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%* Navigate [[..:..:activities:link|back to the activity]].
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\Lab{Divergence and Curl}
\SecMark
\label{curl}
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%*==== DIVERGENCE AND CURL ====
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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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%*-
Visualization of divergence and curl.
%*
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\end{itemize}
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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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%*-
Definition of divergence and curl.
%*
%/*
\item
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%*-
Geometry of divergence and curl, either through a geometric definition or
through Stokes' Theorem and the Divergence Theorem.
%*
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\end{itemize}
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}
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%/*
\subsubsection{Warmup}
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%*== Warmup ==
%/*
\begin{itemize}
\item
%*/
%*-
Students may need to be reminded what
%/*
\textit{circulation}
%*/
%*//circulation//
is.
%*
%/*
\item
%*/
%*-
Students may not have seen
%/*
\textit{flux}
%*/
%*//flux//
in 2 dimensions.
%*
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\end{itemize}
%*/
%/*
\subsubsection{Props}
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%*== Props ==
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\begin{itemize}
\item
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%*-
whiteboards and pens
%*
%/*
\item
%*/
%*-
formula sheet for div and curl in spherical and cylindrical coordinates
(Each group may need its own copy.)
%*
%/*
\item
%*/
%*-
divergence and curl transparency
%/*
(master on page \pageref{curlmaster})
%*/
%*(master available
%*{{private:bridge:activities:content:guides:curl.pdf|here}})
%*
%/*
\item
%*/
%*-
blank transparencies and pens
%*
%/*
\end{itemize}
%*/
%/*
\subsubsection{Wrapup}
%*/
%*== Wrapup ==
%/*
\begin{itemize}
\item
%*/
%*-
Discuss the effect of choosing loops of different shapes, especially those
adapted to the given vector field.
%*
%/*
\item
%*/
%*-
Talk about the geometry of sinks and sources (for divergence) and paddlewheels
(for curl).
%*
%/*
\end{itemize}
%*/
%/*
\newpage
%*/
%/*
\subsection{Details}
%*/
%*=== Details ===
%/*
\subsubsection{In the Classroom}
%*/
%*== In the Classroom ==
%/*
\begin{itemize}
\itemsep=0pt
\item
%*/
%*-
While students are working on this activity, draw the vector fields on the
board to use during the wrapup. Alternatively, bring an overhead transparency
showing the vector fields (and blank transparencies for students to write on).
%*
%/*
\item
%*/
%*-
Students like this lab; it should flow smoothly and quickly.
%*
%/*
\item
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%*-
Students may need to be reminded what $\OINT$ means, and that the positive
orientation in the plane is counterclockwise.
%*
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\item
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%*-
Yes, two pairs of questions are really the same.
%*
%/*
\item
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%*-
Make sure the paths do
%/*
\textit{not}
%*/
%*//not//
go around the origin.
%*
%/*
\item
%*/
%*-
Encourage each group to work on at least two vector fields, which are in
different rows and columns. Include one vector field from the third column if
time permits.
%*
%/*
\item
%*/
%*-
Encourage each group to consider, for a single vector field, moving their loop
to another location. This is especially effective (and in fact essential) for
the two vector fields in the third column.
%*
%/*
\item
%*/
%*-
See the discussion of using transparencies for
%/*
Group Activity~\ref{hill}.
%*/
%*[[private:bridge:activities:guides:hill|the hill activity]]
%*
%/*
\item
%*/
%*-
Students may eventually realize that the vector fields in the middle column
are linear combinations of the vector fields in the first column, which are in
turn
%/*
``pure curl'' and ``pure divergence'',
%*/
%*"pure curl" and "pure divergence",
respectively.
%*
%/*
\end{itemize}
%*/
%/*
\subsubsection{Subsidiary ideas}
%*/
%*== Subsidiary ideas ==
%/*
\Sub{
%*/
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\begin{itemize}
\item
%*/
%*-
Divergence and curl are not just about the behavior near the origin.
Derivatives are about
%/*
\textit{change} --- the \textit{difference}
%*/
%*//change// --- the //difference//
between nearby vectors.
%*
%/*
\end{itemize}
%*/
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}
%*/
%/*
\subsubsection{Homework}
%*/
%*== Homework ==
(MHG refers to McCallum, Hughes-Hallett, Gleason,
%/*
et al.~\cite{Harvard}.)
