Past exam of the mathematics course of the University of Oxford / 2015 / Prelims / Mathematics V / 1 / a / Solution

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 statesHere $\mathbf{n}$ 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.