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>.
- What is the flux through the coordinate planes?
- Parameterize the outer surface of E.
- Do you want to find F.dS on this surface?
- 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.
- Find int_{C} F.dr.
- Find int int_{dE} F.dS.
ma215-080-f09 lecture outline 2009-12-09
Created: Wed Dec 9 12:51:00 EST 2009
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©2009 Gavin LaRose