%*/
%*et al. [3].)
%/*
\HW{
%*/
%/*
\begin{itemize}
\itemsep=0pt
\item
%*/
%*-
%/*
MHG \S19.1:20
%*/
%*MHG §19.1:20
%*
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\item
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%*-
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MHG \S20.2:16
%*/
%*MHG §20.2:16
%*
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\item
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%*-
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MHG \S20.3:10,12,20
%*/
%*MHG §20.3:10,12,20
%*
%/*
\item
%*/
%*-
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MHG \S20.4:22
%*/
%*MHG §20.4:22
%*
%/*
\end{itemize}
%*/
%/*
}
%*/
%/*
\subsubsection{Essay questions \None}
%*/
%*== Essay questions ==
%* (none yet)
%/*
\Essay{
%*/
%/*
}
%*/
%/*
\subsubsection{Enrichment}
%*/
%*== Enrichment ==
%/*
\Rich{
%*/
%/*
\begin{itemize}
\itemsep=0pt
\item
%*/
%*-
Emphasize the importance of divergence and curl in applications.
%*
%/*
\item
%*/
%*-
Ask students how to determine which vector fields are conservative!
(A single closed path with nonzero circulation suffices to show that a vector
field is
%/*
\textit{not}
%*/
%*//not//
conservative. The best geometric way we know to show that a vector field
%/*
\textit{is}
%*/
%*//is//
conservative is to try to draw the level curves for which the given vector
field would be the gradient.)
%*
%/*
\item
%*/
%*-
Discuss the fact that $\rhat\over r$ and $\phat\over r$ are
%/*
\textit{both}
%*/
%*//both//
curl-free and divergence-free; this is counterintuitive, but crucial for
electromagnetism. (These are, respectively, the electric/magnetic field of a
charged/current-carrying wire along the $z$-axis.)
%*
%/*
\item
%*/
%*-
Discuss the behavior of $\rhat\over r^n$ and $\phat\over r^n$, emphasizing
that
%/*
\textit{both}
%*/
%*//both//
the divergence and curl vanish when $n=1$.
%*
%/*
\item
%*/
%*-
Relate these examples to the magnetic field of a wire ($\BB={\phat\over r}$)
and the electric field of a point charge ($\EE={\rhat\over r^2}$; this is the
%/*
\textit{spherical}
%*/
%*//spherical//
$r$).
%*
%/*
\item
%*/
%*-
Show students how to compute divergence and curl of these vector fields in
cylindrical coordinates.
%*
%/*
\item
%*/
%*-
Trying to estimate divergence and curl from a single plot of a vector field
confronts students with the need to zoom in. Technology can be useful here.
%*
%/*
\item
%*/
%*-
An excellent JAVA applet for analyzing the geometry of vector fields,
including the capability to zoom in, is Matthias Kawski's
%/*
\textit{Vector Field Analyzer},
%*/
%*//Vector Field Analyzer//,
available at
%/*
\texttt{http://math.la.asu.edu/\char'176kawski/vfa2}.
%*/
%*http://math.la.asu.edu/~kawski/vfa2.
%*
%/*
\item
%*/
%*-
Point students to our paper on
%/*
\textit{Electromagnetic Conic Sections},
%*/
%*//Electromagnetic Conic Sections//
which appeared in
%/*
Am.\ J. Phys.\ {\bf 70},
%*/
%*Am. J. Phys. **70**,
1129--1135 (2002), and which is also available on the Bridge Project website.
%*
%/*
\item
%*/
%*-
Most physical applications of the divergence are 3-dimensional, rather than
2-dimensional. Each vector field in this activity could be regarded as a
horizontal 3-dimensional vector field by assuming that there is no
$z$-dependence, in which case the flux can be computed through a
%/*
\hbox{3-dimensional}
%*/
%*3-dimensional
box whose cross-section is the loop, and whose horizontal top and bottom do
not contribute.
%*
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\end{itemize}
%*/
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}
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\newpage
\thispagestyle{empty}
% From divcurl.tex
\begin{figure}
\vspace{-0.5in}
\centerline{\epsffile[107 88 540 715]{divcurl.ps}}
\label{curlmaster}
\end{figure}
%*/
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