= Stokes theorem
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{wiki}
Let $S$ be an oriented smooth surface with piecewise smooth boundary, and let $F$ be a continuously differentiable <vector field> on a neighborhood of $S$. With the compatible boundary direction determined by the <right-hand rule>,
$$
\int_S(\nabla\times F)\cdot n\,dS=\oint_{\partial S}F\cdot d\mathbf x.
$$
For $G=(-y,x,0)$ on the horizontal unit disc with upward normal, the left side is $2$ times its area, giving $2\pi$. The counterclockwise boundary has tangential field of magnitude one and circumference $2\pi$, giving the same <line integral>. Reversing the surface normal reverses the boundary direction and both signs. The theorem replaces a circulation calculation with a <surface integral> of the <curl>, whichever is easier.
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