\nopagenumbers
\def\pt#1{\hbox{#1) }}
\input amssym.def
\input amssym.tex
\input epsf
\def\R{{\Bbb R}}


\headline={ESP Math 408D - AP\hfil Fall 1996}
\footline={\ifnum\pageno>1 \hfil Worksheet 35\hskip.3in Page \folio \fi}
\footnote{}{Mike Oehrtman}

\centerline
{\bf WORKSHEET 35} 
\bigskip

\item{1.}  Let $f\colon\R^2\to\R$ be a differentiable function.  Show
that $\nabla f$ is perpendicular to the level sets of $f$.

\medskip 
\item{2.}  Below are two vector fields drawn in $\R^2$.  Only one of
them can possibly be $\nabla f$ for some function $f\colon\R^2\to\R$.

\centerline{
\hskip.5in
\epsfysize=3.4in\epsfxsize=3.4in\epsfbox{/home/oehrtman/m210s96/vfa.ps}
\hskip.3in
\epsfysize=3.4in\epsfxsize=3.4in\epsfbox{/home/oehrtman/m210s96/vfb.ps}}

\itemitem{a)}  Which of the vector fields is $\nabla f$ for some
function $f\colon\R^2\to\R$?  Why?

\itemitem{b)}  For the vector field you identified in part a), draw
the level sets of $f$.

\itemitem{c)}  Sketch the graph of $f$.

\itemitem{d)}  Now draw a few solution curves to the differential
equations represented by the vector field.  That is, draw lines which
are everywhere tangent to the vector field.  If you followed a path on
the graph of $f$ directly above one of these curves, describe what you
would be doing.

\medskip
\item{3.} Let $f\colon\R^2\to\R$ be given by $f(x,y)=x^2+y^2$.

\medskip
\itemitem{a)}  Find $\nabla f$ and sketch it as a vector field on the
plane. 

\medskip
\itemitem{b)}  Sketch a graph of $f$.

\eject
\item{4.}  A volcano just erupted and lava is streaming down the
mountaintop.  Suppose that the altitude of the mountain is given by
$$
z(x,y)=he^{-(x^2+y^2)}
$$
where $h$ is the maximum height, and suppose also that lava flows in
the direction of steepest descent.

\medskip
\itemitem{a)}  Find the projection on the $xy$-plane of the direction
in which the lava flows away from the point $(1,2,he^{-9})$.

\medskip
\itemitem{b)}  Find the projection on the $xy$-plane of the path
taken by the lava as it flows down the mountaintop.

\medskip
\item{5.}  Below is the graph of a function $f\colon\R^2\to\R$.
Sketch $\nabla f$ as a vector field on $\R^2$.

\centerline{
\epsfxsize=3.5in\epsfbox{/home/oehrtman/m210s96/3dgraph.ps}}


\item{6.}  Suppose that a duck is swimming in the circle $x=\cos t$,
$y=\sin t$ and that the water temperature is given by the formula
$T=x^2e^y-xy^3$.  Find $dT/dt$, the rate of change in temperature the
duck might feel: 

\medskip
\itemitem{a)} by the chain rule; 

\medskip
\itemitem{b)} by expressing $T$ in terms of $t$ and differentiating.




\bye