## 2020年3月31日火曜日

### 数学 - 微分積分学 - 積分法 - 平面曲線の長さ - 正則弧、狭義増加関数、逆関数

1. $s={\int }_{a}^{t}\sqrt{{\left(\phi \left(\tau \right)\right)}^{2}+{\left(\psi \left(\tau \right)\right)}^{2}}d\tau$

の逆関数を

$h\left(s\right)$

とおく。

$\begin{array}{l}x\\ =\phi \left(h\left(s\right)\right)\\ =f\left(s\right)\\ y\\ =\psi \left(h\left(s\right)\right)\\ =g\left(s\right)\\ f\text{'}\left(s\right)\\ =\phi \text{'}\left(h\left(s\right)\right)h\text{'}\left(s\right)\\ =\phi \text{'}\left(t\right)h\text{'}\left(s\right)\\ g\text{'}\left(s\right)\\ =\psi \text{'}\left(h\left(s\right)\right)h\text{'}\left(s\right)\\ =\psi \text{'}\left(t\right)h\text{'}\left(s\right)\\ {\left(f\text{'}\left(s\right)\right)}^{2}+{\left(g\text{'}\left(s\right)\right)}^{2}\\ =\left({\left(\phi \text{'}\left(t\right)\right)}^{2}+{\left(\psi \text{'}\left(t\right)\right)}^{2}\right){\left(h\text{'}\left(s\right)\right)}^{2}\\ =\frac{\phi \text{'}{\left(t\right)}^{2}+\psi \text{'}{\left(t\right)}^{2}}{{\left(\sqrt{\phi \text{'}{\left(t\right)}^{2}+\psi \text{'}{\left(t\right)}^{2}}\right)}^{2}}\\ =1\end{array}$

（証明終）