数学 - Python - 解析学 - 多変数の関数 - 合成微分律と勾配ベクトル - 1変数の関数、ノルム、1変数の微分可能な関数、2変数の微分可能な関数

学習環境

解析入門原書第3版 (S.ラング(著)、松坂和夫(翻訳)、片山孝次(翻訳)、岩波書店)の第Ⅵ部(多変数の関数)、第20章(合成微分律と勾配ベクトル)、1(合成微分律)の練習問題8の解答を求めてみる。

1. $\begin{array}{l} r \\ = ∥X∥ \\ = ∥(x, y, z)∥ \\ = \sqrt{x^{2} + y^{2} + z^{2}} \\ \frac{d f}{dx} \\ = \frac{\partial g}{\partial x} \\ = \frac{dg}{d r} \frac{x}{\sqrt{x^{2} + y^{2} + z^{2}}} \\ \frac{d f}{dy} = \frac{d f}{d r} \frac{y}{\sqrt{x^{2} + y^{2} + z^{2}}} \\ \frac{d f}{dz} = \frac{d f}{d r} \frac{z}{\sqrt{x^{2} + y^{2} + z^{2}}} \\ {(\frac{d f}{dx})}^{2} + {(\frac{d f}{dy})}^{2} + {(\frac{d f}{dz})}^{2} \\ = {(\frac{d f}{d r})}^{2} \frac{x^{2} + y^{2} + z^{2}}{x^{2} + y^{2} + z^{2}} \\ = {(\frac{d f}{d r})}^{2} \\ = {(\frac{d g}{d r})}^{2} \end{array}$
2. $\begin{array}{l} \frac{\partial g}{\partial x} \\ = \frac{\partial f}{\partial x} \\ = (\frac{\partial f}{\partial u}, \frac{\partial f}{\partial v}) \cdot (1, 1) \\ = \frac{\partial f}{\partial u} + \frac{\partial f}{\partial v} \\ \frac{\partial g}{\partial y} \\ = (\frac{\partial f}{\partial u}, \frac{\partial f}{\partial v}) \cdot (1, - 1) \\ = \frac{\partial f}{\partial u} - \frac{\partial f}{\partial v} \\ \frac{\partial g}{\partial x} \frac{\partial g}{\partial y} \\ = {(\frac{\partial f}{\partial u})}^{2} - {(\frac{\partial f}{\partial v})}^{2} \end{array}$
3. $t = 2 x + 7 y$
  
  とおく。
  
  このとき、
  
  $\begin{array}{l} \frac{\partial g}{\partial x} \\ = \frac{\partial f}{\partial x} \\ = \frac{d f}{d s} 2 \\ \frac{\partial g}{\partial y} = \frac{d f}{d s} 7 \end{array}$
  
  よって、
  
  $\begin{array}{l} \frac{1}{2} \frac{\partial g}{\partial x} = \frac{1}{7} \frac{\partial g}{\partial y} \\ 2 \frac{\partial g}{\partial y} = 7 \frac{\partial g}{\partial x} \end{array}$

#!/usr/bin/env python3 from unittest import TestCase, main from sympy import symbols, Function, Matrix from sympy.plotting import plot3d print('8.') x, y, z, r, u, v = symbols('x, y, z, r, u, v', real=True) class TestDerivativeChainRule(TestCase): def test_a(self): X = Matrix([x, y, z]) d = {r: X.norm()} g = Function('g')(r) f = g.subs(d) self.assertEqual( (g.diff(r, 1).subs(d) ** 2).simplify(), (f.diff(x, 1) ** 2 + f.diff(y, 1) ** 2 + f.diff(z, 1) ** 2) .simplify() ) def test_b(self): f = Function('f')(u, v) d = {u: x + y, v: x - y} g = f.subs(d) self.assertEqual( (g.diff(x, 1) * g.diff(y, 1)).simplify(), (f.diff(u) ** 2 - f.diff(v, 1) ** 2).subs(d).simplify() ) def test_c(self): f = Function('f')(r) d = {r: 2 * x + 7 * y} g = f.subs(d) self.assertEqual(2 * g.diff(y), 7 * g.diff(x, 1)) if __name__ == "__main__": main()

% ./sample8.py -v 8. test_a (__main__.TestDerivativeChainRule) ... ok test_b (__main__.TestDerivativeChainRule) ... ok test_c (__main__.TestDerivativeChainRule) ... ok ---------------------------------------------------------------------- Ran 3 tests in 0.608s OK %

Kamimura's blog

2020年4月24日金曜日

数学 - Python - 解析学 - 多変数の関数 - 合成微分律と勾配ベクトル - 1変数の関数、ノルム、1変数の微分可能な関数、2変数の微分可能な関数

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