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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.

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What parametrization did you find? – J. M. Jul 29 '12 at 14:23

Hint: the $e^{z^2}$ makes a direct parameterization very unlikely to work. Do any theorems come to mind when you see an ugly contour integral?

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Seems more of a comment than an answer. – enzotib Jul 29 '12 at 15:22

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