Students compute both sides of Stokes' theorem.
- Evaluate \(\oint\boldsymbol{\vec{F}}\cdot d\boldsymbol{\vec{r}}\) explicitly as a line integral, where \(\boldsymbol{\vec{F}} = r^3 \,\boldsymbol{\hat{\phi}}\) and \(C\) is the circle of radius \(3\) in the \(xy\)-plane, oriented in the usual, counterclockwise direction (as seen from above).
- Stokes' Theorem
- List at least 3 different surfaces which you could use with Stokes' Theorem to evaluate the line integral in the previous problem.
- Evaluate the surface integral for any one of the surfaces on your list.
- If time permits, evaluate the surface integral for other surfaces on your list.
None, but be prepared to talk about appropriate surfaces for Stokes' Theorem (perhaps using a “butterfly net” as a prop).
The derivative of a product is the derivative of the first quantity times the second plus the first quantity times the derivative of the second.(The product rules for derivatives of \(\boldsymbol{\vec{F}}\times\boldsymbol{\vec{G}}\) do not obviously have this form, but can be rewritten in terms of differential forms or covariant differentiation so that they do.) The only complication here is figuring out which derivative to take, and what multiplication to use! A similar product rule holds for the divergence.