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\centerline{Mathematics 241 -- Multivariable Calculus}
\centerline{Exam 3}
\centerline{November 30, 2007}
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{\it Information and Directions}
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Do all work in the blue exam booklet.  Do not place anything you want
considered for credit on this sheet.  There are 100 possible points, plus
the Extra Credit problem.  The Extra Credit problem will be counted only if it raises your
average.
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I.  
\item{A)} (15) Let ${\bf F}$ be the vector field
$${\bf F}(x,y,z) = (xz - y^2, xy + z, 2x + \sin(y)).$$
Compute the divergence ${\rm div}({\bf F})$ and the curl ${\rm curl}({\bf F})$.
\item{B)} (5) Show that if ${\bf F} = (F_1,F_2,F_3)$ is 
any vector field on a domain in ${\bf R}^3$ whose
component functions have continuous second-order partial derivatives,
then  ${\rm div}({\rm curl}({\bf F})) = 0$.  
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II.  All parts of this problem refer to the function 
$$f(x,y) = x^3 + y^2 - 6xy + 6x + 3y.$$
\item{A)} (10) Determine all critical points of $f$.
\item{B)} (15) Use the Hessian Criterion (second derivative test)
to classify each of the critical points as a local maximum, a local
minimum, a saddle point, or a degenerate critical point.  
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III.  (20) A mountain has the shape of the graph 
$z = f(x,y) = 4 - x^2 - {y^2\over 4},$
where $x,y$ are east-west and north-south map coordinates respectively, 
and $z$ is the altitude (in thousands of feet).   
If you hike along a trail on the mountain described by the circle
$g(x,y) = x^2 + y^2 = 1$ on the map, what are the highest and lowest altitudes
that you reach?
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IV.  All parts of this problem refer to the region $D$ in the right half-plane ($x \ge 0$),
bounded by $y = x$, $y = -x$ and $x^2 + y^2 = 9$.
\item{A)} (5) Sketch the region $D$.
\item{B)} (10) What would the limits of integration be 
to compute $\int\!\int_D f(x,y)\ dx\ dy$ integrating with respect to $y$ first, then $x$?
\item{C)} (10) What would the limits of integration be 
to compute $\int\!\int_D f(x,y)\ dx\ dy$ integrating with respect to $x$ first, then $y$?
\item{D)} (10) Using any applicable method, compute  $\int\!\int_D x\ dx\ dy.$


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{\it Extra Credit}  (10) Compute the triple integral of the function 
$f(x,y,z) = e^{(x^2 + y^2 + z^2)^{3/2}}$ over the unit ball
$x^2 + y^2 + z^2 \le 1$ in ${\bf R}^3$.  (A good choice of coordinates
is essential here!)
\bye
