The proof starts by considering a spherical region and firstly performing the z-integral of the z-derivative, from the bottom hemisphere to the top hemisphere [Transparancy 1]. The dxdy integral over the top and bottom hemispheres is then evaluated [Transparancy 2] using the area cosine principle (see lecture 28). A symmetry argument is used to obtain a similar result for the x-integral of the x-derivative and the y-integral of the y-derivative. This proves the divergence theorem on a sphere.
The proof is then generalized [Transparancy 3] to non-spherical regions, including the dumbell shape, which must be split into two different regions.
We end with a rigorous justification of our original definition (lecture 14) of the divergence of a vector field being the net outflux per unit volume. Using the divergence theorem, we can see that this definition is true [Transparancy 4] for general regions, and not just for rectangular boxes.
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