SUNY Geneseo Department of Mathematics
Monday, May 1
Math 223
Spring 2023
Prof. Doug Baldwin
(No.)
This Friday (May 5), 5:00 - 7:00 PM, Highland Park.
The last day for grading is Thursday, May 18 (the last day of finals).
If you have lots of grading to catch up on, plan and schedule it now.
If you know you won’t use an appointment you make, please cancel it so other people can have the time.
To the extent that you can, please don’t leave little (e.g., under 20 minutes) gaps of time between appointments. Those are blocks too short for other people to use, but if a couple of them combine they become very useful.
Base on “Conservative Vector Fields and Potential Functions” in section 5.3 of the textbook.
The strategy for finding potential functions (see the examples below or the textbook).
A conservative vector field has multiple potential functions, differing by “+ C”
Friday we used the cross partials property to discover that G(x, y) = ⟨ -y, -x ⟩ might be conservative. Try to find a potential function for it to verify that it is.
The book’s strategy for finding potential functions boils down to alternating integrations and differentiations, which move back and forth between the known vector field and increasingly better-known ideas about what the potential function is.
So in this example, start by integrating the first (x) component of the vector field, to get an imprecise version of the potential function:
Then find the derivative of that function with respect to y, and compare it to the second (y) component of the vector field:
We don’t need anything from the derivative of g to make the derivative of f equal the second component of the field, so the potential function is finished. (It seems conventional not to explicitly include the “+ C” in potential functions.)
Does F(x, y, z) = ⟨ y2 + zex, 2xy + sin z, y cos z + ex ⟩ have a potential function? If so, what is it?
Finding a potential function for a 3-dimensional vector field is very similar to finding one in 2 dimensions, except the process may take longer, and the functions produced as “constants” of integration have more variables:
The cross partials property also showed that F(x, y) = ⟨ y, -2x ⟩ is not conservative. So what happens if you try to find a potential function for it?
The process starts similarly to the earlier examples, but the function h(y) arising from the first integration turns out to be a function of both y and x. Were you to turn this into a complete “potential function,” it would turn out that it’s derivative with respect to x was no longer equal to the x component of the field.
Line integrals of conservative fields.
Please read “Curves and Regions” and “Fundamental Theorem for Line Integrals” in section 5.3 of the textbook.