数学 - Python - 解析学 - 多変数の関数 - 多変数の関数 - 偏微分 - 偏導関数、積、累乗、三角関数(正弦、余弦)、逆正弦関数、指数関数

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19章(多変数の関数)、2(偏微分)の練習問題1、2、3、4、5、6、7、8、9、10の解答を求めてみる。

$\begin{array}{l} \frac{\partial f}{\partial x} \\ = \frac{\partial}{\partial x} (x y + z) \\ = y \\ \frac{\partial}{\partial y} (x y + z) = x \\ \frac{\partial}{\partial z} (x y + z) = 1 \end{array}$
$\begin{array}{l} \frac{\partial}{\partial x} (x^{2} y^{5} + 1) = 2 x y^{5} \\ \frac{\partial f}{\partial y} = 5 x^{2} y^{4} \\ \frac{\partial f}{\partial z} = 0 \end{array}$
$\begin{array}{l} \frac{\partial}{\partial x} (\sin (x y) + \cos z) \\ = (\cos (x y)) y \\ \frac{\partial f}{\partial y} = (\cos (x y)) x \\ \frac{\partial f}{\partial z} = - \sin z \end{array}$
$\begin{array}{l} \frac{\partial}{\partial x} \cos (x y) = - (\sin x y) y \\ \frac{\partial f}{\partial y} = - (\sin x y) x \\ \frac{\partial f}{\partial z} = 0 \end{array}$
$\begin{array}{l} \frac{\partial}{\partial x} \sin (x y z) = (\cos (x y z)) y z \\ \frac{\partial f}{\partial y} = (\cos (x y z)) x z \\ \frac{\partial f}{\partial z} = (\cos (x y z)) x y \end{array}$
$\begin{array}{l} \frac{\partial}{\partial x} e^{x y z} = e^{x y z} y z \\ \frac{\partial f}{\partial y} = e^{x y z} x z \\ \frac{\partial f}{\partial z} = e^{x y z} x y \end{array}$
$\begin{array}{l} \frac{\partial}{\partial x} (x^{2} \sin (y z)) = 2 x \sin (y z) \\ \frac{\partial f}{\partial y} = x^{2} (\cos (y z)) z \\ \frac{\partial f}{\partial z} = x^{2} (\cos (y z)) y \end{array}$
$\begin{array}{l} \frac{\partial}{\partial x} (x y z) = y z \\ \frac{\partial f}{\partial y} = x z \\ \frac{\partial f}{\partial z} = x y \end{array}$
$\begin{array}{l} \frac{\partial}{\partial x} (x z + y z + x y) = z + y \\ \frac{\partial f}{\partial y} = z + x \\ \frac{\partial f}{\partial z} = x + y \end{array}$
$\begin{array}{l} \frac{\partial}{\partial x} (x \cos (y - 3 z) + \arcsin (x y)) \\ = \cos (y - 3 z) + \frac{y}{\sqrt{1 - x^{2} y^{2}}} \\ \frac{\partial f}{\partial y} = x (- \sin (y - 3 z)) + \frac{x}{\sqrt{1 - x^{2} y^{2}}} \\ \frac{\partial f}{\partial z} = x (- \sin (y - 3 z)) (- 3) \end{array}$

#!/usr/bin/env python3 from unittest import TestCase, main from sympy import symbols, plot, sin, cos, exp, asin, Derivative, sqrt from sympy.plotting import plot3d print('1, 2, 3, 4, 5, 6, 7, 8, 9, 10.') x, y, z = symbols('x, y, z', real=True) fs = [x * y + z, x ** 2 * y ** 5 + 1, sin(x * y) + cos(z), cos(x * y), sin(x * y * z), exp(x * y * z), x ** 2 * sin(y * z), x * y * z, x * z + y * z + x * y, x * cos(y - 3 * z) + asin(x * y)] fxyzs = [(y, x, 1), (2 * x * y ** 5, 5 * x ** 2 * y ** 4, 0), (cos(x * y) * y, cos(x * y) * x, -sin(z)), (-sin(x * y) * y, -sin(x * y) * x, 0), (cos(x * y * z) * y * z, cos(x * y * z) * x * z, cos(x * y * z) * x * y), (exp(x * y * z) * y * z, exp(x * y * z) * x * z, exp(x * y * z) * x * y), (2 * x * sin(y * z), x ** 2 * cos(y * z) * z, x ** 2 * cos(y * z) * y), (y * z, x * z, x * y), (z + y, z + x, x + y), (cos(y - 3 * z) + y / sqrt(1 - x ** 2 * y ** 2), x * (- sin(y - 3 * z)) + x / sqrt(1 - x ** 2 * y ** 2), x * (-sin(y - 3 * z)) * (-3))] class TestPartialDerivative(TestCase): def test(self): for f, fxyz in zip(fs, fxyzs): for fo, o in zip(fxyz, [x, y, z]): self.assertEqual( Derivative(f, o, 1).doit(), fo) for i, f in enumerate(fs, 1): p = plot3d(f.subs({z: 1}), show=False) p.save(f'sample{i}.png') p.show() if __name__ == "__main__": main()