Originally Posted by

**HallsofIvy** Irrelevant! This is a surface and so can be written in terms of two parameters. Here, they have chosen to use x and y as those parameters- z is then a function of x and y.

You are completely misunderstanding the problem and making it much harder than it is. You are NOT to integrate over the ellipsoid or even the top part of it. You are to integrate over the region S with is the part of the ellipsoid directly above the square bounded by x= 0, y= 0, x= 1, y= 1. That does NOT make a closed surface even adding the "base" (by which I take it you mean the ellipse where the ellipsoid passes through the xy plane) so the divergence theorem does not apply. Just do what they say:

integrate F dot dS as given, with limits of integration, x= 0 to 1, y= 0 to 1.

Many thanks!