= Solution
For a continuously differentiable <vector field> $\mathbf F$ near a piecewise smooth <oriented surface> $S$, <Stokes theorem> states
$$
\boxed{\oint_{\partial S}\mathbf F\cdot d\mathbf r=\iint_S(\nabla\times\mathbf F)\cdot\hat{\mathbf n}\,dS.}
$$
The <circulation> around the whole boundary equals the <flux> of the <curl> through the surface. The <unit normal vector> and boundary direction must be compatible with the <right-hand rule>: curling the right-hand fingers in the boundary direction points the thumb along the normal. Reversing both orientations reverses both integrals.
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