Divergence theorem

The divergence theorem is the most general theorem relating an integral over a region with an integral over its boundary. Its special forms include Green’s theorem, Stoke’s theorem, and the fundamental theorem of calculus.

If we have some region $\Omega$ in $\R^k$, and $\partial\Omega$ is its boundary, the divergence theorem states that for some operation $(*)$:

The operation $(*)$ here might for example be a dot product with some vector field $\mathbf F$, in which case $\int_{\partial\Omega} \hat{\mathbf n}\cdot \mathbf F d^{k-1}x = \int_\Omega \nabla \cdot \mathbf F d^k x$. Or it might be the cross product with $\mathbf F$, in which case $\int_{\partial\Omega} \hat{\mathbf n}\times\mathbf F d^{k-1}x = \int_\Omega \nabla \times\mathbf F d^k x$.

The most commonly used form of the divergence theorem is where $(*) = \cdot \mathbf F$. In this case, it can be written as: