Section 6.3 Flux Across a Surface
We've seen the Flux Across a Curve back in
Section 2.7. That was true for 2D vector fields. Now we upgrade to 3D vector fields and Flux Across a Surface. Before watching, can you guess how we will define this based on analogy with the 2D case?
Learning Objectives:
Geometrically interpret the definition of Flux across a surface
State formula to compute flux for parametric or implicit surface (as before, any explicit surface can be converted to an implicit one)
TYPOS: At 5:57 I'm doing the implicit formula. I kept using the same dudv from parametric. But the differentials should be dA here where if the surface is living above the xy-plane, say, dA=dxdy. Secondly, it is conventional to convert from S (in the definition of flux) to R (in the computation) for flux where R is the region in, say, the xy-plane beneath the surface. Put together the corrected formula for Flux implicitly is \(\iint_R\vec{F}\cdot \frac{\nabla h}{\|\nabla g\cdot\vec{p}\|}dA\)
Post-Video Activities Post-Video Activities
1.
This discussion of Flux across a surface is very closely related to the flux in the plane across a curve (right??). But in the plane we ALSO had a notion of circulation which is noticeably absent in this module. Why isn't there a direct parallel of circulation in the 3D case the way there is for flux? What is the geometric problem that prevents that?
