SUNY Geneseo Department of Mathematics

The Flux Form of Green’s Theorem

Tuesday, May 9

Math 223
Spring 2023
Prof. Doug Baldwin

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Warning!

Don’t try to use Green’s Theorem for the circulation and flux integrals in problem set 13. It doesn’t apply!

Why not? Because the vector field doesn’t meet the conditions for it to apply, i.e., the “fine print.” For one thing the derivatives of the vector field aren’t continuous over the regions in question (they’re undefined at the origin); for another the components of the vector field themselves are undefined at the origin, so their domain isn’t simply connected.

As well as heading off some possibly incorrect answers on the problem set, this is also a good example of the general need to know when a theorem applies before you use it.

The Flux Form of Green’s Theorem

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

Key Ideas

The equation:

Integral around C of F dot N equals double integral over D of P sub X plus Q sub Y, where D is region inside C

Examples

An example of a typical use: find the flux of F(x, y) = ⟨ x, y2 ⟩ across the triangle with vertices (0,0), (1,0), and (1,1).

Coordinate axes with triangle enclosing region D and vector field X comma Y squared

Now applying the double integral from Green’s Theorem (flux form) and integrating gives…

Integrating P sub X plus Q sub Y over D yields 5 sixths

An example with an unusual motivation: this is one I was thinking of bringing to class, but couldn’t make “work,” i.e., I couldn’t get the answer from Green’s Theorem to agree with the answer from the straightforward line integral. So let’s see if collectively we can get them to agree (i.e., avoid whatever I was overlooking when I tried it): Find the flux of field F(x, y) = ⟨ x, y2 ⟩ across the curve consisting of the part of the parabola y = x2 - 1 below the x axis, and the x axis itself; check that the value produced by Green’s Theorem equals the value produced by a standard flux line integral:

Region D between curve Y equals X squared minus 1 and X axis, field X comma Y squared

We started with the double integral from Green’s Theorem:

Green double integral from negative 1 to 1 of integral from X squared minus 1 to 0 of 1 plus 2 Y yields 4 over 15

Then we turned to the line integral. This is the first example we’ve done of a line integral where the path to integrate over has to be broken into 2 parts. You can do that by integrating over each part separately and adding the integrals. So we came up with parametric forms for each part of the path, being careful that the forms were such that one moves counterclockwise around the path as t increases:

Integral around region between X squared minus 1 and X axis as 2 integrals along 2 parametric curves

This is as far as we had time to get today. We’ll finish this example tomorrow.

Next

Finish checking the flux form of Green’s Theorem with the second example above.

Time permitting, introduce two common operations on vector fields, divergence and curl.

Please read “Divergence” (but you can skip the paragraphs about source-free vector fields) and “Curl” in section 5.5 of the textbook.

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