Stokes の定理 - yoshitake-hのブログ

有向 $C^2$ 曲面 $S\subset \mathbb{R}^3$ とその上の $C^1$ 関数 $f$ ，　 $C^2$ 関数 $g$ に対し，

$\int_{\partial S}f\,\mathrm{d}g=\int\!\!\!\int_S\mathrm{d}f\,\mathrm{d}g.$

証明
$\int_{\partial S}f\,\mathrm{d}g =\int_If\,\frac{\mathrm{d}g}{\mathrm{d}t}\,\mathrm{d}t$
$=\int_If\left( \frac{\partial g}{\partial u}\,\frac{\mathrm{d}u}{\mathrm{d}t}+\frac{\partial g}{\partial v}\,\frac{\mathrm{d}v}{\mathrm{d}t}\right)\,\mathrm{d}t$
$=\int_{\partial D} f\,\frac{\partial g}{\partial u}\,\mathrm{d}u+f\,\frac{\partial g}{\partial v}\,\mathrm{d}v$
Green の定理より，
$=\int\!\!\!\int_D\left(\frac{\partial}{\partial u}\left( f\,\frac{\partial g}{\partial v}\right) -\frac{\partial}{\partial v}\left( f\,\frac{\partial g}{\partial u}\right)\right)\,\mathrm{d}u\,\mathrm{d}v$
$g$ が $C^2$ なので，
$=\int\!\!\!\int_D\left(\frac{\partial f}{\partial u}\,\frac{\partial g}{\partial v} -\frac{\partial f}{\partial v}\,\frac{\partial g}{\partial u}\right)\,\mathrm{d}u\,\mathrm{d}v$
$=\int\!\!\!\int_S\mathrm{d}f\,\mathrm{d}g.\quad //$