Consider the vector field $\mathbf H(x,y,z)=(-x,y,e^{z^2}).$ Let $V\subset\mathbb R^3$ be the region inside the cylinder $x^2+y^2\le1$ and between the surfaces $z=-2$ and $z=xy$. Let $C$ be the closed curve that lies on the intersection of $V$ and $z=xy$.
How can I compute $\oint_C\mathbf H\;d\mathbf r?$ I;ve tried to find a direct parametrization, but that makes the resulting integral quite difficult to compute.