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\centerline{Mathematics 241 -- Multivariable Calculus}
\centerline{Exam 2}
\centerline{October 26, 2007}
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\noindent
{\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.  Extra Credit points will be awarded as on the 
first exam -- the Extra Credit problem will be counted only if it raises your
average.
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\noindent
I.  Let $F(x,y) = (x - y^2, xy)$ and let $g(t) = (\sin(t), \cos(t) + \sin(t))$
\item{A)} (10)  Compute the derivative matrices $D(F)$ and $D(g)$.  
\item{B)} (10)  Use the Chain Rule to compute $D(F \circ g)(\pi)$.
\item{C)} (10)  Compute the composition $F \circ g$ and find its 
derivative matrix directly to check your answer from part B. 
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\noindent
II. (20) Let $z = f(x + y, x - y)$ have continuous partial derivatives
with respect to $u = x + y$ and $v = x - y$.  Show that 
$${\partial z\over \partial x} {\partial z\over \partial y} = 
\left({\partial z\over \partial u}\right)^2 - \left({\partial z\over\partial v}\right)^2.$$ 

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\noindent
III.  A mountain has the shape of the graph 
$$z = \sqrt{4 - x^2 - {y^2\over 4}},$$ 
where $x,y$ are east-west and north-south map coordinates respectively, and $z$
is the altitude.
\item{A)} (10) If you are standing at the point on the mountain
with $(x,y,z) = (1,2,\sqrt{2})$, which direction should you move (on the 
map) to increase your altitude the fastest?
\item{B)} (10) At the same point as in part A, in which direction should 
you move (on the map) to remain at a constant altitude?

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\noindent
IV.  Let $x(t) = (t^2/2, t, t)$ and $y(t) = (t\cos(t), t\sin(t), t)$ 
be two curves in ${\bf R}^3$.  
\item{A)}  (10) Does the tangent line to $x(t)$
at $t = 2$ cross the $xy$-plane?  If so, where does this occur?  
\item{B)}  (10) Show that $y(t)$ (for $t > 0$) is a flow line for the 
vector field
$$F(x,y,z) = \left(-y + {x \over z}, x + {y \over z}, 1\right).$$
\item{C)}  (10) Show that the segments of $x(t)$ and $y(t)$
for $a \le t \le b$ have the {\it same arclength} for all $a,b$.  ({\it Hint}:
Think about how the arclength would be computed, but it's not necessary to 
actually do the computation.)



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\noindent
{\it Extra Credit}  (10) What is the {\it curvature function} for the curve
$x(t)$ from problem IV above?
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