MathHelp

Divergence 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.

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

Return to Calculus 3 Notes