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I have seen problems similar to this and know how to solve for the flux of the field through it, but each of those gave shadows of easily integrated shapes when scaled down to 2space --- we would have cylinders cut by planes, giving us perfect circles when looking only at the shape's shadow (z=0) --- but this one would give an ellipse and would be very tough to integrate unless I am missing something.

Here is a quick paint I made (imagine the red circle on the plane is actually a perfect circle on said plane and that a complex vector field F(x,y,z)=<F1,F2,F3> is defined at every point): http://imgur.com/fn0dFRN

By complex vector field I just mean that you can't give me any shortcuts that might exist if the divF=0 or the curlF=<0,0,0>

Edit: Maybe someone tell me if there is such a thing as parametric shadow method? There surely is but I have no idea what the Jacobian would be or any of that.