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\Lab{Stokes' Theorem}
\SecMark
\label{stokes}
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%*==== STOKES' THEOREM ====
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\subsection{Essentials}
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%*=== Essentials ===
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\subsubsection{Main ideas}
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%*== Main ideas ==
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\Goal{
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Practice visualizing surfaces
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\item
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Stokes' Theorem
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}
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\subsubsection{Prerequisites}
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%*== Prerequisites ==
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\Req{
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Ability to do line and surface integrals
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\item
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Definition of curl
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\item
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Statement of Stokes' Theorem
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\end{itemize}
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}
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\subsubsection{Warmup}
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%*== Warmup ==
None, but be prepared to talk about appropriate surfaces for Stokes' Theorem
(perhaps using a
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``butterfly net''
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%*"butterfly net"
as a prop).
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\subsubsection{Props}
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%*== Props ==
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whiteboards and pens
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a butterfly net; homemade is fine, such as a plastic bag on a wire rim
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\item
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formula sheet for div and curl in spherical and cylindrical coordinates
(Each group may need its own copy.)
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\end{itemize}
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\subsubsection{Wrapup}
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%*== Wrapup ==
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Discuss the various surfaces one could use for the second question.
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Discuss the various ways one could compute the curl.
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This could be a good time to emphasize the similarity between the basic
theorems.
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\end{itemize}
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\subsection{Details}
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%*=== Details ===
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\subsubsection{In the Classroom}
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%*== In the Classroom ==
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Students like this lab; it should flow smoothly and quickly.
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Make sure students choose surfaces which can catch butterflies!
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The curl is easy but slightly messy in rectangular coordinates, starting from
the formula $\DS\phat={-y\,\ii+x\,\jj\over\sqrt{x^2+y^2}}$.
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It is easier to factor $\FF$ as $(r^2)(r\,\phat)$ than as $(r^3)(\phat)$.
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The (curl and the) resulting surface integrals are much easier in cylindrical
(or possibly spherical) coordinates.
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Some students want to write
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``$\FF\times\grad$''
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rather than $\grad\times\FF$.
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A possibly related problem is that students will often write $\grad\times\FF$
even when the vector field is called something else, such as $\GG$.
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Students using a disk or cylinder may well want to use cylindrical basis
vectors here; this should be encouraged.
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Some students will draw a cone whose tip is at the origin; this is wrong.
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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.
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\subsubsection{Subsidiary ideas}
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%*== Subsidiary ideas ==
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\Sub{
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Different ways of calculating curl.
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Calculating the curl in curvilinear coordinates.
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}
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\subsubsection{Homework \None}
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%*== Homework ==
%* (none yet)
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\HW{
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}
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\subsubsection{Essay questions \None}
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%*== Essay questions ==
%* (none yet)
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\Essay{
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}
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\subsubsection{Enrichment}
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%*== Enrichment ==
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\Rich{
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\begin{itemize}
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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.)
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\item
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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
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\textit{all}
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%*//all//
product rules take the form
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\footnote{
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%*((
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.
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}
\begin{quote}
\it
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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.
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The only complication here is figuring out which derivative to take, and what
multiplication to use! A similar product rule holds for the divergence.
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The vector field is deliberately given in
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\textit{polar}
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%*//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!
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\textit{Any}
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%*//Any
vector field which has the correct limit to the circle (and is differentiable)
will work!
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The
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``wire''
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%*"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.
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\item
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Ask students how to apply Stokes' Theorem to an open cylinder, with neither
top nor bottom.
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\end{itemize}
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}
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