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(三角関数(正弦、余弦)、極座標、グラディエント(gradient)、内積(ドット積、スカラー積))

解析入門〈3〉

学習環境

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• 参考書籍
  • Pythonからはじめる数学入門
  • JavaScriptリファレンス 第6版
  • インタラクティブ・データビジュアライゼーション

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4.を取り組んでみる。

4. 
   
   $\begin{array}{}\frac{\partial \; <!--\; partial\; differential\; -->g}{\partial \; <!--\; partial\; differential\; -->r}\\ =gradf\left(x,y\right).\left(\cos \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->},\sin \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\right)\\ =\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->x}\cos \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}+\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->y}\sin \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\end{array}$
   $\begin{array}{}\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}g}{\partial \; <!--\; partial\; differential\; -->r^2}\\ =\left(\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}f}{\partial \; <!--\; partial\; differential\; -->x^2}\cos \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}+\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}f}{\partial \; <!--\; partial\; differential\; -->x\partial \; <!--\; partial\; differential\; -->y}\sin \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->},\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}f}{\partial \; <!--\; partial\; differential\; -->y\partial \; <!--\; partial\; differential\; -->x}\cos \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}+\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}f}{\partial \; <!--\; partial\; differential\; -->y^2}\sin \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\right)·\; <!--\; middle\; dot\; -->\left(\cos \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->},\sin \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\right)\\ =\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}f}{\partial \; <!--\; partial\; differential\; -->x^2}{\cos }^2\mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}+2\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}f}{\partial \; <!--\; partial\; differential\; -->x\partial \; <!--\; partial\; differential\; -->y}\sin \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\cos \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}+\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}f}{\partial \; <!--\; partial\; differential\; -->y^2}{\sin }^2\mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\end{array}$
   $\begin{array}{}\frac{\partial \; <!--\; partial\; differential\; -->g}{\partial \; <!--\; partial\; differential\; -->\mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}}\\ =\left(\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->x},\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->y}\right)·\; <!--\; middle\; dot\; -->\left(-r\sin \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->},r\cos \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\right)\\ =-\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->x}r\sin \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}+\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->y}r\cos \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\end{array}$
   $\begin{array}{}\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}g}{\partial \; <!--\; partial\; differential\; -->{\mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}}^2}\\ =-\left(\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}f}{\partial \; <!--\; partial\; differential\; -->x^2},\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}f}{\partial \; <!--\; partial\; differential\; -->y\partial \; <!--\; partial\; differential\; -->x}\right)·\; <!--\; middle\; dot\; -->\left(-r\sin \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->},r\cos \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\right)·\; <!--\; middle\; dot\; -->r\sin \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}-\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->x}r\cos \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\\ +\left(\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}f}{\partial \; <!--\; partial\; differential\; -->x\partial \; <!--\; partial\; differential\; -->y},\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}f}{\partial \; <!--\; partial\; differential\; -->y^2}\right)·\; <!--\; middle\; dot\; -->\left(-r\sin \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->},r\cos \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\right)·\; <!--\; middle\; dot\; -->r\cos \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}-\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->y}r\sin \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\\ =\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}f}{\partial \; <!--\; partial\; differential\; -->x^2}{r}^2{\sin }^2\mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}-\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}f}{\partial \; <!--\; partial\; differential\; -->x\partial \; <!--\; partial\; differential\; -->y}{r}^2\sin \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\cos \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}-\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->x}r\cos \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\\ -\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}f}{\partial \; <!--\; partial\; differential\; -->x\partial \; <!--\; partial\; differential\; -->y}{r}^2\sin \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\cos \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}+\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}f}{\partial \; <!--\; partial\; differential\; -->y^2}{r}^2{\cos }^2\mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}-\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->y}r\sin \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\\ =\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}f}{\partial \; <!--\; partial\; differential\; -->x^2}{r}^2{\sin }^2\mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}-2\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}f}{\partial \; <!--\; partial\; differential\; -->x\partial \; <!--\; partial\; differential\; -->y}{r}^2\sin \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\cos \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}+\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}f}{\partial \; <!--\; partial\; differential\; -->y^2}{r}^2{\cos }^2\mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}-\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->x}r\cos \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\\ -\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->y}r\sin \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\end{array}$

