This template computes the flux of a vector field through a surface defined over the interior of an ellipse. The field can be specified in its own coordinate system. The surface parameters can be two of the coordinate variables, or all three coordinates for the surface can be functions of two other parameters. The ellipse is specified by giving its rectangular equation in the coordinates of the parameter space. Thus, if the surface is parametrized as $x$($u$, $v$), $y$($u$, $v$), $z$($u$, $v$), the ellipse supporting the surface is a quadratic equation of the form $a u^2 \&plus; b u v \&plus; c v^2 \&plus; d u \&plus; e v \&plus; f\&equals;0$, where $b^2-4 \mathrm{ac}<0$, and the resulting ellipse is real and non-degenerate.

VectorCalculus[Flux]

VectorCalculus[SurfaceInt]