数学 - Python - 解析学 - ベクトルの微分 - 微分係数 - 2曲線、交点、接線のなす角、指数関数、三角関数(正弦と余弦)、内積

続 解析入門 (原書第2版)
楽天ブックス Yahoo!
原著: Calculus of Several Variables

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• 参考書籍
  • Pythonからはじめる数学入門 (楽天ブックス Yahoo!)

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18の解答を求めてみる。

18. 
    
    $\begin{array}{l}{e}^t=1-\theta \\ e^{2t}=\cos \theta \\ 1-e^{-t}=\sin \theta \end{array}$

    $\begin{array}{l}{e}^t=-1\\ 1=\cos \theta \\ \theta =0\\ 1-e^{-t}=0\\ t=0\end{array}$

    よって、 2曲線は

    $(1,1,0)$

    で交わる。

    $\frac{d}{\mathrm{dt}}(e^t,e^{2t},1-e^{-t})=(e^t,2e^{2t},e^{-t})$

    $\frac{d}{\mathrm{dt}}(1-\theta ,\cos \theta ,\sin \theta )=(-1,-\sin \theta ,\cos \theta )$

    $(e^t,2e^{2t},e^{-t})·(-1,-\sin \theta ,\cos \theta )$

    $=-e^t-2e^{2t}\sin \theta +e^{-t}\cos \theta$

    $\begin{array}{l}t=0,\theta =0\\ -1+0+1=0\end{array}$

    よって、接線の間の角の余弦は

    $0$

    よって、求める角は

    $\frac{\pi }{2}$

コード

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

print('18.')

t, theta = symbols('t, θ', real=True)
x1 = Matrix([exp(t), exp(2 * t), 1 - exp(-t)])
x2 = Matrix([1 - theta, cos(theta), sin(theta)])
x11 = Derivative(x1, t, 1).doit()
x21 = Derivative(x2, theta, 1).doit()
t0 = 0
theta0 = solve((x1 - x2).subs({t: t0}), theta)[0][0]


class Test(TestCase):
    def test1(self):
        self.assertEqual(x1.subs({t: t0}), x2.subs({theta: theta0}))

    def test2(self):
        a = x11.subs({t: t0})
        b = x21.subs({theta: theta0})
        self.assertEqual(a.dot(b) / (a.norm() * b.norm()), 0)


colors = ['red', 'green', 'blue', 'brown', 'orange',
          'purple', 'pink', 'gray', 'skyblue', 'yellow']
p = plot3d_parametric_line(
    (*x1, (t, -1, 1)),
    (*x2.subs({theta: t}), (t, -1, 1)),
    (*(x1.subs({t: t0}) + t * x11.subs({t: t0})), (t, 0, 1)),
    (*(x2.subs({theta: theta0}) + t * x21.subs({theta: theta0})), (t, 0, 1)),
    show=False,
    legend=False,
)
for o, color in zip(p, colors):
    o.line_color = color
p.save(f'sample18.png')
p.show()


if __name__ == "__main__":
    main()

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

% ./sample18.py -v
18.
/opt/local/Library/Frameworks/Python.framework/Versions/3.8/lib/python3.8/site-packages/numpy/ma/core.py:2831: VisibleDeprecationWarning: Creating an ndarray from ragged nested sequences (which is a list-or-tuple of lists-or-tuples-or ndarrays with different lengths or shapes) is deprecated. If you meant to do this, you must specify 'dtype=object' when creating the ndarray
  _data = np.array(data, dtype=dtype, copy=copy,
test1 (__main__.Test) ... ok
test2 (__main__.Test) ... ok

----------------------------------------------------------------------
Ran 2 tests in 0.003s

OK
%