= Solution
{c}
Let $S$ be a piecewise smooth <oriented surface> with piecewise smooth boundary $\partial S$, and let $\mathbf F$ be a <continuously differentiable vector field> on a neighbourhood of $S$. The <Stokes theorem> states
$$
\boxed{\oint_{\partial S}\mathbf F\cdot d\mathbf r
=\iint_S(\nabla\times\mathbf F)\cdot\mathbf n\,dS}.
$$
The <boundary orientation> is induced by the chosen <unit normal vector> $\mathbf n$ through the <right-hand rule>: with the right thumb along $\mathbf n$, the curled fingers indicate positive boundary traversal. Equivalently, when walking along the boundary with one's head pointing along $\mathbf n$, the <surface> lies to the left. An upward oriented <disk> therefore has counterclockwise boundary traversal when viewed from above. Reversing the <orientation> reverses both the <line integral> and the <surface integral>.
Back to article page
