= Circulation around a hole in an oriented surface
{title2=$\oint_{\partial S'}\mathbf v\cdot d\mathbf r=-\int_H(\nabla\times\mathbf v)\cdot\mathbf n\,dS$}
Let a closed oriented surface $S$ lose a patch $H$, leaving $S'$. Assume a smooth <vector field> on the relevant surfaces and that its total <curl> flux through $S$ is zero. The <Stokes theorem> gives the title formula when the edge is oriented as the boundary of the remaining surface $S'$. The boundary of the removed patch $H$, with its original normal, has the opposite direction and hence the opposite <circulation>.
For a cylindrical side of radius $R$, if $(\nabla\times\mathbf v)\cdot\mathbf n=-\alpha R$ everywhere on the removed patch, the <circulation> around the boundary of the remaining surface is $\alpha R\operatorname{area}(H)$. This uses actual curved area, rather than its planar projection, and does not depend on the patch shape. For multiple boundary components, the formula concerns the sum with their induced orientations.
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