$M\subset\mathbb R^3$ beschränktes Gebiet, $\partial M\subset\mathbb R^3$ glatt orientiert durch $\nu\colon\partial M \to\mathbb R^3$ Normalenfeld
$$
L &\colon& M\to\mathbb R^3\text{ glattes Vektorfeld }
\\ L&=&CL_x, Ly, Lz
$$
$$
\int_{\partial M}\langle L, \nu\rangle\intd\underbrace{S}_{\text{ Flächenelement }}=\int_M \operatorname{div} L \intd x\intd y\intd z
$$
$$
\operatorname{div} L =\frac{\partial L_x}{\partial x}+\frac{\partial L_y}{\partial y}+\frac{\partial L_z}{\partial z}
$$
Wenn $\partial M$ parametrisiert ist:
$$
x &=& x(u,v)
\\ y &=& y(u,v)
\\ z &=& z(u,v)
$$
$$
\int_{\partial M} f(x,y,z)\intd S :=\int_v f(x(u,v), y(u,v), z(u,v))\cdot\sqrt{\det G(u,v)}\intd u \intd v
$$
$$
G(u,v)=(D_{(u,v)}\psi)^T D_{(u,v)}\psi
$$
die Gram-Matrix der Koordinatenbasis ($\psi\colon U \to\partial M$, $(u,v)\mapsto(x(u,v)\ldots)$)
$$
=\int f(x(u,v), y(u,v), z(u,v))\lVert e_u \times e_v \rVert\intd u \intd v
$$
Stokes:
$$
\int_{\partial M}\omega=\int_M \intd\omega
$$
wollen:
$$
\diffd\omega=\operatorname{div} L \intd x \wedge\diffd y \wedge\diffd z
$$
$$
\omega= L_x \intd y\wedge\diffd z + L_y \intd z\wedge\diffd z + L_z \diffd x \wedge\diffd y
$$
$$
\diffd\omega&=&\frac{\partial L_x}{\partial x}\intd x\wedge\diffd y \wedge\diffd z +\frac{\partial L_y}{\partial y}\intd y\wedge\diffd z \wedge\diffd x +\frac{\partial L_z}{\partial z}\intd z\wedge\diffd x \wedge\diffd y +0+\ldots+0
\\&=&(\operatorname{div}L)\intd x \wedge\diffd y \wedge\diffd z
$$
$$
\int_{partial M}\omega=\int_{\partial M} L_x \intd y\wedge\diffd z + L_y \intd z\wedge\diffd x + L_z \intd x\wedge\diffd y
\\&\overset{?}=&\int_{\partial M}\langle L, \nu\rangle\intd S
$$
$$
\\&&\int_{\partial M} L_x \intd y\wedge\diffd z
\\&=&\int_U L_x (x(u,v), y(u,v), z(u,v))\left[\frac{\partial y}{\partial u}\intd u +\frac{\partial y}{\partial v}\intd v \right]\wedge
\wedge\left(\frac{\partial z}{\partial u}\intd u +\frac{\partial z}{\partial v}\intd v\right)