%exam3.tex Annie 4-17-00
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\centerline{Exam III\hfil MA 174\hfil 4/21/00\hfil D. Gottlieb\hfil}
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\item{\quad 1a.}Suppose it is $2$ miles from town $A$ to town $B$ along a 
certain road.  If you drive from $A$ to $B$ without stopping or turning around,
what is the difference on your odometer between the mileage at town $A$ and 
the mileage at town $B$?
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\item{\quad b.}The above simple question illustrates what equation 
expressing length as a line integral
of a curve in terms of a parameter $t$?
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\item{\quad 2.}Write down the vector field $\vec{F}$ for the vector 
field such that $\vec{F}(P)$ points towards the origin and is of unit 
length for any point $P$ in $\Bbb R^3$.  Use the $\vec{i}, 
\vec{j}, \vec{k}$ vectors.
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\item{\quad 3.}Find a potential function for the vector field $\vec{F}
(x, y, z)= z \vec{k}$.
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\item{\quad 4.}Find the circulation around the ellipse $x^2 + 
\ds{z^2 \over 2} =3$ of the vector field $\vec{F}$ in Problem 3.
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\item{\quad 5.}Suppose an umbrella's shadow is the unit circle $x^2 + y^2 =
1$ when the sun is directly overhead on the $z$-axis.  The umbrella can be
described by the equation $f(x,y, z)=0$ for some $f$ whose gradient is given
by $x \vec{i} + y^2 \vec{j} + z^3 \vec{k}$.  Set up an 
integral giving the surface area of the umbrella. (You may leave $z$ in your
answer, since the information given does not completely describe $z$ as a 
function of $x$ and $y$.)
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\item{\quad 6.}State Stokes's theorem with all its hypotheses.
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\item{\quad 7.}Find the curl of the vector field in Problem 3.
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\item{\quad 8.}Is a sphere $(x^2 + y^2 + z^2 = r^2)$ simply connected?
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\item{\quad 9.}  Let 
$$\vec{F} = \ds{-y \over{x^2 + y^2}} \ \ \vec{i}\ \ +\ \  \ds{x \over{x^2  + y^2}} \ \ \vec{j} \ \ +\ \  z \ \ \vec{k}$$
Now the curl of $\vec{F}$ is zero, and the circulation of $\vec{F}$ is not
zero around the unit circle in the $xy$ plane.
Why can't you apply Stokes's Theorem to calculate the circulation on the 
unit circle? (If you are unsure of the reason, then show the curl of $\vec{F}$
is in fact zero and calculate the circulation of $\vec{F}$ around the unit circle in the $xy$ plane.)
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\item{\quad 10.}You want to use Stokes's theorem to  calculate the
circulation on the boundary of the surface $S$  shown below.  
Can you choose a unit normal
$\vec{n}$ to use in the surface integral of Stokes's theorem integrated over
$S$?  If so, what is it?

\bye

