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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?

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    @anon You should have posted this as answer, this question still shows up as unanswered2019-04-29

1 Answers 1

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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.