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\centerline{Mathematics 241, section 1 -- Multivariable Calculus}
\centerline{Final Examination -- December 14, 2010}
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{\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 10 Extra Credit points, distributed 
as indicated next to the questions.  You have until 5:30pm 
to work on these problems.  
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\noindent
I.  In this problem,
$P = (1,-1,3),\ Q = (0, 3, 1),\ {\rm and}\ R = (-4,1,2).$
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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 vector
from $P$ to $Q$ and ${\bf w}$ is the vector
from $P$ to $R$, at what angle do ${\bf v},{\bf w}$
meet?
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\noindent
II.
Suppose you follow a flow
line of the vector field $\nabla f$ for $f(x,y)$ in the $xy$-plane.
\item{A)} (5)  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.  
\item{B)}  (5) What does the vector field $\nabla f$ look like
near a local maximum of $f$?  Near a local minimum of $f$?
Make rough sketches by hand to illustrate.
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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 contours of $f(x,y)$ for
the values $c = -1, 0, 1$.
\item{B)}  (10) At the point $(1,2)$, in which
direction is $f$ increasing the fastest? Express your
answer as a unit direction vector.
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\noindent
IV. (20) After an ill-fated ``three hour tour'' goes awry, you 
are stranded on a island at the point with coordinates
$(1,1)$.  Fortunately, you have a radio transmitter with you.  
Unfortunately, it has a limited range -- its signal can only be 
received at distances less than or equal to $2/3$ from its position.  
You know that there is a Coast Guard patrol boat that makes a circuit of 
the path $x^2 + y^2 = 4$ every day, and they always carry
a radio receiver and listen for transmissions.  Will the patrol boat 
ever get within $2/3$ of your position and receive your signal?
(Note: Minimizing the distance from a point $(a,b)$ is 
the same as minimizing the function $f(x,y) = \|(x,y) - (a,b)\|^2.$)
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V.  Let $f(x,y) = y e^{-x^2-2y^2}$.
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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 
to determine the type of each critical point you found in part B.
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VI.  The region $R$ in ${\bf R}^2$ is the set of points
$$R = \{(x,y) \in {\bf R}^2 \mid x^2 + y^2 \le 25, y \ge x, x \ge 0, {\rm\ and\ }
y \le 4\}
$$
and let $f(x,y)$ be some continuous function defined on $R$.  
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\item{A)} (5) Sketch $R$.
\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
VII.  (20)  The metal making up a thin plate
with the shape of the region in
${\bf R}^2$ with $x^2 + y^2 \le 4$ has density $\delta(x,y) = 6 + y$
at all points.  Determine the coordinates of its center of
mass.
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VIII.    Consider the following triple integral:
$$\int_{-3}^3\int_{-\sqrt{9-x^2}}^{\sqrt{9-x^2}}
\int_{-\sqrt{9-x^2-y^2}}^{\sqrt{9-x^2-y^2}} 1 + x\ dz\ dy\ dx$$
\item{A)} (5) Describe the solid over which you are 
integrating here.
\item{B)} (20) Using any convenient coordinate system,
evaluate this integral.
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IX.
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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
$D = \{(x,y) \in {\bf R}^2 : x^2 + y^2 \le 9\}$ 
by computing both sides of the equation in the 
theorem and showing that they are equal.  
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{\it Extra Credit}  
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\noindent
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 $U$.  (Here $f_x, f_y, f_{xx}, f_{yy}$
are the partial derivatives of the harmonic function $f$ with
respect to the indicated variables.)
\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 Second Derivative Test.
\item{B)} (5)  If $f$ is harmonic, what is true about the 
{\it total flux} of the vector field
$${\bf F} (x,y) = \nabla f (x,y) $$
across any simple closed curve in $U$?  Explain.
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\centerline{\it Have a peaceful and joyous holiday season!}
\bye 
