SUNY Geneseo Department of Mathematics

Divergence and Curl

Wednesday, May 10

Math 223
Spring 2023
Prof. Doug Baldwin

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Flux Form of Green’s Theorem

Finishing yesterday’s example checking that Green’s Theorem and a traditional flux line integral give the same results for a certain problem.

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 (Green’s Theorem says the flux is 4/15):

Region between X squared minus 1 and X axis, parametric forms for boundary and corresponding line integral

We started with the flux across r₂, because it’s quick to compute. Once you work out what the normal vector n₂ is, the product of F(r₂(t))•n₂ turns out to be 0:

Integral of F of R 2 of T dot N 2 is integral of 1 minus T comma 0 dot 0 comma 1, or 0

Then we had to integrate across the parabola. The integral isn’t hard to calculate, although it’s important to keep track of the signs of things:

Integral of T comma T squared minus 1 all squared dot 2 T comma negative 1 is 4 fifteenths

In the end, the flux line integral and the double integral from Green’s Theorem both produced values of 4/15. And as has been sort of a theme with our discussion of Green’s Theorem, the integral from Green’s Theorem was significantly easier to evaluate.

Divergence and Curl

Based on “Divergence” and “Curl” in section 5.5 of the textbook.

Suppose vector field G(x, y, z) = ⟨ x sin y, y cos z, z tan x ⟩

Divergence Example

Find ∇•G

What?

How are you supposed to do this with no preliminary background!?

Two things: first, you likely can do something with the problem based on the book’s explanation, even with no-one reviewing it with you. Second, if you can’t get anywhere with answering the question, you can at least start forming the questions you’d need answered in order to make progress. These are important bits of mindset moving forward into other math (or non-math) courses.

Key Ideas/Questions

The meaning of the notation, particularly its use of gradient and dot product to remind you of how to calculate divergence. (But that notation is purely mnemonic, differentiation operations aren’t really something you can put into a vector, and applying them to functions isn’t multiplication.)

The equation for divergence (see below)

Solution

The solved divergence:

Divergence is D P D X plus D Q D Y plus D R D Z, or in this case sine Y plus cosine Z plus tangent X

Curl Example

Find ∇×G

Based on the interpretation of ∇•G, you can figure out what this operation is going to be like, too:

Curl is a vector whose component calculations follow a pattern similar to cross product with derivative operations

Congratulations!

You’re done with calculus!

(Well, almost, you might still have some grading to finish.)

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