2009-12-09: The Divergence Theorem

Stewart section 17.9

• We've been looking at Stokes' theorem, which relates the flux of the curl of a vector field F through a surface S to the line integral of F on the boundary of S. Here S is an oriented surface and the boundary of S is a simple (doesn't cross itself) closed curve with positive (counterclockwise around the normal) orientation.
• Key Point Our last theorem is the Divergence Theorem: the triple integral over a volume E of div F = the flux integral of F over the surface bounding E. Here E is a simple ("convex") volume and the boundary of E is taken to have outward (positive) orientation.
• As before, the goal is to evaluate the easier integral. Usually this will be the volume integral.
• Game: Consider the volume E inside a sphere of radius 3 in the first octant, with F = <x,0,z>.
1. What is the flux through the coordinate planes?
2. Parameterize the outer surface of E.
3. Do you want to find F.dS on this surface?
4. Find the flux through the boundary with the divergence theorem.
• Note: we said that div F is the tendency of F to push things apart; this argues that the divergence theorem should make sense.
• Game: Let F = <3xy, 2y, -xz>, let E be the region of height z = x + y over the triangular region with corners (0,0,0), (2,0,0), and (1,1,0), and C be the boundary of the triangular region.
1. Find intC F.dr.
2. Find int intdE F.dS.
ma215-080-f09 lecture outline 2009-12-09
Created: Wed Dec 9 12:51:00 EST 2009