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