# Gauss’, Green’s and Stokes’ Theorems

If $\Omega$ is a domain in $R^3$ with boundary $\Gamma$ with outward unit normal $n=(n_1,n_2,n_3)$, and $u:\Omega\rightarrow R^3$ and $v,w:\Omega\rightarrow R$, then we obtain applying the Divergence Theorem to the product $vw$,

• $\int_\Omega \frac{\partial v}{\partial x_i}w\,dx=\int_\Gamma vw\,n_{i}\,ds-\int_\Omega v\frac{\partial w}{\partial x_i}\,dx,\quad i=1,2,3$.

Further, similarly,

• $\int_\Omega\nabla v\cdot\nabla w\, dx =\int_\Gamma v\partial_nw\, ds - \int_\Omega v\Delta w\, dx$

and

• $\int_\Omega v\Delta w\, dx -\int_\Omega \Delta v w\, dx =\int_\Gamma v\partial_nw\, ds -\int_\Gamma\partial_nvw\, ds$.

These formulas are referred to as Green’s Formulas and express 3d analogs to integration by parts in 1d.

If $S$ is a surface in $R^3$ bounded by a closed curve $\Gamma$ $n$ is a unit normal to $S$, $\Gamma$ is oriented in a clockwise direction following the positive direction of the normal $n$, and $u:R^3\rightarrow R^3$ is differentiable, then

• $\int_S(\nabla\times u)\cdot n\, ds=\int_\Gamma u\cdot ds$,

which is Stokes’ Theorem.