A cell is roughly shaped like a solid ellipsoid E

x2+4y2+9z21.

We can parameterize it's boundary surface B=E by a stretched version of spherical coordinates:

x(,)=cossin,y(,)=2sinsin,z(,)3cos .

Suppose water is flowing out of the cell with velocity vector field

F=x,y,z. Calculate the rate that the water is flowing out of the cell in two different ways. Do not use the divergence theorem in this question.

1. Compute the flux over the boundary of the cell

BFndS. Make sure your normal vector points out of the cell.

1. Compute the "volume total of all sources and sinks" inside the cell:

EdivFdV.