ベクトルの微分曲線の長さ、三角関数、正弦と余弦、加法定理、倍角、速度ベクトル、積分

続解析入門 (原書第2版)
楽天ブックス
Yahoo!ショッピング
au PAY マーケット
原著: Calculus of Several Variables

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)、2(曲線の長さ)の練習問題4-a、b.の解答を求めてみる。

X' (t) = (1 - \cos t, \sin t)

∥ X' (t) ∥

= \sqrt{1 - 2 \cos t + \cos^{2} t + \sin^{2} t}

= \sqrt{2 - 2 \cos t}

= \sqrt{2} \sqrt{1 - \cos t}

= \sqrt{2} \sqrt{\cos (\frac{t}{2} - \frac{t}{2}) - \cos (\frac{t}{2} + \frac{t}{2})}

= \sqrt{2} \sqrt{2 \sin^{2} \frac{t}{2}}

= 2 \sqrt{\sin^{2} \frac{t}{2}}

求める曲線の長さ。

2 \int_{0}^{2 π} \sqrt{\sin^{2} \frac{t}{2}} dt

= 2 (\int_{0}^{π} \sin \frac{t}{2} dt - \int_{π}^{2 π} \sin \frac{t}{2} dt)

= 4 {[- 2 \cos \frac{t}{2}]}_{0}^{π}

= 8

ここから

2 \int_{3}^{5} \sqrt{\sin^{2} \frac{t}{2}} dt

= 2 (\int_{3}^{π} \sin \frac{t}{2} dt - \int_{π}^{5} \sin \frac{t}{2} dt)

= 2 ({[- 2 \cos \frac{t}{2}]}_{3}^{π} - {[- 2 \cos \frac{t}{2}]}_{π}^{5})

= 2 (2 \cos \frac{3}{2} + 2 \cos \frac{5}{2})

= 4 (\cos \frac{3}{2} + \cos \frac{5}{2})

コード

#!/usr/bin/env python3
from unittest import TestCase, main
from sympy import Matrix, Derivative, sin, cos, Integral, pi, sqrt
from sympy import Rational
from sympy.plotting import plot_parametric
from sympy.abc import t

print('4.')

X = Matrix([t - sin(t), 1 - cos(t)])
X1 = Derivative(X, t, 1).doit()


class Test(TestCase):
    def test_a(self):
        self.assertEqual(
            Integral(2 * sqrt(sin(t / 2) ** 2), (t, 0, 2 * pi)).doit(),
            8
        )

    def test_b(self):
        self.assertEqual(
            Integral(2 * sqrt(sin(t / 2) ** 2), (t, 3, 5)).doit(),
            4 * (cos(Rational(3, 2)) - cos(Rational(5, 2)))
        )


p = plot_parametric(
    (*X, (t, -pi, 0)),
    (*X, (t, 0, 2 * pi)),
    (*X, (t, 2 * pi, 3 * pi)),
    show=False,
)
colors = ['red', 'green', 'blue', 'brown', 'orange',
          'purple', 'pink', 'gray', 'skyblue', 'yellow']
for o, color in zip(p, colors):
    o.line_color = color
    print(o, color)
p.save('sample4_a.png')
p = plot_parametric(
    (*X, (t, 2, 3)),
    (*X, (t, 3, 5)),
    (*X, (t, 5, 6)),
    show=False,
)
for o, color in zip(p, colors):
    o.line_color = color
    print(o, color)
p.save('sample4_b.png')
p.show()


if __name__ == "__main__":
    main()

入出力結果

 % ./sample4.py -v
4.
parametric cartesian line: (t - sin(t), 1 - cos(t)) for t over (-3.141592653589793, 0.0) red
parametric cartesian line: (t - sin(t), 1 - cos(t)) for t over (0.0, 6.283185307179586) green
parametric cartesian line: (t - sin(t), 1 - cos(t)) for t over (6.283185307179586, 9.42477796076938) blue
parametric cartesian line: (t - sin(t), 1 - cos(t)) for t over (2.0, 3.0) red
parametric cartesian line: (t - sin(t), 1 - cos(t)) for t over (3.0, 5.0) green
parametric cartesian line: (t - sin(t), 1 - cos(t)) for t over (5.0, 6.0) blue
test_a (__main__.Test) ... ok
test_b (__main__.Test) ... ok

----------------------------------------------------------------------
Ran 2 tests in 0.671s

OK
%