Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2004 / Paper 1 / 3 / Solution

For an oriented piecewise smooth surface $S$ with boundary $C=\partial S$ and a vector field $\mathbf{A}$ with continuous first derivatives near $S$, the Stokes theorem statesHere $\mathbf{n}$ is the chosen unit normal vector, $dS$ the positive area element, and $d\mathbf{r}$ the oriented tangent line element. The traversal of $C$ follows the right-hand rule relative to the surface orientation.