Divergence Theorem (Gauss's Theorem)

Background: Recall that Green's theorem gives the relationship between a line integral

around a simple closed curve, C, and a double integral over the plane region R bounded

by C. See the figure below.

Also recall that the “divergence form” of Green’s theorem is

∫ F . n ds = ∫ ∫ div F dA

C R

where R is a region in the x-y plane enclosed by a piecewise-smooth, positively

oriented (keep region to your left as you travel around the simple closed curve C)

F(x,y) is a vector field F = P(x,y) i + Q(x,y) j,

n(x,y) is a unit vector to the curve C

ds = arc length along curve C

div F = ∂P/∂x + ∂Q/∂y

dA = element of area in R

Click here to continue with discussion of the Divergence Theorem.

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