Past exam of the mathematical methods II course of the University of Oxford / 2016 / 8 / d / Solution

Choose the oriented surface parametrization $\mathbf{r}(x,y)=(x,y,1-x)$, with $0\le x,y\le 1$. Then ${\mathbf{r}}_x\times {\mathbf{r}}_y=(1,0,1)$ and $\nabla \times \mathbf{A}=(-3z,x^2,y)$. The surface integral isFor the compatible boundary direction, use successive vertices $(0,0,1)$, $(1,0,0)$, $(1,1,0)$, $(0,1,1)$ and back. The four line integrals are respectivelyTheir sum is $-1$, agreeing with the surface integral. Choosing the opposite normal and reversing the boundary direction gives $+1$ on both sides. Stating an orientation is necessary because the question does not prescribe one.