SUNY Geneseo Department of Mathematics
Tuesday, December 6
Math 223 01
Fall 2022
Prof. Doug Baldwin
How to set up the integral in homework 11, problem 3 (a line integral around half a circle)?
The main thing you need is a parameterization of the half circle. You can base this on a standard parameterization for circles, namely s(t) = ⟨ r cos(t), r sin(t) ⟩, where r is the radius of the circle. In this problem the circle has radius 4, so we get r(t) = ⟨ 4 cos(t), 4 sin(t) ⟩.
Can you use Green’s Theorem for the circulation and flux parts of homework 12, problem 2?
Actually, you can’t, because the vector field in the problem doesn’t have continuous partial derivatives over the regions in the problem (the derivatives aren’t defined at the origin).
Geneseo’s Association for Women in Mathematics (AWM) chapter is hosting a study night this Thursday, December 8th from 5:30-8pm in South 328. All are welcome.
PRISM is looking for people who might be interested in being officers next year, so that those people can start seeing how the club runs during this spring, while the current officers are still around.
Anyone can do this, you don’t have to be any particular year, major, etc., just interested in math and other people with similar interests.
If interested, let the current officers know by emailing prism@geneseo.edu.
Based on “Using Divergence and Curl” in section 5.5 of the textbook.
Some properties of and relationships between divergence and curl:
You can use these properties to disprove claims that certain field are curls, etc.
The proofs of the properties tend to be via Clairaut’s Theorem.
Verify that the formula for the curl of a 2-dimensional field F = ⟨ P, Q ⟩ is really the formula for the curl of 3-dimensional field ⟨ P, Q, 0 ⟩. This makes it plausible that 2-dimensional curls are really just a special case of 3-dimensional ones, and so properties discussed in 3 dimensions should apply in 2.
Write out the formula for the 3-dimensional curl of ⟨ P, Q, 0 ⟩, keeping in mind that derivatives of P and Q with respect to z will be 0 because P and Q are functions of only x and y in this case, and any derivative of the 0 term will be 0:
Check the book’s claim that the curl of a conservative vector field is always 0 by working out what the relevant mixed partial derivatives are.
We started with a potential function, f(x,y,z), and worked out from there what its gradient field would be, and then what the various derivatives in the curl would be:
In working with line integrals, Green’s Theorem, etc., we’ve been talking about 2-dimensional regions enclosed in or bounded by curves or line segments. There’s another way of enclosing regions in 3 dimensions, namely within surfaces. I’d like to spend the last few days of this course looking at how some of the things we’ve done generalize to that realm.
To start, we’ll talk about parametric surfaces.
Please read “Parametric Surfaces” in section 5.6 of the textbook.