## Divergence Theorem

A.k.a. Gauss's Theorem. Let be a region in space with boundary . Then

 (1)

Let be a region in the plane with boundary .
 (2)

If the Vector Field satisfies certain constraints, simplified forms can be used. If where is a constant vector , then
 (3)

But
 (4)

so
 (5)

 (6)

But , and must vary with so that cannot always equal zero. Therefore,
 (7)

If , where is a constant vector , then
 (8)

Arfken, G. Gauss's Theorem.'' §1.11 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 57-61, 1985.