2020年7月18日土曜日

学習環境

続 解析入門 (原書第2版) (S.ラング(著)、松坂 和夫(翻訳)、片山 孝次(翻訳)、岩波書店)の第2章(ベクトルの微分)、1(微分係数)の練習問題1、2、3、4の解答を求めてみる。


  1. d dt ( e t , cos t , sin t ) = ( e t , - sin t , cos t )

  2. d dt ( sin ( 2 t ) , log ( 1 + t ) , t ) = ( 2 cos ( 2 t ) , 1 1 + t , 1 )

  3. d dt ( cos t , sin t ) = ( - sin t , cos t )

  4. d dt ( cos 3 t , sin 3 t ) = ( - 3 sin 3 t , 3 cos 3 t )

コード

#!/usr/bin/env python3
from unittest import TestCase, main
from sympy import Matrix, sin, cos, exp, log, Derivative
from sympy.abc import t
from sympy.plotting import plot3d_parametric_line

print('1, 2, 3, 4.')


class Test(TestCase):
    def test1(self):
        self.assertEqual(
            Derivative(Matrix([exp(t), cos(t), sin(t)]), t, 1).doit(),
            Matrix([exp(t), -sin(t), cos(t)])
        )

    def test2(self):
        self.assertEqual(
            Derivative(Matrix([sin(2 * t), log(1 + t), t]), t, 1).doit(),
            Matrix([2 * cos(2 * t), 1 / (1 + t), 1])
        )

    def test3(self):
        self.assertEqual(
            Derivative(Matrix([cos(t), sin(t)]), t, 1).doit(),
            Matrix([-sin(t), cos(t)])
        )

    def test4(self):
        self.assertEqual(
            Derivative(Matrix([cos(3 * t), sin(3 * t)]), t, 1).doit(),
            Matrix([-3 * sin(3 * t), 3 * cos(3 * t)])
        )


p = plot3d_parametric_line(
    (exp(t), cos(t), sin(t), (t, -5, 5)),
    *[(exp(t0) + t * exp(t0),
       cos(t0) - t * sin(t0),
       sin(t0) + t * cos(t0),
       (t, 0, 1))
      for t0 in range(-5, 6) if t0 != 0],
    legend=False,
    show=False
)

colors = ['red', 'green', 'blue', 'brown', 'orange',
          'purple', 'pink', 'gray', 'skyblue', 'yellow']
for o, color in zip(p, colors):
    o.line_color = color
p.save('sample1.png')
p.show()

if __name__ == "__main__":
    main()

入出力結果(Zsh、PowerShell、Terminal、Jupyter(IPython))

% ./sample1.py -v 
1, 2, 3, 4.
test1 (__main__.Test) ... ok
test2 (__main__.Test) ... ok
test3 (__main__.Test) ... ok
test4 (__main__.Test) ... ok

----------------------------------------------------------------------
Ran 4 tests in 0.013s

OK
%

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