= Solution
For an oriented piecewise smooth surface $S$ with boundary curve $C=\partial S$, and a continuously differentiable vector field $\mathbf F$ near $S$, <Stokes theorem> states
$$
\boxed{\int_S(\nabla\times\mathbf F)\cdot\mathbf n\,dS=\oint_C\mathbf F\cdot d\mathbf r.}
$$
Here $\mathbf n$ is the chosen <unit normal>, $dS$ is positive surface area, and $d\mathbf r$ is the directed tangent <line element>. The positive traversal of $C$ is fixed by the right-hand rule relative to $\mathbf n$. Thus the <flux> of the <curl> through the surface equals the circulation along its oriented boundary.
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