜ ∂  ⎛ ⎛ u    v    ⎞⎞⎟│            ⎟  -u     ⎜ v ⎜ ∂  ⎛ ⎛     -v    -u⎞⎞⎟│    
⎜────⎝f⎝ℯ  + ℯ , ξ₂⎠⎠⎟│            ⎟⋅ℯ   + 2⋅⎜ℯ ⋅⎜────⎝f⎝ξ₁, ℯ   + ℯ  ⎠⎠⎟│    
⎜   2                ⎟│    -v    -u⎟         ⎜   ⎜   2                  ⎟│    
⎝∂ξ₂                 ⎠│ξ₂=ℯ   + ℯ  ⎠         ⎝   ⎝∂ξ₁                   ⎠│ξ₁=ℯ

             ⎛⎛    2             ⎞│          ⎞│            ⎞      ⎛   ⎛  2    
          -v ⎜⎜   ∂              ⎟│          ⎟│            ⎟  u   ⎜ v ⎜ ∂  ⎛ ⎛
       - ℯ  ⋅⎜⎜───────(f(ξ₁, ξ₂))⎟│    u    v⎟│    -v    -u⎟⋅ℯ  + ⎜ℯ ⋅⎜────⎝f⎝
u    v       ⎝⎝∂ξ₂ ∂ξ₁           ⎠│ξ₁=ℯ  + ℯ ⎠│ξ₂=ℯ   + ℯ  ⎟      ⎜   ⎜   2   
  + ℯ                                                      ⎠      ⎝   ⎝∂ξ₁    

               ⎞│                 ⎛⎛    2             ⎞│          ⎞│          
     -v    -u⎞⎞⎟│              -v ⎜⎜   ∂              ⎟│          ⎟│          
ξ₁, ℯ   + ℯ  ⎠⎠⎟│           - ℯ  ⋅⎜⎜───────(f(ξ₁, ξ₂))⎟│    u    v⎟│    -v    
               ⎟│    u    v       ⎝⎝∂ξ₂ ∂ξ₁           ⎠│ξ₁=ℯ  + ℯ ⎠│ξ₂=ℯ   + ℯ
               ⎠│ξ₁=ℯ  + ℯ                                                    

  ⎞      ⎛   ⎛⎛    2             ⎞│            ⎞│                 ⎛  2        
  ⎟  v   ⎜ v ⎜⎜   ∂              ⎟│            ⎟│              -v ⎜ ∂  ⎛ ⎛ u  
-u⎟⋅ℯ  - ⎜ℯ ⋅⎜⎜───────(f(ξ₁, ξ₂))⎟│    -v    -u⎟│    u    v - ℯ  ⋅⎜────⎝f⎝ℯ  +
  ⎟      ⎜   ⎝⎝∂ξ₂ ∂ξ₁           ⎠│ξ₂=ℯ   + ℯ  ⎠│ξ₁=ℯ  + ℯ        ⎜   2       
  ⎠      ⎝                                                        ⎝∂ξ₂        

         ⎞│            ⎞         ⎛   ⎛⎛    2             ⎞│            ⎞│     
  v    ⎞⎞⎟│            ⎟  -v     ⎜ v ⎜⎜   ∂              ⎟│            ⎟│     
 ℯ , ξ₂⎠⎠⎟│            ⎟⋅ℯ   - 2⋅⎜ℯ ⋅⎜⎜───────(f(ξ₁, ξ₂))⎟│    -v    -u⎟│    u
         ⎟│    -v    -u⎟         ⎜   ⎝⎝∂ξ₂ ∂ξ₁           ⎠│ξ₂=ℯ   + ℯ  ⎠│ξ₁=ℯ 
         ⎠│ξ₂=ℯ   + ℯ  ⎠         ⎝                                            

            ⎛  2                 ⎞│            ⎞                              
         -v ⎜ ∂  ⎛ ⎛ u    v    ⎞⎞⎟│            ⎟  -u    u ⎛ ∂ ⎛ ⎛     -v    -u
    v - ℯ  ⋅⎜────⎝f⎝ℯ  + ℯ , ξ₂⎠⎠⎟│            ⎟⋅ℯ   + ℯ ⋅⎜───⎝f⎝ξ₁, ℯ   + ℯ  
 + ℯ        ⎜   2                ⎟│    -v    -u⎟          ⎝∂ξ₁                
            ⎝∂ξ₂                 ⎠│ξ₂=ℯ   + ℯ  ⎠                              

