1.
This time, for each field, is that divergence positive, zero, or negative ? Try visualizing each and guess the result, and then compute it out from the formula to check your intuition.
Section 4.3 Divergence and Green's Theorem (Divergence Form)
Just as circulation density was like zooming in locally on circulation, we're now going to learn about divergence which is the corresponding local property of flux. It measures the degree to which the field is spreading out at any given point. We will then have the second half of Green's Theorem, in its so called Divergence Form, which relates the local property of divergence over an entire region to the global property of flux across the boundary.
Learning Objectives:
- State Green's Theorem in its Divergence Form
- Geometricaly Interpret Green's Theorem in its Divergence Form
Post-Video Activities Post-Video Activities
We've previously studied 4.1 these two types of rotation:
Uniform Rotation: \(\vec{F}=-y\hat{i}+x\hat{j}\)
Whirlpool rotation: \(\vec{F}=\frac{-y}{x^2+y^2}\hat{i}+\frac{x}{x^2+y^2}\hat{j}\)
2.
At around the 3:00 mark I very quickly skim the derivation that compute the divergence at a point as it was analogous to the derivation for circulation density, based on computing for an infinitesimal rectangle in a a limit. Carefully repeat that computation in full for divergence.
