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### STOKES' THEOREM

#### Essentials

##### Main ideas

- Practice visualizing surfaces
- Stokes' Theorem

##### Prerequisites

- Ability to do line and surface integrals
- Definition of curl
- Statement of Stokes' Theorem

##### Warmup

None, but be prepared to talk about appropriate surfaces for Stokes' Theorem (perhaps using a “butterfly net” as a prop).

##### Props

- whiteboards and pens
- a butterfly net; homemade is fine, such as a plastic bag on a wire rim
- formula sheet for div and curl in spherical and cylindrical coordinates (Each group may need its own copy.)

##### Wrapup

- Discuss the various surfaces one could use for the second question.
- Discuss the various ways one could compute the curl.
- This could be a good time to emphasize the similarity between the basic theorems.

#### Details

##### In the Classroom

- Students like this lab; it should flow smoothly and quickly.
- Make sure students choose surfaces which can catch butterflies!
- The curl is easy but slightly messy in rectangular coordinates, starting from the formula $\DS\phat={-y\,\ii+x\,\jj\over\sqrt{x^2+y^2}}$.
- It is easier to factor $\FF$ as $(r^2)(r\,\phat)$ than as $(r^3)(\phat)$.
- The (curl and the) resulting surface integrals are much easier in cylindrical (or possibly spherical) coordinates.
- Some students want to write “$\FF\times\grad$” rather than $\grad\times\FF$.
- A possibly related problem is that students will often write $\grad\times\FF$ even when the vector field is called something else, such as $\GG$.
- Students using a disk or cylinder may well want to use cylindrical basis vectors here; this should be encouraged.
- Some students will draw a cone whose tip is at the origin; this is wrong.
- Students using a hemisphere will probably reinterpret $r$ as the spherical radial coordinate; this is fine, although the instructor needs to be prepared to help students understand why they get a different answer for curl; see below.

##### Subsidiary ideas

- Different ways of calculating curl.
- Calculating the curl in curvilinear coordinates.

##### Homework

(none yet)

##### Essay questions

(none yet)

##### Enrichment

- Many students who try the paraboloid will discover that they don't in fact need to substitute the equation of the paraboloid! That is, leaving both $dr$ and $dz$ intact results in the $dz$ term canceling anyway. Such students have in fact done a nearly arbitrary surface! (If it's not the graph of a function a further argument is needed.)
- Mention the product rule for curl, namely \begin{eqnarray*} \grad\times(f\GG) = (\grad f) \times \GG + f (\grad\times\GG) \end{eqnarray*} Discuss the fact that
*all*product rules take the form 1)*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 vector field is deliberately given in
*polar*coordinates; the extension off the plane (or for that matter off the circle) doesn't matter! Most students will assume there is no $z$-dependence without thinking about it; this is fine, and does not need to be discussed. But students using spherical coordinates will most likely interpret $r$ as the spherical radial coordinate, thus obtaining a different vector field than the above (which would be $r^3\sin^3\theta\,\phat$). It is important to realize that this is fine! vector field which has the correct limit to the circle (and is differentiable) will work! - The “wire” singularity for the vector field $\phat\over r$ from an earlier activity can in fact be handled by interpreting $r$ as the spherical radial coordinate, and using Stokes' Theorem on a hemisphere. This is of course no longer the magnetic field of a wire carrying a steady current, and the curl of this vector field isn't zero.
- Ask students how to apply Stokes' Theorem to an open cylinder, with neither top nor bottom.

1)
The product rules for derivatives of $\FF\times\GG$ do not obviously
have this form, but can be rewritten (in terms of differential forms or
covariant differentiation) so that they do.