%% -*- latex -*-
%/*
\input ../macros/Header
\if\Book0
\usepackage{epsfig}
\begin{document}
\setcounter{section}{10}
\setcounter{page}{109}
\fi
%*/
%*{{page>wiki:headers:hheader}}
%* Navigate [[..:..:activities:link|back to the activity]].
%/*
\Lab{The Fishing Net}
\SecMark
\label{div}
%*/
%*==== THE FISHING NET ====
%/*
\subsection{Essentials}
%*/
%*=== Essentials ===
%/*
\subsubsection{Main ideas}
%*/
%*== Main ideas ==
%/*
\Goal{
%*/
%/*
\begin{itemize}
\item
%*/
%*-
Practice doing surface integrals
%*
%/*
\item
%*/
%*-
The Divergence Theorem
%*
%/*
\end{itemize}
%*/
%/*
}
%*/
%/*
\subsubsection{Prerequisites}
%*/
%*== Prerequisites ==
%/*
\Req{
%*/
%/*
\begin{itemize}
\item
%*/
%*-
Ability to do flux integrals
%*
%/*
\item
%*/
%*-
Definition of divergence
%*
%/*
\item
%*/
%*-
Statement of Divergence Theorem
%/*
\hfill\break
\textit{
%*/
%*//\\\\
This lab can be used prior to covering the Divergence Theorem in class
with either a minimal introduction or a restatement of the last question based
on the assumption that the given vector field doesn't
%/*
``lose''
%*/
%*"lose"
anything going through the net.
%*//
%/*
}
%*/
%*
%/*
\end{itemize}
%*/
%/*
}
%*/
%/*
\subsubsection{Warmup}
%*/
%*== Warmup ==
%/*
\begin{itemize}
\item
%*/
%*-
Perhaps a reminder about what the Divergence Theorem is.
%*
%/*
\end{itemize}
%*/
%/*
\subsubsection{Props}
%*/
%*== Props ==
%/*
\begin{itemize}
\item
%*/
%*-
whiteboards and pens
%*
%/*
\item
%*/
%*-
a model of the fishing net, made from any children's building set
%*
%/*
\end{itemize}
%*/
%/*
\subsubsection{Wrapup}
%*/
%*== Wrapup ==
%/*
\begin{itemize}
\item
%*/
%*-
Reiterate that the Divergence Theorem only applies to closed surfaces.
%*
%/*
\item
%*/
%*-
Emphasize that the Divergence Theorem is one of several astonishing theorems
relating what happens inside to what happens outside.
%*
%/*
\item
%*/
%*-
Have several students show how they computed $d\SS$, since most likely
different choices were made for $d\rr_i$ and hence the limits.
%*
%/*
\end{itemize}
%*/
%/*
\newpage
%*/
%/*
\subsection{Details}
%*/
%*=== Details ===
%/*
\subsubsection{In the Classroom}
%*/
%*== In the Classroom ==
%/*
\begin{itemize}
\item
%*/
%*-
By now the groups should be working well. Sit back and watch!
%*
%/*
\item
%*/
%*-
The main thing to watch out for is whether students choose the correct signs,
both for the normal vectors and the limits of integration. Reiterate that one
should
%/*
\textit{always}
%*/
%*//always//
write $d\rr=dx\,\ii+dy\,\jj+dz\,\kk$; there should
%/*
\textit{never}
%*/
%*//never//
be minus signs in this equation. The signs will come out right provided one
integrates in the direction of the vectors chosen.
%*
%/*
\item
%*/
%*-
Most students will realize quickly that there is no flux through the
triangular sides.
%*
%/*
\item
%*/
%*-
Some students will try to do the surface integrals! Point out that this isn't
possible --- and that the instructions say not to.
%*
%/*
\item
%*/
%*-
Student may be surprised at first when they calculate $\grad\cdot\FF=0$,
especially since they (correctly) won't think that the surface integrals will
add to zero. Use this to motivate the
%/*
``missing top''.
%*/
%*"missing top".
%*
%/*
\item
%*/
%*-
Some students incorrectly think that $d|z|=|dz|$.
%*
%/*
\end{itemize}
%*/
%/*
\subsubsection{Subsidiary ideas}
%*/
%*== Subsidiary ideas ==
%/*
\Sub{
%*/
%/*
\begin{itemize}
\item
%*/
%*
%/*
\end{itemize}
%*/
%/*
}
%*/
%/*
\subsubsection{Homework \None}
%*/
%*== Homework ==
%* (none yet)
%/*
\HW{
%*/
%/*
}
%*/
%/*
\subsubsection{Essay questions \None}
%*/
%*== Essay questions ==
%* (none yet)
%/*
\Essay{
%*/
%/*
}
%*/
%/*
\subsubsection{Enrichment}
%*/
%*== Enrichment ==
%/*
\Rich{
%*/
%/*
\begin{itemize}
\item
%*/
%*-
The surface integrals can in fact be done --- provided one adds them up prior
to evaluating the integrals.
%*
%/*
\item
%*/
%*-
This lab provides a good opportunity for students to visualize the flux: It's
easy to see that the flux of the horizontal component of this vector field
must be zero geometrically. (It's even easier to see that the vertical flux
must be zero.)
%*
%/*
\begin{figure}
\centerline{\epsfysize=2in\epsffile{net.ps}}
\end{figure}
%*/
%*{{private:bridge:activities:content:guides:net.jpg}}
%/*
\item
%*/
%*-
During the wrapup (or the following lecture), draw a picture such as the one
above of one of the rectangular faces, showing all 4 possible choices for
$d\rr_1$ and $d\rr_2$ (and which is which!), and discuss the integration
limits in each case.
%*
%/*
\item
%*/
%*-
An alternative approach to this problem is to determine $\dS$ geometrically,
compute $\FF\cdot\nn$ explicitly, and then do the integral using
%/*
``standard''
%*/
%*"standard"
(increasing) limits. There is nothing wrong with this approach, but we would
discourage the use of the $d\rr$ notation here for fear of making sign errors.
%*
%/*
\item
%*/
%*-
One could show students the remarkable trick for integrating $e^{-x^2}$ from
$0$ to $\infty$, by squaring and evaluating in polar coordinates.
%*
%/*
\end{itemize}
%*/
%/*
}
%*/
%/*
\input ../macros/footer
%*/