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\centerline{Mathematics 241 -- Multivariable Calculus}
\centerline{Final Examination -- December 11, 2007}
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\noindent
{\it Directions}
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Do all work in the blue exam booklet.  Do not place anything
you wish to have considered for credit on the exam sheet.
There are 200 regular and 20 extra credit points, distributed as indicated next
to the questions.
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\noindent
{\it Notation:}  In the following, for simplicity, the
subscript notation is used for partial derivatives:
$f_x = {\partial f\over \partial x}, f_y = {\partial f\over \partial y},
f_{xx} = {\partial^2 f\over\partial x^2},$ and so forth.
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\noindent
I.  In this problem,
$P = (1,0,1),\ Q = (-2, 3, 2),\ {\rm and}\ R = (1,2,0).$
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\item{A)} (10)  Find the equation of the plane
containing the points $P,Q,R$ in ${\bf R}^3$.
\item{B)} (10)  At what point does the line containing
$P,Q$ meet the $xy$-plane?
\item{C)}  (5)  If ${\bf v}$ is the displacement vector
from $P$ to $Q$ and ${\bf w}$ is the displacement vector
from $P$ to $R$, at what angle do ${\bf v},{\bf w}$
meet?
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\noindent
II.
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\item{A)} (7.5) Let $F: {\bf R}^2_{u,v} \to {\bf R}^2_{x,y}$
be $F(u,v) = (x(u,v),y(u,v)) = (u^2 - v^2, 2uv)$
and let $g: {\bf R}^2_{x,y} \to {\bf R}$ be
$g(x,y) = \sin(x)\cos(y).$  Find the derivative matrix
$D(g\circ F)$ by direct substitution and differentiation.
\item{B)} (7.5) Compute $D(g\circ F)$ for
the functions in part A using the Chain Rule
and show you get the same result as in part A.
\item{C)} (10)  Now let $z = g(x,y)$ for a {\it general
function $g$ -- not specifically $g(x,y)$ as above in part A},
and $x = u^2 - v^2$,
$y = 2uv$ as in the definition of $F$ above. Show
that
$${1\over 4(u^2+v^2)}\left((z_u)^2 + (z_v)^2\right)
= (z_x)^2 + (z_y)^2.$$

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III.  All parts of this problem refer to $f(x,y) = (x+1)^2 + y^2$.
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\item{A)} (5) Sketch the level curves of $f(x,y)$ for
the values $c = 1, 4, 9$.
\item{B)}  (10) At the point $(1,2)$, in which
direction is $f$ increasing the fastest? Express your
answer as a unit direction vector.
\item{C)}  (15)  Find the points on the curve
$g(x,y) = {x^2\over 4} + y^2 = 1$
at which $f(x,y)$ takes its largest and smallest
values.  What is true about the vectors
$\nabla f$ and $\nabla g$ at your points?
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\noindent
IV.  Let $f(x,y) = x e^{-2x^2 - y^2}$.
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\item{A)} (10) Find the equation of the tangent plane
to the graph $z = f(x,y)$ at the point $(1,1,e^{-3})$.
\item{B)} (10) Find all the critical points of
$f(x,y)$.
\item{C)}  (20) Use the second derivative test (Hessian
criterion) to determine the type of each critical point
you found in part B.
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\noindent
V.  A region $R$ in ${\bf R}^2$ is the set of points
satisfying $x^2 + y^2 \ge 1$, $y \ge x$, $x \ge 0$, and
$y \le 4$.
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\item{A)} (5) Sketch the region $R$.
\item{B)}  (10) Set up the limits of integration
of iterated integral(s) to
compute $\int\!\int_R f(x,y)\ dA$ integrating with
respect to $x$ first, then $y$.
\item{C)}  (10) Now reverse the order of the
variables and set up iterated integral(s) to compute
the same integral, but integrating with respect to $y$ first, then
$x$.
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\noindent
VI.  (20)  The metal making up a solid slug
having the shape of the region in
${\bf R}^3$ with $x^2 + y^2 \le 4$ and
$-1\le z \le 1$ has density $\delta(x,y,z) = 5 + x$
at all points.  Determine the coordinates of its center of
mass.
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\noindent
VII.
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\item{A)} (10)  State Green's Theorem.
\item{B)} (10) Let ${\bf F}(x,y) = (x - y^2, x^2 + y)$.
Verify that Green's Theorem holds for the region
$D = \{(x,y) \in {\bf R}^2 : x^2 + y^2 \le 9\}$.
\item{C)} (5) We can think of a vector field on ${\bf R}^2$ 
as a vector field on ${\bf R}^3$ by
making the last component equal to {\it zero} at all points.
Compute ${\rm curl}({\bf F})$ for ${\bf F} = (M(x,y),N(x,y),0)$ 
and relate your result to
the statement of Green's Theorem.
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\noindent
VIII.  A function $f(x,y)$ is said to be {\it harmonic}
on an open set $U$ in ${\bf R}^2$ if it satisfies the equation
$$f_{xx} + f_{yy} = 0$$
at all points in $U$.
\item{A)} (5)  How does a nondegenerate
critical point of a harmonic function
fit into our classification?  Is it a local maximum,
local minimum, or a saddle point?  Explain how you can
tell from the Hessian criterion.
\item{B)} (5)  If $f$ is harmonic, what is true about the line
integral of the vector field
$${\bf F}(x,y) =
\left(-f_y, f_x\right)$$
around any simple closed curve in $U$?
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\noindent
{\bf Extra Credit}  (20)
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Suppose you follow a flow
line of the vector field $\nabla f$ for $f(x,y)$ in the $xy$-plane.
As you traverse the flow line in the increasing
$t$-direction, is the corresponding
path on the graph $z = f(x,y)$ going uphill or downhill?
Explain.  What does the vector field $\nabla f$ look like
near a local maxmimum of $f$?  near a local minimum of $f$?
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\centerline{\it Have a peaceful and joyous holiday season!}
\bye 
