1.
First try visualizing the two fields. What's the difference? (you might like to use the vector field plotter we've seen before 1 )
Section 4.1 Circulation Density and Curl
We've preciously studied Circulation (i.e. the Flow around a closed loop). That is a global property, i.e. something that takes place over a big region. Now we are going to study Circulation Density. This measures the degree to which there is spinning locally at a point. We'll derive a formula for this by computing a sort of limit of circulations around smaller and smaller loops until we can say something about what happens at a point. Note that this is also known as kth component of curl, which is mainly going to be relevant later when we see the full curl.
Learning Objectives:
- Define Circulation Density
- Geometrically interpret Circulation Density.
Post-Video Activities Post-Video Activities
Consider 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.
Fortuitously, both fields have the same circulation density at all points (the Whirlpool one isn't defined at the origin). For each field, is that constant circulation density positive, zero, or negative ? Try visualizing each and guess the result, and then compute it out from the formula to check your intuition.
www.geogebra.org/m/QPE4PaDZ
