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\headline={ESP Math 408D - AP\hfil Fall 1996}
\footline={\ifnum\pageno>1 \hfil Worksheet 38\hskip.3in Page \folio \fi}
\footnote{}{Mike Oehrtman}

\centerline
{\bf WORKSHEET 38} 
\bigskip

\item{1.}  Recall that when integrating a function over a region in
the plane using polar coordinates, a ``magical'' factor of $r$ appears
in the integrand.  What about other coordinate systems?  In general,
we must include some factor which makes up for the stretching and
shrinking that occurs in such a transformation.

\medskip
\itemitem{a)}  For the following transformation, draw the region
of the $xy$-plane corresponding to $0\leq u,v\leq1$:
$$
x=u+2v \qquad y=-u.
$$

\itemitem{b)} What is the area in the $xy$-plane of a region for which
$$
u_0\leq u\leq u_0+\Delta u 
\qquad\hbox{and}\qquad
v_0\leq v\leq v_0+\Delta v?
$$ 

\itemitem{c)}  Does this depend on where in the plane this takes
place (i.e., does it depend on $u_0$ and $v_0$)?  Guess what
``correction factor'' must be included when converting to an integral
in the $uv$-coordinate system.

\medskip
\itemitem{2.\hskip12pt a)}  Sketch the set of all points $(r,\theta)$
in polar coordinates on the plane that satisfy the inequalities
$$
r_0\leq r\leq r_0+\Delta r
\qquad\hbox{and}\qquad 
\theta_0\leq \theta\leq \theta_0+\Delta \theta.
$$

\itemitem{b)}  Find the area between the circles of radii $r_0$ and
$r_0+\Delta r$. 

\medskip
\itemitem{c)}  What fraction of the area in b) falls between two
rays at angles differing by $\Delta\theta$?

\medskip
\itemitem{d)}  Show that the area of the region in part a) is
precisely
$$
\left(r_0+{\Delta r\over2}\right)\Delta r\Delta\theta.
$$

\itemitem{e)}  Use the result from part d) to justify the correction
factor $r$ that appears when integrating in polar coordinates.

\medskip
\item{3.}  In general, we may wish to convert an integral from
Euclidean $(x,y)$ coordinates to some other coordinate system $(u,v)$.
Note that $x$ and $y$ can be computed in terms of $u$ and $v$, that
is, $x=x(u,v)$ and $y=y(u,v)$.  Then the correction factor which we
must include in the integral for the $(u,v)$-coordinate system is
given by
$$
|J|=
\left|\matrix{{\partial x\over\partial u}&{\partial x\over\partial v}\cr\cr
{\partial y\over\partial u}&{\partial y\over\partial v}\cr}\right|
={\partial x\over\partial u}{\partial y\over\partial v}
-{\partial x\over\partial v}{\partial y\over\partial u}{\atop.}
$$
Here, $J$ is called the {\it Jacobian} matrix.  The vertical bars
indicate to take the {\it determinant} of $J$, which for the
2-dimensional case, is what is written out on the right-hand side
above. 

\medskip
\itemitem{a)}  Compute $|J|$ for the coordinate system in Problem 1.
Was your guess correct?

\medskip
\itemitem{b)}  Compute $|J|$ for the polar coordinate system.

\eject
\itemitem{4.\hskip12pt a)}  Sketch the region in the $xy$-plane
corresponding to $0\leq a,t\leq1$ where $(a,t)$ is the coordinate
system given by
$$
x=4a\cos2\pi t \qquad y=a\sin2\pi t.
$$
(Hint:  For a fixed value of $a$, what object is described by these
equations?)

\medskip
\itemitem{b)}  In what circumstances might these coordinates be useful?

\medskip
\itemitem{c)}  Compute $|J|$ for this coordinate system.

\medskip
\item{5.}  Convert the function $f(x,y)=x$ into $(u,v)$-coordinates
(from Problem 1) and into $(a,t)$-coordinates (from Problem 4).
Compute the integral of this function over the regions you sketched in
Problems 1 and 4 using BOTH the new coordinates and Euclidean
coordinates.

\medskip
\item{6.}  To take advantage of spherical symmetries (which appear
when modelling many physical situations) spherical coordinates are
often used.  Let $(\rho,\theta,\phi)$ be coordinates on three-space as
follows:  
$$
\eqalign{
\rho &\hbox{ - distance from the origin;}\cr
\theta &\hbox{ - polar angle (measured in the $xy$-plane);}\cr
\phi &\hbox{ - zenith angle (measured from the $z$-axis).}\cr}
$$

\centerline{
\epsfysize=1.7in\epsfbox{/home/oehrtman/m210f96/spherical_coords.ps}}

\medskip
\itemitem{a)}  Determine $x$, $y$, and $z$ as functions of $\rho$,
$\theta$, and $\phi$.

\medskip
\itemitem{b)}  Compute the $3\times3$ determinant $|J|$ for spherical
coordinates.

\medskip
\itemitem{c)}  Use your answer to part b) to prove that the volume of
a ball of radius $R$ is $4/3\pi R^3$.

\medskip
\item{7.}  Recal that the area bounded by the polar graph
$r=f(\theta)$ between the rays $\theta=a$ and $\theta=b$ is given by
the definite integral
$$
\int_a^b{1\over2}(f(\theta))^2\,d\theta.
$$
Prove this using the double integral in polar coordinates.

\medskip
\item{}  How did we prove this earlier in the semester?





\bye