Section 8.1 The Divergence Theorem
When we first saw Green's Theorem back in section 16.4 there were two sides, a Circulation Form and a Divergence Form. Stokes' Theorem generalized the Circulation Form and today we will generalize the Divergence Form.
Learning Objectives:
State the Divergence Theorem
Geometrically interpret both sides of the Divergence Theorem
Pre-Video Activity: Before you watch, can you try and figure out the core ideas that will be in the Divergence Theorem? I've hinted it is a generalization of
Green's Theorem Divergence Form 4.3. But that was 2D. What would it even mean to have an analogous theorem in 3D?
Post-Video Activities Post-Video Activities
1.
Show that \(div(curl \vec{F})=0\text{,}\) analogous to a previous computation we have done that \(curl(grad f)=0\text{.}\) More generally, what are all the ways you can combine the div, curl, and grad operators so they make sense? Which of those are zero?
2.
Stokes' Theorem and the Divergence Theorem both generalize two sides of Green's Theorem which was about a region in the 2D plane with a boundary. However, they generalize in different ways. Stokes' theorem is still comparing a surface integral to a line integral along the boundary, it is just the surface lives in 3D not 2D. But for the Divergence theorem, we for the first time get a triple integral. Try to visualize what is going on here and why the generalization to 3D for Stokes' didn't give a triple integral but Divergence theorem did.
