数学 - Python - 解析学 - 多変数の関数 - 合成微分律と勾配ベクトル - 偏導関数、変数、内積

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2の解答を求めてみる。

1. $\begin{array}{l} \frac{d}{dx} f (x, y, z) \\ = \frac{d}{dx} (x^{3} + 3 x y z - y^{2} z) \\ = 3 x^{2} + 3 y z \\ \frac{\partial}{\partial y} f (x, y, z) = 3 x z - 2 y z \\ \frac{\partial}{\partial s} f (x, y, z) \\ = (3 x^{2} + 3 y z, 3 x z - 2 y z, 3 x y - y^{2}) \cdot (1, - 1, 2 s) \\ = 3 x^{2} + 3 y z - 3 x z + 2 z + 6 s x y - 2 s y^{2} \\ = 3 x^{2} + 5 y z - 3 x z + 6 s x y - 2 s y^{2} \\ \frac{\partial}{\partial t} f (x, y, z) \\ = (g r a d f) \cdot (2, - 1, 2 t) \\ = 6 x^{2} + 6 y z - 3 x z + 2 y z + 6 t x y - 2 t y^{2} \\ = 6 x^{2} + 8 y z - 3 x z + 6 t x y - 2 t y^{2} \end{array}$
2. $\begin{array}{l} \frac{d}{dx} f (x, y, z) \\ = \frac{d}{dx} \frac{x + y}{1 - x y} \\ = \frac{(1 - x y) - (x + y) (- y)}{{(1 - x y)}^{2}} \\ = \frac{y^{2} + 1}{{(1 - x y)}^{2}} \\ \frac{d f}{dy} = \frac{x^{2} + 1}{{(1 - x y)}^{2}} \\ \frac{d f}{d s} \\ = (g r a d f) \cdot (0, (3 - s)) \\ = \frac{(1^{2} + 1) (3 - s)}{{(1 - x y)}^{2}} \\ \frac{d f}{dt} \\ = (g r a d f) \cdot (2 \cos 2 t, - 3 \sin (3 t - s)) \\ = \frac{2 (y^{2} + 1) \cos 2 t - 3 (x^{2} + 1) \sin (3 t - s)}{{(1 - x y)}^{2}} \end{array}$

#!/usr/bin/env python3 from unittest import TestCase, main from sympy import symbols, sin, cos from sympy.plotting import plot3d print('2.') x, y, z, s, t = symbols('x, y, z, s, t') fa = x ** 3 + 3 * x * y * z - y ** 2 * z xa = 2 * t + s ya = -t - s za = t ** 2 + s ** 2 da = {x: xa, y: ya, z: za} fb = (x + y) / (1 - x * y) xb = sin(2 * t) yb = cos(3 * t - s) db = {x: xb, y: yb} class TestSyntheticDerivative(TestCase): def test_a_dx(self): self.assertEqual(fa.diff(x, 1), 3 * x ** 2 + 3 * y * z) def test_a_dy(self): self.assertEqual(fa.diff(y, 1), 3 * x * z - 2 * y * z) def test_a_ds(self): self.assertEqual( fa.subs(da).diff(s, 1).simplify(), (3 * x ** 2 + 5 * y * z - 3 * x * z + 6 * s * x * y - 2 * s * y ** 2).subs(da).simplify() ) def test_a_dt(self): self.assertEqual( fa.subs(da).diff(t, 1).simplify(), (6 * x ** 2 + 8 * y * z - 3 * x * z + 6 * t * x * y - 2 * t * y ** 2).subs(da).simplify() ) def test_b_dx(self): self.assertEqual(fb.diff(x, 1).factor(), ((y ** 2 + 1) / (1 - x * y) ** 2).factor()) def test_b_dy(self): self.assertEqual(fb.diff(y, 1).factor(), ((x ** 2 + 1) / (1 - x * y) ** 2).factor()) def test_b_ds(self): self.assertEqual( fb.subs(db).diff(s, 1).factor(), ((x ** 2 + 1) * sin(3 * t - s) / (1 - x * y) ** 2).subs(db).factor()) def test_b_dt(self): self.assertEqual( fb.subs(db).diff(t, 1).factor(), ((2 * (y ** 2 + 1) * cos(2 * t) - 3 * (x ** 2 + 1) * sin(3 * t - s)) / (1 - x * y) ** 2).subs(db).factor() ) for i, f in enumerate([fa.subs(da), fb.subs(db)]): p = plot3d(f, show=False) p.save('sample2_{i}.png') p.show() if __name__ == "__main__": main()

% ./sample2.py -v 2. test_a_ds (__main__.TestSyntheticDerivative) ... ok test_a_dt (__main__.TestSyntheticDerivative) ... ok test_a_dx (__main__.TestSyntheticDerivative) ... ok test_a_dy (__main__.TestSyntheticDerivative) ... ok test_b_ds (__main__.TestSyntheticDerivative) ... ok test_b_dt (__main__.TestSyntheticDerivative) ... ok test_b_dx (__main__.TestSyntheticDerivative) ... ok test_b_dy (__main__.TestSyntheticDerivative) ... ok ---------------------------------------------------------------------- Ran 8 tests in 0.636s OK %

Kamimura's blog

2020年4月18日土曜日

数学 - Python - 解析学 - 多変数の関数 - 合成微分律と勾配ベクトル - 偏導関数、変数、内積

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