Geneseo Math 223 01 3D Potential Functions

Anything You Want to Talk About?

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Misc

End of Semester

The last day for grading anything will be Tuesday, December 20 (the last day of finals). I’ll accept grading appointments up to 5:00 PM that day.

What do you want to do with our final exam period (8:00 - 11:20 AM on the 20th)? (For example, use it for review for people finishing problem sets or redos? Block it out as meeting times reserved for this course? Something else?)

My preference would be to block it off for this course, but that’s easily changed if you have ideas you like better. So let me know if you do.

History of Calculus Colloquium

The Cultural Context of Calculus and the Curious Complications of Convergence

by Prof. Jeff Johannes

Friday, December 2 3:30 - 4:20 PM, Newton 204

Conservative Fields and Potential Functions

Continuing last Tuesday’s examples, particularly concentrating on more than 2 dimensions.

Example

Find a potential function for F(x,y,z) = ⟨ 2xy+z², x²+2yz, y²+2xz ⟩

Follow the same plan of alternating integrations and differentiations that we used for 2 dimensions, but this time with more alternations. In particular…

Start by integrating the first component of F with respect to x, because if there’s a potential function, f, at all, that component is f’s derivative with respect to x. While integrating, remember that there are 2 other variables, y, and z, so the “constant” of integration can be any function of those two variables; call it g(y,z):

Integrate 2 X Y plus Z squared to get potential function X squared Y plus X Z squared plus G of Y and Z

To figure out what g(y,z) is, differentiate f as known so far with respect to y; setting this derivative equal to the second component of F tells you what the derivative of g with respect to y must be. Integrate that with respect to y to find g, possibly with a “constant” of integration that is a function of just z:

Derivative of little F with respect to Y should be X squared plus 2 Y Z but is X squared plus derivative of G

Finally, differentiate the now-better-known f with respect to z, and set the result equal to the third component of F to determine what h(z) is. In this case it turned out to be 0, but had it been non-0 you could have integrated its derivative to find it, plus a truly constant constant of integration:

2 rounds of integrate and differentiate yield potential function X squared Y plus X Z squared plus Y squared Z

Use that potential function to evaluate the integral of F(x,y,z)•T(x,y,z) over the path r(t) = ⟨ cos(4πt), sin(4πt), t ⟩, 0 ≤ t ≤ 1.

Since F is a conservative field, you don’t need to work out F( r(t) ) • r′(t), you can just evaluate the potential function at the ends of the path and subtract, by the fundamental theorem of line integrals:

Evaluating a line integral via the fundamental theorem gives 1

Problem Set

The last problem set, (mostly) on conservative vector fields and potential functions, is ready.

The plan is to work on it this week and grade it next, although since there’s currently no time to speak of available on my calendar next week, the official “grade by” date is December 16.

See the handout for details.

Green’s Theorem, an equivalence between certain line integrals and certain double integrals.

Please read “Extending the Fundamental Theorem of Calculus” and “Circulation Form of Green’s Theorem” in section 5.4 of the textbook.

Potential Functions in 3 Dimensions