TITLE: Un-curl operator?
QUESTION [11 upvotes]: I would like to use Stokes' Theorem to find the area of a surface over a given region. This is given by:  
$A = \oint\vec{F}\centerdot d\vec{r}$
but only if the following condition holds:  
$(\vec{\triangledown}\times\vec{F})\centerdot \vec{n} = 1$  
where $\vec{n}$ is the normal to the surface. How do I come up with a vector field, $\vec{F}$, that satisfies this condition?  I found a paper that discusses an inverse-curl operator here, but this is only useful if I know what $(\vec{\triangledown}\times\vec{F})$ is and need to find $\vec{F}$.  Any ideas?

REPLY [6 votes]: As anon put it in a comment:

Essentially this is a vector boundary problem in $\mathbf G = \nabla \times \mathbf F$ defined by
  $$\begin{align}
\nabla \cdot \mathbf G &=0\\
\mathbf G \vert_{\partial \Omega}\cdot \mathbf n &= 1
\end{align}$$
  The vector field $\mathbf F$ can be recovered from $\bf G$ via the Helmholtz theorem.