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\centerline{Mathematics 241 -- Multivariable Calculus}
\centerline{Final Examination -- December 14, 2013}
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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 10 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)}  (10) 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.  All parts of this problem refer to the parametric curve
$$\alpha(t) = \left({\cos(t)\over \sin^2(t) + 1},{\cos(t)\sin(t)\over 1 + \sin^2(t)}\right)$$
defined for all $t \in [0,2\pi]$, called a {\it lemniscate}.
\item{A)}  (5) Is $\alpha(t)$ a simple closed curve?  (Hint: 
Thinking of $\alpha(t)$ as the position of a moving
object as a function of time, are there different times $t \in [0,2\pi)$ 
at which the object is at the location $(x,y) = (0,0)$?)
\item{B)} (10) What is the tangent vector to the curve at $t = \pi$?
\item{C)} (5) A thin wire has the shape of the portion of the 
curve $\alpha$ for $t \in [0,1]$. 
What integral would you use to compute its arc length?  (You do not need
to evaluate!)
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\noindent
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.  (If you have difficulty, partial 
credit will be given for a correct statement of the second 
derivative test even if you cannot see how to apply it.)
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\noindent
V.  A region $R$ in ${\bf R}^2$ is the set of points
$$R = \{(x,y) \mid x^2 + y^2 \ge 1, y \ge x, x \ge 0,\ {\rm and}\ 
y \le 4.\}$$
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\item{A)} (10) 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 
Note: In parts B and C, you are just to set up integrals
with a ``dummy function'' $f(x,y)$, there is no evaluation 
of integrals involved.
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\noindent
VI.  (20)  The metal making up a solid half-cone in the shape of
$$H = \{(x,y,z) \in {\bf R}^3 \mid z^2 \ge x^2 + y^2, 0 \le z \le 1, y \ge 0\}$$
has density $\delta(x,y,z) = \sqrt{x^2 + y^2 + z^2}$
at all points.  Determine its total mass.  
(Hint: a wise choice of coordinate system will simplify
this one immensely!)
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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
$D = \{(x,y) \in {\bf R}^2 : x^2 + y^2 \le 9\}$.
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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 second derivative test.
\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}  (10)
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Suppose you follow a flow
line of the vector field $-\nabla f$ for a differentiable function 
$f(x,y)$ 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 
