Multivariable calculus, class #32: conservative vector fields

Mathematician spotlight: Harrison Bray

We briefly discuss the Thurston set in the complex plane. We do an example where Green's Theorem "fails" because the vector field is not continuous at a point inside the curve. We do an example of closing off a curve and applying Green's Theorem inside, and then subtracting off the line integral over the added curve. We give the Fundamental Theorem of Line Integrals, for conservative vector fields and their associated potential functions. We do two examples of vector line integrals over conservative vector fields, the second of which is a closed curve.

[Erican]

Thank you soooo much for the whole course, it is truly a life-saving series of videos that helped me to understand multivariable culculus thorouly and systematically.

[georgesadler]

Professor Diana Davis, thank you for a powerful analysis of Conservative vector fields and Green's Theorem, however there is an error in your first example. There is a square missing in the denominator of the Q sub x for both P sub y and Qsub x to be equal.Please correct this error in the video. Calculus Three uses every mathematical tool/trick in mathematics.