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\Lab{Potential Functions}
\SecMark
\label{mmmlab}
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%*==== POTENTIAL FUNCTIONS ====
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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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\begin{itemize}
\item
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%*-
Finding potential functions.
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\end{itemize}
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}
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\noindent
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Students love this activity. Some groups will finish in 10 minutes or less;
few will require as much as 30 minutes.
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\footnote{
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%*((
More accurately, students love the Murder Mystery Method! We often
incorporate this activity into an exam review, rather than devoting an entire
period to it.
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}
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%*))
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\subsubsection{Prerequisites}
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%*== Prerequisites ==
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\Req{
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\begin{itemize}
\item
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%*-
Fundamental Theorem for line integrals
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\item
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The Murder Mystery Method
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\end{itemize}
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}
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\subsubsection{Warmup}
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%*== Warmup ==
none
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\subsubsection{Props}
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%*== Props ==
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\begin{itemize}
\item
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%*
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\end{itemize}
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\subsubsection{Wrapup}
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%*== Wrapup ==
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\begin{itemize}
\item
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%*-
Revisit integrating conservative vector fields along various paths, including
reversing the orientation and integrating around closed paths.
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\end{itemize}
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\newpage
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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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\begin{itemize}
\item
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%*-
We recommend having the students work in groups of 2 on this activity, and not
having them turn anything in.
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\item
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%*-
Most students will treat the last example as 2-dimensional, giving the answer
$xyz$. Ask these students to check their work by taking the gradient; most
will include a $\kk$ term. Let them think this through. The correct answer
of course depends on whether one assumes that $z$ is constant; we have
deliberately left this ambiguous.
%*
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\end{itemize}
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\subsubsection{Subsidiary ideas}
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%*== Subsidiary ideas ==
%/*
\Sub{
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%/*
\begin{itemize}
\item
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%*-
3-d vector fields do not necessarily have a $\kk$-component!
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\end{itemize}
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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 ==
%/*
\begin{itemize}
\item
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%*-
The derivative check for conservative vector fields can be described using the
same type of diagrams as used in the Murder Mystery Method; this is just
moving down the diagram (via differentiation) from the row containing the
components of the vector field, rather than moving up (via integration). We
believe this should not be mentioned until after this lab.
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%*//\\\\
When done in 3-d, this makes a nice introduction to curl --- which
however is not needed until one is ready to do Stokes' Theorem. We would
therefore recommend delaying this entire discussion, including the 2-d case,
until then.
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}
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%*//
%*
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\end{itemize}
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\Rich{
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\begin{itemize}
\item
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%*-
Work out the Murder Mystery Method using polar basis vectors, by reversing the
process of taking the gradient in this basis.
%*
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\item
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%*-
Use pictures of vector fields to investigate whether they are conservative.
If not, find a closed curve with nonzero circulation. If yes, find level
curves. See the examples {{vcconservative.pdf|here}}.
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\item
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%*
-
Revisit the example in the
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\textit{Amp\`ere's Law}
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%*Ampère's Law
lab, using the Fundamental Theorem to explain the results. This can be done
without reference to a basis, but it is worth computing $\grad\phi$ in a polar
basis.
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\end{itemize}
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}
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