   よって、

   $\begin{array}{}\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}g}{\partial \; <!--\; partial\; differential\; -->r^2}+\frac{1}{r}\frac{\partial \; <!--\; partial\; differential\; -->g}{\partial \; <!--\; partial\; differential\; -->r}+\frac{1}{r^2}\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}g}{\partial \; <!--\; partial\; differential\; -->{\mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}}^2}\\ =\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}f}{\partial \; <!--\; partial\; differential\; -->x^2}\left({\cos }^2\mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}+{\sin }^2\mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\right)+\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}f}{\partial \; <!--\; partial\; differential\; -->y^2}\left({\sin }^2\mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}+{\cos }^2\mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\right)\\ +\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}f}{\partial \; <!--\; partial\; differential\; -->x\partial \; <!--\; partial\; differential\; -->y}\left(2\sin \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\cos \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}-2\sin \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\cos \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\right)\\ +\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->x}\left(\frac{1}{r}\cos \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}-\frac{1}{r}\cos \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\right)+\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->y}\left(\frac{1}{r}\sin \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}-\frac{1}{r}\sin \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\right)\\ =\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}f}{\partial \; <!--\; partial\; differential\; -->x^2}+\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}f}{\partial \; <!--\; partial\; differential\; -->y^2}\end{array}$

コード(Emacs)

Python 3

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

r, θ = symbols('r, θ', real=True)
x = r * sin(θ)
y = r * cos(θ)
f = Function('f')(x, y)
Df = Derivative(f, r, 2) + 1 / r * Derivative(f, r, 1) + \
    1 / (r ** 2) * Derivative(f, r, 2)

for t in [Df, Df.doit(), Df.doit().factor()]:
    pprint(t.factor())
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample4.py
     2                                                          2             
 2  ∂                             ∂                            ∂              
r ⋅───(f(r⋅sin(θ), r⋅cos(θ))) + r⋅──(f(r⋅sin(θ), r⋅cos(θ))) + ───(f(r⋅sin(θ), 
     2                            ∂r                            2             
   ∂r                                                         ∂r              
──────────────────────────────────────────────────────────────────────────────
                                            2                                 
                                           r                                  

          
          
r⋅cos(θ)))
          
          
──────────
          
          

           ⎛  2                  ⎞│                                 ⎛⎛    2   
 2    2    ⎜ ∂                   ⎟│                 2               ⎜⎜   ∂    
r ⋅sin (θ)⋅⎜────(f(ξ₁, r⋅cos(θ)))⎟│            + 2⋅r ⋅sin(θ)⋅cos(θ)⋅⎜⎜───────(
           ⎜   2                 ⎟│                                 ⎝⎝∂ξ₂ ∂ξ₁ 
           ⎝∂ξ₁                  ⎠│ξ₁=r⋅sin(θ)                                
──────────────────────────────────────────────────────────────────────────────
                                                                              
                                                                              

          ⎞│           ⎞│                         ⎛  2                  ⎞│    
          ⎟│           ⎟│               2    2    ⎜ ∂                   ⎟│    
f(ξ₁, ξ₂))⎟│           ⎟│            + r ⋅cos (θ)⋅⎜────(f(r⋅sin(θ), ξ₂))⎟│    
          ⎠│ξ₁=r⋅sin(θ)⎠│ξ₂=r⋅cos(θ)              ⎜   2                 ⎟│    
                                                  ⎝∂ξ₂                  ⎠│ξ₂=r
──────────────────────────────────────────────────────────────────────────────
                                                                              
                                                                              

                                                                              
                   ⎛ ∂                  ⎞│                       ⎛ ∂          
        + r⋅sin(θ)⋅⎜───(f(ξ₁, r⋅cos(θ)))⎟│            + r⋅cos(θ)⋅⎜───(f(r⋅sin(
                   ⎝∂ξ₁                 ⎠│ξ₁=r⋅sin(θ)            ⎝∂ξ₂         
⋅cos(θ)                                                                       
──────────────────────────────────────────────────────────────────────────────
                                                  2                           
                                                 r                            

                                ⎛  2                  ⎞│                      
        ⎞│                 2    ⎜ ∂                   ⎟│                      
θ), ξ₂))⎟│            + sin (θ)⋅⎜────(f(ξ₁, r⋅cos(θ)))⎟│            + 2⋅sin(θ)
        ⎠│ξ₂=r⋅cos(θ)           ⎜   2                 ⎟│                      
                                ⎝∂ξ₁                  ⎠│ξ₁=r⋅sin(θ)           
──────────────────────────────────────────────────────────────────────────────
                                                                              
