簡にして要を得る〜弧長パラメーターと曲率〜弧長パラメーター曲線の長さ、逆関数、置換積分法、対数関数

学習環境

新装版　幾何学は微分しないと　〜微分幾何学入門〜 (中内伸光(著)、現代数学社)の第1章(簡にして要を得る〜弧長パラメーターと曲率〜)、1.3(弧長パラメーター)の練習問題1.2の解答を求めてみる。

s (t)

= \int_{0}^{t} ∥ C' (t) ∥ dt

= \int_{0}^{t} ∥ (1, 2 t) ∥ dt

= \int_{0}^{t} \sqrt{1 + 4 t^{2}} dt

= \int_{0}^{t} \sqrt{1 + {(2 t)}^{2}} dt

置換積分法。

\begin{array}{l} u = 2 t \\ \frac{d u}{dt} = 2 \\ t = 0, 1 \\ u = 0, 2 t \end{array}

\frac{1}{2} \int_{0}^{2 t} \sqrt{1 + u^{2}} d u

= \frac{1}{2} \cdot \frac{1}{2} {[u \sqrt{1 + u^{2}} + \log (u + \sqrt{1 + u^{2}})]}_{0}^{2 t}

= \frac{1}{2} \cdot \frac{1}{2} (2 t \sqrt{1 + 4 t^{2}} + \log (2 t + \sqrt{1 + 4 t^{2}}))

= \frac{1}{2} t \sqrt{1 + 4 t^{2}} + \frac{1}{4} \log (2 t + \sqrt{1 + 4 t^{2}})

この関数の逆関数を

t = t (s)

とおくと、求める問題の放物線を弧長パラメーターで表示したものは、

C (s) = (t (s), t {(s)}^{2})

コード(Wolfram Language, Jupyter)

ParametricPlot[{t, t^2}, {t, -5, 5}]

Integrate[Norm[D[{t, t^2}, t]], {t, 0, t}]

D[{t, t^2}, t]

Integrate[Sqrt[1 + 4t^2], {t, 0, t}]

Solve[s == 1/4(2t Sqrt[1+4t^2] + ArcSinh[2t]), t]

This system cannot be solved with the methods available to Solve.: This system cannot be solved with the methods available to Solve.

Solve[s == 1/4(2t Sqrt[1+4t^2] + Log[2t + Sqrt[1+4t^2]]), t]

This system cannot be solved with the methods available to Solve.: This system cannot be solved with the methods available to Solve.

Mathematica(Wolfram Engine)でも逆関数は求められないみたい。