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(偏微分、グラディエント(gradient)、三角関数(正弦、余弦)、累乗(平方)、等式)

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11.を取り組んでみる。

11. 
    
    $\begin{array}{}\frac{\partial \; <!--\; partial\; differential\; -->g}{\partial \; <!--\; partial\; differential\; -->r}=gradf\left(x,y,z\right)·\; <!--\; middle\; dot\; -->\left(\sin \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\cos \mathrm{\phi \; <!--\; greek\; small\; letter\; phi\; -->},\sin \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\sin \mathrm{\phi \; <!--\; greek\; small\; letter\; phi\; -->},\cos \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\right)\\ =\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->x}\sin \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\cos \mathrm{\phi \; <!--\; greek\; small\; letter\; phi\; -->}+\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->y}\sin \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\sin \mathrm{\phi \; <!--\; greek\; small\; letter\; phi\; -->}+\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->z}\cos \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\end{array}$
    $\frac{\partial \; <!--\; partial\; differential\; -->g}{\partial \; <!--\; partial\; differential\; -->\mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}}=\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->x}r\cos \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\cos \mathrm{\phi \; <!--\; greek\; small\; letter\; phi\; -->}+\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->y}r\cos \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\sin \mathrm{\phi \; <!--\; greek\; small\; letter\; phi\; -->}-\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->z}r\sin \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}$
    $\frac{\partial \; <!--\; partial\; differential\; -->g}{\partial \; <!--\; partial\; differential\; -->\mathrm{\phi \; <!--\; greek\; small\; letter\; phi\; -->}}=-\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->x}r\sin \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\sin \mathrm{\phi \; <!--\; greek\; small\; letter\; phi\; -->}+\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->y}r\sin \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\cos \mathrm{\phi \; <!--\; greek\; small\; letter\; phi\; -->}$
    $\frac{1}{r}\frac{\partial \; <!--\; partial\; differential\; -->g}{\partial \; <!--\; partial\; differential\; -->\mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}}=\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->x}\cos \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\cos \mathrm{\phi \; <!--\; greek\; small\; letter\; phi\; -->}+\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->y}\cos \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\sin \mathrm{\phi \; <!--\; greek\; small\; letter\; phi\; -->}-\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->z}\sin \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}$
    $\frac{1}{r\sin \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}}\frac{\partial \; <!--\; partial\; differential\; -->g}{\partial \; <!--\; partial\; differential\; -->\mathrm{\phi \; <!--\; greek\; small\; letter\; phi\; -->}}=-\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->x}\sin \mathrm{\phi \; <!--\; greek\; small\; letter\; phi\; -->}+\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->y}\cos \mathrm{\phi \; <!--\; greek\; small\; letter\; phi\; -->}$
    $\begin{array}{}{\left(\frac{\partial \; <!--\; partial\; differential\; -->g}{\partial \; <!--\; partial\; differential\; -->r}\right)}^2\\ ={\left(\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->x}\right)}^2{\sin }^2\mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}{\cos }^2\mathrm{\phi \; <!--\; greek\; small\; letter\; phi\; -->}+{\left(\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->y}\right)}^2{\sin }^2\mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}{\sin }^2\mathrm{\phi \; <!--\; greek\; small\; letter\; phi\; -->}+{\left(\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->z}\right)}^2{\cos }^2\mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\\ +2\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->x}·\; <!--\; middle\; dot\; -->\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->y}{\sin }^2\mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\sin \mathrm{\phi \; <!--\; greek\; small\; letter\; phi\; -->}\cos \mathrm{\phi \; <!--\; greek\; small\; letter\; phi\; -->}+2\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->y}·\; <!--\; middle\; dot\; -->\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->z}\sin \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\cos \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\sin \mathrm{\phi \; <!--\; greek\; small\; letter\; phi\; -->}\\ +2\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->z}·\; <!--\; middle\; dot\; -->\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->x}\sin \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\cos \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\cos \mathrm{\phi \; <!--\; greek\; small\; letter\; phi\; -->}\end{array}$
    $\begin{array}{}{\left(\frac{1}{r}\frac{\partial \; <!--\; partial\; differential\; -->g}{\partial \; <!--\; partial\; differential\; -->\mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}}\right)}^2\\ ={\left(\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->x}\right)}^2{\cos }^2\mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}{\cos }^2\mathrm{\phi \; <!--\; greek\; small\; letter\; phi\; -->}+{\left(\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->y}\right)}^2{\cos }^2\mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}{\sin }^2\mathrm{\phi \; <!--\; greek\; small\; letter\; phi\; -->}+{\left(\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->z}\right)}^2{\sin }^2\mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\\ +2\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->x}·\; <!--\; middle\; dot\; -->\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->y}{\cos }^2\mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\sin \mathrm{\phi \; <!--\; greek\; small\; letter\; phi\; -->}\cos \mathrm{\phi \; <!--\; greek\; small\; letter\; phi\; -->}-2\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->y}·\; <!--\; middle\; dot\; -->\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->z}\sin \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\cos \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\sin \mathrm{\phi \; <!--\; greek\; small\; letter\; phi\; -->}\\ -2\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->z}\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->x}\sin \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\cos \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\cos \mathrm{\phi \; <!--\; greek\; small\; letter\; phi\; -->}\end{array}$
    ${\left(\frac{1}{r\sin \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}}\frac{\partial \; <!--\; partial\; differential\; -->g}{\partial \; <!--\; partial\; differential\; -->\mathrm{\phi \; <!--\; greek\; small\; letter\; phi\; -->}}\right)}^2={\left(\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->x}\right)}^2{\sin }^2\mathrm{\phi \; <!--\; greek\; small\; letter\; phi\; -->}+{\left(\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->y}\right)}^2{\cos }^2\mathrm{\phi \; <!--\; greek\; small\; letter\; phi\; -->}-2\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->x}\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->y}\sin \mathrm{\phi \; <!--\; greek\; small\; letter\; phi\; -->}\cos \mathrm{\phi \; <!--\; greek\; small\; letter\; phi\; -->}$

    よって、

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

コード(Emacs)

Python 3

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

r, a, b = symbols('r, a, b', nonzero=True)
x = r * sin(a) * cos(b)
y = r * sin(a) * sin(b)
z = r * cos(b)
g = Function('g')(r, a, b)
expr = Derivative(g, r, 1) ** 2 + (1 / r * Derivative(g, a, 1)
                                   ) ** 2 + (1 / (r * sin(a)) * Derivative(g, b, 1)) ** 2
for t in [expr, expr.doit()]:
    pprint(t)
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample11.py
                                    2                   2
                    ⎛∂             ⎞    ⎛∂             ⎞ 
                2   ⎜──(g(r, a, b))⎟    ⎜──(g(r, a, b))⎟ 
⎛∂             ⎞    ⎝∂a            ⎠    ⎝∂b            ⎠ 
⎜──(g(r, a, b))⎟  + ───────────────── + ─────────────────
⎝∂r            ⎠             2               2    2      
                            r               r ⋅sin (a)   

                                    2                   2
                    ⎛∂             ⎞    ⎛∂             ⎞ 
                2   ⎜──(g(r, a, b))⎟    ⎜──(g(r, a, b))⎟ 
⎛∂             ⎞    ⎝∂a            ⎠    ⎝∂b            ⎠ 
⎜──(g(r, a, b))⎟  + ───────────────── + ─────────────────
⎝∂r            ⎠             2               2    2      
                            r               r ⋅sin (a)   

$