                                                                              
⎞⎞⎞│              v ⎛ ∂ ⎛ ⎛     -v    -u⎞⎞⎞│              -v ⎛ ∂ ⎛ ⎛ u    v   
⎠⎠⎟│    u    v + ℯ ⋅⎜───⎝f⎝ξ₁, ℯ   + ℯ  ⎠⎠⎟│    u    v + ℯ  ⋅⎜───⎝f⎝ℯ  + ℯ , ξ
  ⎠│ξ₁=ℯ  + ℯ       ⎝∂ξ₁                  ⎠│ξ₁=ℯ  + ℯ        ⎝∂ξ₂             
                                                                              

                                                          
 ⎞⎞⎞│                -u ⎛ ∂ ⎛ ⎛ u    v    ⎞⎞⎞│            
₂⎠⎠⎟│    -v    -u + ℯ  ⋅⎜───⎝f⎝ℯ  + ℯ , ξ₂⎠⎠⎟│    -v    -u
   ⎠│ξ₂=ℯ   + ℯ         ⎝∂ξ₂                ⎠│ξ₂=ℯ   + ℯ  
                                                          

          ⎛          ⎛  2                   ⎞│                       ⎛  2     
⎛ u    v⎞ ⎜ 3⋅u  2⋅v ⎜ ∂  ⎛ ⎛     -v    -u⎞⎞⎟│              2⋅u  3⋅v ⎜ ∂  ⎛ ⎛ 
⎝ℯ  + ℯ ⎠⋅⎜ℯ   ⋅ℯ   ⋅⎜────⎝f⎝ξ₁, ℯ   + ℯ  ⎠⎠⎟│           + ℯ   ⋅ℯ   ⋅⎜────⎝f⎝ξ
          ⎜          ⎜   2                  ⎟│    u    v             ⎜   2    
          ⎝          ⎝∂ξ₁                   ⎠│ξ₁=ℯ  + ℯ              ⎝∂ξ₁     

              ⎞│                                                              
    -v    -u⎞⎞⎟│              2⋅u  2⋅v ⎛ ∂ ⎛ ⎛     -v    -u⎞⎞⎞│               
₁, ℯ   + ℯ  ⎠⎠⎟│           + ℯ   ⋅ℯ   ⋅⎜───⎝f⎝ξ₁, ℯ   + ℯ  ⎠⎠⎟│    u    v - 2⋅
              ⎟│    u    v             ⎝∂ξ₁                  ⎠│ξ₁=ℯ  + ℯ      
              ⎠│ξ₁=ℯ  + ℯ                                                     

        ⎛⎛    2             ⎞│            ⎞│                       ⎛⎛    2    
 2⋅u  v ⎜⎜   ∂              ⎟│            ⎟│                u  2⋅v ⎜⎜   ∂     
ℯ   ⋅ℯ ⋅⎜⎜───────(f(ξ₁, ξ₂))⎟│    -v    -u⎟│    u    v - 2⋅ℯ ⋅ℯ   ⋅⎜⎜───────(f
        ⎝⎝∂ξ₂ ∂ξ₁           ⎠│ξ₂=ℯ   + ℯ  ⎠│ξ₁=ℯ  + ℯ              ⎝⎝∂ξ₂ ∂ξ₁  
                                                                              

         ⎞│            ⎞│                                                     
         ⎟│            ⎟│              u  v ⎛ ∂ ⎛ ⎛ u    v    ⎞⎞⎞│            
(ξ₁, ξ₂))⎟│    -v    -u⎟│    u    v + ℯ ⋅ℯ ⋅⎜───⎝f⎝ℯ  + ℯ , ξ₂⎠⎠⎟│    -v    -u
         ⎠│ξ₂=ℯ   + ℯ  ⎠│ξ₁=ℯ  + ℯ          ⎝∂ξ₂                ⎠│ξ₂=ℯ   + ℯ  
                                                                              

      ⎛  2                 ⎞│                  ⎛  2                 ⎞│        
    u ⎜ ∂  ⎛ ⎛ u    v    ⎞⎞⎟│                v ⎜ ∂  ⎛ ⎛ u    v    ⎞⎞⎟│        
 + ℯ ⋅⎜────⎝f⎝ℯ  + ℯ , ξ₂⎠⎠⎟│             + ℯ ⋅⎜────⎝f⎝ℯ  + ℯ , ξ₂⎠⎠⎟│        
      ⎜   2                ⎟│    -v    -u      ⎜   2                ⎟│    -v  
      ⎝∂ξ₂                 ⎠│ξ₂=ℯ   + ℯ        ⎝∂ξ₂                 ⎠│ξ₂=ℯ   +

    ⎞            
    ⎟  -2⋅u  -2⋅v
    ⎟⋅ℯ    ⋅ℯ    
  -u⎟            
 ℯ  ⎠            

$