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\input ../macros/Header
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\newlabel{master}{{4}{21}}
\newlabel{implicit}{{11.2}{59}}
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%*{{page>wiki:headers:hheader}}
%* Navigate [[..:..:activities:link|back to the activity]].
%/*
\Lab{The Valley}
\SecMark
\label{Gradient}
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%*==== THE VALLEY ====
%/*
\subsection{Essentials}
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%*=== Essentials ===
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\subsubsection{Main ideas}
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%*== Main ideas ==
%/*
\Goal{
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%/*
\begin{itemize}
\itemsep=0pt
\item
%*/
%*-
Reinforces both the Master Formula and differentials.
%*
%/*
\item
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%*-
Sets the stage for path-independence.
%*
%/*
\end{itemize}
%*/
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}
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%/*
%\textbf{Don't skip this lab!}
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\subsubsection{Prerequisites}
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%*== Prerequisites ==
%/*
\Req{
%*/
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\begin{itemize}
\itemsep=0pt
\item
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%*-
Some familiarity with differentials.
%*
%/*
\item
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%*-
Familiarity with the gradient.
%*
%/*
\end{itemize}
%*/
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}
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%/*
\subsubsection{Warmup}
%*/
%*== Warmup ==
A brief derivation of the master formula from the expression for the
differential of a function of two variables.
%/*
\subsubsection{Props}
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%*== Props ==
%/*
\begin{itemize}
\itemsep=0pt
\item
%*/
%*-
whiteboards and pens
%*
%/*
\item
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%*-
valley transparency
%/*
(master on page \pageref{valleymaster})
%*/
%*(master available
%*{{private:bridge:activities:content:guides:valley.pdf|here}})
%*
%/*
\item
%*/
%*-
blank transparencies and pens
%/*
%*
%/*
\end{itemize}
%*/
%/*
\subsubsection{Wrapup}
%*/
%*== Wrapup ==
%/*
\begin{itemize}
\itemsep=0pt
\item
%*/
%*-
Call someone from each group to the board to draw both their path and $d\rr$
on the topo map and show how they found $d\rr$. Discuss the different methods
used by different groups. The idea here is that
%/*
\textit{\bfseries on a curve}
%*/
%*//**on a curve**//
$dy$ is related to $dx$. Students are being asked to find this relationship,
and plug it into the general expression for $d\rr$.
%/*
\hfill\break \null\quad
%*/
%*\\\\
%/*
\textit{``Use what you know! Any (algebraically correct) method will work.''}
%*/
%*//"Use what you know! Any (algebraically correct) method will work."//
%*
%/*
\item
%*/
%*-
Emphasize that $\grad h$ is a property of the hill, while $d\rr$ is a property
of the curve. The point of the master formula is that it naturally separates
the information in $dh$ into these quite different geometric ideas.
%*
%/*
\item
%*/
%*-
Have the class discuss why the answer to the second integral is in fact easy
to find without integration.
%*
%/*
\end{itemize}
%*/
%/*
\newpage
%*/
%/*
\subsection{Details}
%*/
%*=== Details ===
%/*
\subsubsection{In the Classroom}
%*/
%*== In the Classroom ==
%/*
\begin{itemize}
\item
%*/
%*-
This lab is on the long side; don't plan to do
%/*
\textit{anything}
%*/
%*//anything//
else in a 50-minute period. The wrapup alone easily requires 20 minutes to do
properly; you may wish to do part of it in a subsequent class period.
%*
%/*
\item
%*/
%*-
Some students may not realize that $(1,1)$ is on the given circle!
%*
%/*
\item
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%*-
Ask the students if their level curves are equally spaced.
%/*
\hfill\break
%*/
%*\\\\
(They shouldn't be.)
%*
%/*
\item
%*/
%*-
Initially assign each group one of the curves; groups which finish quickly can
try other curves. The first curve, the circle, is qualitatively different
from the others, and more difficult; see
%/*
Section~\ref{implicit}.
%*/
%*Section 11.2.
Furthermore, the instructions do not uniquely determine the curve in this case
--- although the final answer is unaffected. You may wish to assign this
curve to a strong group, or not let any group try the circle until they have
first done one of the other curves.
%*
%/*
\item
%*/
%*-
Some students substitute the given point into the height function before
computing the gradient! Perhaps asking for a sketch of $\grad h$ at several
points rather than just one would discourage this.
%*
%/*
\item
%*/
%*-
Ensure that students reduce to one variable before integrating.
%*
%/*
\item
%*/
%*-
Emphasize that one can plug in the relationship between $x$ and $y$ either
before or after computing the differential of $h$. Which choice is easiest
depends on the circumstances; both will work.
%*
%/*
\item
%*/
%*-
In the next-to-last question, groups may need to be reminded that they need to
plug in information about their curve in order to find $dh$. They should use
the expression for the differential of $h$ as a function of either one or two
variables, rather than the master formula (which should not be used until the
last question).
%*
%/*
\item
%*/
%*-
Some students will realize that the integrals must be the same because of the
master formula before ever trying to compute the second integral. Such
students should be praised --- but still encouraged to compute the second
integral without using the master formula.
