Divergence over a Region

The Divergence Theorem says that if a region, D, is bound by a closed curve, then the integral of the divergence over that region is equal to the flux across the boundary. This is often written:

.

But what does this mean? Let g(t), t=0..T, parameterize the boundary in a counter-clockwise fashion. If n(t) is the unit normal to g(t), then we can make sense of the right-hand integral by writing

Question #2

In this question we will evaluate the double integral on the left by computing the line integral on the right.
  1. Parametrize the unit circle about the stable equilibrium (the sink) by g(t), t=0..2*Pi. Note: The sink is not at the origin!
  2. Compute the tangent vector g'(t) and use this to write down the unit normal vector to the circle.
  3. Use Maple to help you compute the dot product of f(g(t)) and n(t) and the integral of this dot product times the length of g'(t) from t=0..2*Pi.

We've provided a hint in case you get stuck.

Next: The Average Divergence
Up: Introduction
Previous: Divergence at a Point

Frederick J. Wicklin <fjw@geom.umn.edu>
Brian Burt<burt@geom.umn.edu>
Document Created: Wed Jan 25 CST
Last modified: Tue Mar 14 10:27:25 1995