PX275 - D3e - divergence theorem in 2D

$d \vec{n}$ is normal, and $d \vec{r}$ is tangent to the curve:

\begin{aligned} d \vec{r} & = d x \hat{i} + d y \hat{j} \\ d \vec{n} & = d y \hat{i} + d x \hat{j} \end{aligned}

for the vectors:

\vec{F} = F_{x} \hat{i} + F_{y} \hat{j}

then:

\oint_{C} \vec{F} \cdot d \vec{n} = \oint F_{x} d y - F_{y} d x

this is the $L H S$ of green's theorem:

\begin{aligned} P & = - F_{y} \\ Q & = F_{x} \end{aligned}

green's theorem:

\oint P d x + Q d y = \iint_{R} (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}) d A

using the comparison with green's theorem:

\oint F_{x} d x - F_{y} d y = \iint_{R} \underset{\vec{\nabla} \cdot F}{\underset{⏟}{(\frac{\partial F_{x}}{\partial x} + \frac{\partial F_{y}}{\partial y})}} d A

therefore, the divergence theorem in 2D (AKA integral form of the divergence):

\oint_{C} \vec{F} \cdot d \vec{n} = \iint_{R} \vec{\nabla} \cdot \vec{F} d A