Green’s Theorem, Flux Form

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The Flux Form of Green’s Theorem

From “Flux Form of Green’s Theorem” in section 5.4 of the textbook.

Key Ideas

This form of Green’s Theorem gives a formula for flux line integrals, similar but not identical to the one for circulation integrals:

This has a proof from the circulation form.

Source free vector fields and stream functions are flux versions of conservative fields and potential functions.

Examples

When we first talked about flux, we used the example of my beer brewing and the flux of water with a velocity field F(x,y) = ⟨ -y, x ⟩ through a bag of radius 1 centered at (2,0). Then we used a line integral to find that the total flux is 0. Can you get the same result with Green’s Theorem?

Both of the relevant partial derivatives are 0, leading immediately to a double integral that equals 0:

What about the flux of G(x,y) = ⟨ xy, y ⟩ across the rectangle with vertices (0,0), (1,0), (1,2), and (0,2)?

This time the double integral is a little more complicated…

Source Free Vector Fields

Just as conservative vector fields lead to particularly well-behaved circulation line integrals, “source free” fields lead to well-behaved flux line integrals. The “good” behaviors are similar, i.e., line integrals around closed curves are 0, line integrals along a curve can be found by taking the difference of an antiderivative-like function at the endpoints of the curve, etc. The “antiderivative-like” function for a conservative field is the potential function, for a source free field it’s the “stream function,” which has a definition very similar to that of the potential function:

Given a vector field, you can test to see if it’s source free by trying to find a stream function for it. For example, we showed that the field from the beer brewing example is source free by finding a stream function as follows:

Next

Divergence.

Please read “Divergence” in section 5.5 of the textbook.

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