= Solution
{c}
For an <oriented surface> $\Sigma$ with piecewise smooth closed <boundary> $\partial\Sigma$, and a <vector field> $\mathbf F$ that is <continuously differentiable> on an open neighborhood of $\Sigma$, <Stokes theorem> states
$$
\boxed{\iint_\Sigma(\nabla\times\mathbf F)\cdot\mathbf n\,dS
=\oint_{\partial\Sigma}\mathbf F\cdot d\mathbf r.}
$$
Here $\mathbf n$ is the chosen <unit normal vector>. The <orientation> of $\partial\Sigma$ follows the <right-hand rule>: with the right thumb pointing along $\mathbf n$, the fingers curl in the positive direction. Equivalently, walking along the <boundary> with one's head along $\mathbf n$ leaves the <surface> on the left. Each <boundary component> receives this induced <orientation>.
Back to article page