                                                                              

        ⎛⎛    2             ⎞│           ⎞│                      ⎛  2         
        ⎜⎜   ∂              ⎟│           ⎟│                 2    ⎜ ∂          
⋅cos(θ)⋅⎜⎜───────(f(ξ₁, ξ₂))⎟│           ⎟│            + cos (θ)⋅⎜────(f(r⋅sin
        ⎝⎝∂ξ₂ ∂ξ₁           ⎠│ξ₁=r⋅sin(θ)⎠│ξ₂=r⋅cos(θ)           ⎜   2        
                                                                 ⎝∂ξ₂         
──────────────────────────────────────────────────────────────────────────────
                                                                              
                                                                              

         ⎞│           
         ⎟│           
(θ), ξ₂))⎟│           
         ⎟│           
         ⎠│ξ₂=r⋅cos(θ)
──────────────────────
                      
                      

           ⎛  2                  ⎞│                                 ⎛⎛    2   
 2    2    ⎜ ∂                   ⎟│                 2               ⎜⎜   ∂    
r ⋅sin (θ)⋅⎜────(f(ξ₁, r⋅cos(θ)))⎟│            + 2⋅r ⋅sin(θ)⋅cos(θ)⋅⎜⎜───────(
           ⎜   2                 ⎟│                                 ⎝⎝∂ξ₂ ∂ξ₁ 
           ⎝∂ξ₁                  ⎠│ξ₁=r⋅sin(θ)                                
──────────────────────────────────────────────────────────────────────────────
                                                                              
                                                                              

          ⎞│           ⎞│                         ⎛  2                  ⎞│    
          ⎟│           ⎟│               2    2    ⎜ ∂                   ⎟│    
f(ξ₁, ξ₂))⎟│           ⎟│            + r ⋅cos (θ)⋅⎜────(f(r⋅sin(θ), ξ₂))⎟│    
          ⎠│ξ₁=r⋅sin(θ)⎠│ξ₂=r⋅cos(θ)              ⎜   2                 ⎟│    
                                                  ⎝∂ξ₂                  ⎠│ξ₂=r
──────────────────────────────────────────────────────────────────────────────
                                                                              
                                                                              

                                                                              
                   ⎛ ∂                  ⎞│                       ⎛ ∂          
        + r⋅sin(θ)⋅⎜───(f(ξ₁, r⋅cos(θ)))⎟│            + r⋅cos(θ)⋅⎜───(f(r⋅sin(
                   ⎝∂ξ₁                 ⎠│ξ₁=r⋅sin(θ)            ⎝∂ξ₂         
⋅cos(θ)                                                                       
──────────────────────────────────────────────────────────────────────────────
                                                  2                           
                                                 r                            

                                ⎛  2                  ⎞│                      
        ⎞│                 2    ⎜ ∂                   ⎟│                      
θ), ξ₂))⎟│            + sin (θ)⋅⎜────(f(ξ₁, r⋅cos(θ)))⎟│            + 2⋅sin(θ)
        ⎠│ξ₂=r⋅cos(θ)           ⎜   2                 ⎟│                      
                                ⎝∂ξ₁                  ⎠│ξ₁=r⋅sin(θ)           
──────────────────────────────────────────────────────────────────────────────
                                                                              
                                                                              

        ⎛⎛    2             ⎞│           ⎞│                      ⎛  2         
        ⎜⎜   ∂              ⎟│           ⎟│                 2    ⎜ ∂          
⋅cos(θ)⋅⎜⎜───────(f(ξ₁, ξ₂))⎟│           ⎟│            + cos (θ)⋅⎜────(f(r⋅sin
        ⎝⎝∂ξ₂ ∂ξ₁           ⎠│ξ₁=r⋅sin(θ)⎠│ξ₂=r⋅cos(θ)           ⎜   2        
                                                                 ⎝∂ξ₂         
──────────────────────────────────────────────────────────────────────────────
                                                                              
                                                                              

         ⎞│           
         ⎟│           
(θ), ξ₂))⎟│           
         ⎟│           
         ⎠│ξ₂=r⋅cos(θ)
──────────────────────
                      
                      

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