%% -*- latex -*-
%/*
\input ../macros/Header
\if\Book0
\usepackage{exscale}
\begin{document}
\setcounter{section}{8}
\setcounter{page}{105}
\fi
%*/
%*{{page>wiki:headers:hheader}}
%* Navigate [[..:..:activities:link|back to the activity]].
%/*
\Lab{The Grid}
\SecMark
\label{grid}
%*/
%*==== THE GRID ====
%/*
\subsection{Essentials}
%*/
%*=== Essentials ===
%/*
\subsubsection{Main ideas}
%*/
%*== Main ideas ==
%/*
\Goal{
%*/
%/*
\begin{itemize}
\item
%*/
%*<html><ul><li></html>
Understanding different ways of expressing area using integration.
%*<html></li></html>
%/*
\item
%*/
%*<html><li></html>
Concrete example of Area Corollary to Green's/Stokes' Theorem.
%*<html></li></ul></html>
%/*
\end{itemize}
%*/
%/*
}
%*/
%/*
\noindent
%*/
We originally used this activity after covering Green's Theorem; we now skip
Green's Theorem and do this activity shortly before Stokes' Theorem.
%/*
\subsubsection{Prerequisites}
%*/
%*== Prerequisites ==
%/*
\Req{
%*/
%/*
\begin{itemize}
\item
%*/
%*<html><ul><li></html>
Familiarity with line integrals.
%*<html></li></html>
%/*
\item
%*/
%*<html><li></html>
%/*
\textit{Green's Theorem is not a prerequisite!}
%*/
%*//Green's Theorem is not a prerequisite!//
%*<html></li></ul></html>
%/*
\end{itemize}
%*/
%/*
}
%*/
%/*
\subsubsection{Warmup}
%*/
%*== Warmup ==
%/*
\begin{itemize}
\item
%*/
%*<html><ul><li></html>
The first problem is a good warmup.
%*<html></li></ul></html>
%/*
\end{itemize}
%*/
%/*
\subsubsection{Props}
%*/
%*== Props ==
%/*
\begin{itemize}
\item
%*/
%*<html><ul><li></html>
whiteboards and pens
%*<html></li></html>
%/*
\item
%*/
%*<html><li></html>
a planimeter if available
%*<html></li></ul></html>
%/*
\end{itemize}
%*/
%/*
\subsubsection{Wrapup}
%*/
%*== Wrapup ==
%/*
\begin{itemize}
\item
%*/
%*<html><ul><li></html>
Emphasize the magic -- finding area by walking around the boundary!
%*<html></li></html>
%/*
\item
%*/
%*<html><li></html>
Point out that this works for any closed curve, not just the rectangular
regions considered here.
%*<html></li></html>
%/*
\item
%*/
%*<html><li></html>
Demonstrate or describe a planimeter, used for instance to measure the area of
a region on a map by tracing the boundary.
%*<html></li></ul></html>
%/*
\end{itemize}
%*/
%/*
\newpage
%*/
%/*
\subsection{Details}
%*/
%*=== Details ===
%/*
\subsubsection{In the Classroom}
%*/
%*== In the Classroom ==
%/*
\begin{itemize}
\item
%*/
%*<html><ul><li></html>
Make sure students use a consistent orientation on their path.
%*<html></li></html>
%/*
\item
%*/
%*<html><li></html>
Make sure students explicitly include all segments of their path, including
those which obviously yield zero.
%*<html></li></html>
%/*
\item
%*/
%*<html><li></html>
Students in a given group should all use the same curve.
%*<html></li></html>
%/*
\item
%*/
%*<html><li></html>
Students should be discouraged from drawing a curve whose longest side is
along a coordinate axis.
%*<html></li></html>
%/*
\item
%*/
%*<html><li></html>
Students may need to be reminded that $\OINT$ implies the counterclockwise
orientation. But it doesn't matter what orientation students use so long as
they are consistent!
%*<html></li></html>
%/*
\item
%*/
%*<html><li></html>
A geometric argument that the orientation should be reversed when
interchanging $x$ and $y$ is to rotate the $xy$-plane about the line $y=x$.
(This explains the minus sign in Green's Theorem.)
%*<html></li></html>
%/*
\item
%*/
%*<html><li></html>
Students may not have seen line integrals of this form (see below).
%*<html></li></ul></html>
%/*
\end{itemize}
%*/
%/*
\subsubsection{Subsidiary ideas}
%*/
%*== Subsidiary ideas ==
%/*
\Sub{
%*/
%/*
\begin{itemize}
\item
%*/
%*<html><ul><li></html>
Orientation of closed paths.
%*<html></li></html>
%/*
\item
%*/
%*<html><li></html>
Line integrals of the form $\INT P\,dx+Q\,dy$.
%/*
\hfill\break
\textit{
%*/
%*//
We do not discuss such integrals in class! Integrals of this form
almost always arise in applications as $\INT\FF\cdot d\rr$.
%*//
%/*
}
%*/
%*<html></li></ul></html>
%/*
\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
%*/
%*<html><ul><li></html>
Write down Green's Theorem.
%*<html></li></html>
%/*
\item
%*/
%*<html><li></html>
Go to 3 dimensions --- bend the curve out of the plane and stretch the region
like a butterfly net or rubber sheet. This is the setting for Stokes'
Theorem!
%*<html></li></ul></html>
%/*
\end{itemize}
%*/
%/*
}
%*/
%/*
\input ../macros/footer
%*/