%*
%/*
\item
%*/
%*-
On the circle, some students go from $x^2+y^2=a^2$ directly to
%/*
``$d\rr=2x\,dx\,\ii+2y\,dy\,\jj$''!
%/*
%*"$d\rr=2x\,dx\,\ii+2y\,dy\,\jj$"!
One way to push students away from this mistake is to emphasize that one
%/*
\textit{always}
%*/
%*//always//
has $d\rr = dx\,\ii+dy\,\jj$
(or a similar expression in other coordinate systems). We literally stomp
our feet when insisting that students start problems involving $d\rr$ by
writing down one of these expressions! A discussion of this point works well
as part of the wrapup.
%*
%/*
\item
%*/
%*-
See the discussion of using transparencies for
%/*
Group Activity~\ref{hill}.
%*/
%*[[private:bridge:activities:guides:hill|the hill activity]]
%*
%/*
\item
%*/
%*-
Emphasize that $\DS\Lint$ is a definite integral, and that $\DS\Lint 0\,dx=0$
(not 1).
%*
%/*
\end{itemize}
%*/
%/*
\subsubsection{Subsidiary ideas}
%*/
%*== Subsidiary ideas ==
%/*
\Sub{
%*/
%/*
\begin{itemize}
\item
%*/
%*-
The gradient is perpendicular to level curves.
%*
%/*
\item
%*/
%*-
Emphasize that $df=\Partial{f}{x}\,dx+\Partial{f}{y}\,dy$ is a
coordinate-dependent expression for $df$, whereas writing $df=\grad f\cdot
d\rr$ is coordinate independent.
%*
%/*
\end{itemize}
%*/
%/*
}
%*/
%/*
\subsubsection{Homework}
%*/
%*== Homework ==
%/*
\HW{
%*/
%/*
\begin{enumerate}
\item
%*/
%*-
Consider the valley in this group activity, whose height $h$ in meters is
given by $h={~x^2\over10}+{~y^2\over10}$, with $x$ and $y$ (and 10!) in
meters. Suppose you are hiking through this valley on a trail given by
\begin{eqnarray*}
x=3t \qquad y=2t^2
\end{eqnarray*}
with $t$ in seconds (and where
%/*
``3'' and ``2''
%*/
%*"3" and "2"
have appropriate units).
%/*
\begin{enumerate}
\item
%*/
%*
-
Starting from the master formula, determine how fast you are climbing (rate of
change of $h$)
%/*
{\it per meter\/}
%*/
%*//per meter//
along the trail when $t=1$.
%/*
\hfill\break {/it
%*/
%*//
You may find it helpful to recall that $ds=|d\rr|$.
%*//
%/*
}
%*/
%*
%/*
\item
%*/
%*-
Starting from the master formula, determine how fast you are climbing
%/*
{\it per second\/}
%*/
%*//per second//
when $t=1$.
%*
%/*
\end{enumerate}\end{enumerate}
%*/
%/*
}
%*/
%/*
\subsubsection{Essay questions}
%*/
%*== Essay questions ==
%/*
\Essay{
%*/
%/*
\begin{itemize}
\item
%*/
%*-
During this activity, you drew a gradient vector on a topographic map.
Can you draw this vector to scale? Explain.
%*
%/*
\item
%*/
%*-
What properties of your path are needed to compute the integrals in this
activity? To determine the answer?
%*
%/*
\end{itemize}
%*/
%/*
}
%*/
%/*
\subsubsection{Enrichment}
%*/
%*== Enrichment ==
%/*
\Rich{
%*/
%/*
\begin{itemize}
\item
%*/
%*-
Discuss the relationship between the master formula, the gradient, topographic
maps, and path-independence.
%*
%/*
\item
%*/
%*-
Discuss the fundamental theorem for gradients, namely that the line integral
of a gradient is just an obvious antiderivative. Relate this to the geometry,
especially the existence of a topo map.
%*
%/*
\item
%*/
%*-
Many students will integrate the two pieces of $dh=2x\,dx+2y\,dy$ separately,
without worrying about the path. What path is implicitly being used?
%*
%/*
\item
%*/
%*-
We strongly discourage students from inserting artificial signs into
expressions such as $d\rr = dx\,\ii + dy\,\jj$. This forces $dy<0$, and in
some cases also $dx<0$, so that one must integrate from $1$ to $0$. By all
means discuss the alternative convention with students, which requires $dx$
and $dy$ to always be positive, and then forces one to insert (and keep track
of) appropriate signs by hand.
%*
%/*
\item
%*/
%*-
Following this lab is a good time to introduce or review the proof, using the
master formula, that the gradient is perpendicular to level curves and that
it points in the direction of maximal increase.
%*
%/*
\item
%*/
%*-
A great followup to this activity is a discussion of what questions you can
answer using the master formula.
%*
%/*
\item
%*/
%*-
It is immediately obvious in polar coordinates that these integrals do not
depend on $\phi$, and hence are independent of path.
%*
%/*
\end{itemize}
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}
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\newpage
\thispagestyle{empty}
\begin{figure}
\epsfxsize=6.5in
\centerline{\epsffile[61 151 523 613]{valley.ps}}
\label{valleymaster}
\end{figure}
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