Divergence Theorem

The divergence theorem, more commonly known especially in older literature as Gauss's theorem (e.g., Arfken 1985) and also known as the Gauss-Ostrogradsky theorem, is a theorem in vector calculus that can be stated as follows. Let be a region in space with boundary . Then the volume integral of the divergence of over and the surface integral of over the boundary of are related by

(1)

The divergence theorem is a mathematical statement of the physical fact that, in the absence of the creation or destruction of matter, the density within a region of space can change only by having it flow into or away from the region through its boundary.

A special case of the divergence theorem follows by specializing to the plane. Letting be a region in the plane with boundary , equation (1) then collapses to

(2)

If the vector field satisfies certain constraints, simplified forms can be used. For example, if where is a constant vector , then

(3)

But

(4)

so

			(5)
			(6)

and

(7)

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

(8)

Similarly, if , where is a constant vector , then

(9)