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\centerline{Mathematics 173 -- Applied Mathematics 1}
\centerline{Review Sheet for Final Exam}
\centerline{November 30, 1999}
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\noindent
{\it General Information}
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The final exam for this course will be given at 8:30 am on 
Friday, December 17 (not quite the last possible time!)  The
exam will be comprehensive, covering all the material we have
discussed this semester.  I will write the exam so that you
should be able to complete it in 2 hours if you work steadily.
However, you will have the full 3 hour period if you need more 
time.  Because there are a relatively large number of different
PDE, integral formulas, etc. you will need to apply, this will be an 
{\it open book, open class notes} exam.  I will also 
supply a table of integrals for you to use.  Be sure to prepare carefully, 
though, because {\it you will not have enough time\/} if you need to start 
every problem by looking up the topic in the book and/or notes.

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\noindent
{\it Topics to Be Covered}
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\noindent
These are:
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\item{1)} Vector Calculus -- line integrals, Green's Theorem, and the 
``alternate form'', surface integrals, the Divergence Theorem.  Know the 
precise statements of the two theorems and how to compute both sides in the 
equalities.  (You should know the standard parametrizations for curves like
lines, graphs ($y = f(x), x = g(y)$), circles, and ellipses, and surfaces 
like planes and spheres.   Also be prepared
to use polar, cylindrical, or spherical coordinates to 
evaluate some of the double and triple integrals that come up.)
\item{2)}  PDE in general. 
Know how to verify that given functions are solutions.
\item{3)} The 1D, 2D, and 3D heat equations. 
Equilibrium solutions for heat equations
(know how to derive them, and be able to interpret  what they mean).
Laplace's Equation for equilibrium solutions of the
2D heat equation.  Know the forms of the Laplacian in rectangular
and polar coordinates and how to apply them.  
\item{4)}  Boundary value problems for the 1D Heat Equation -- 
know how to express boundary conditions such as ``the ends of the rod 
are maintained at zero temperature'', ``the ends of the rod are insulated'', 
``the heat flux at the ends is ... '' as equations.
\item{5)} Separation of variables for general homogenous linear PDE, 
solutions of the ordinary differential equation
${d^2 \phi \over dx^2} = -\lambda \phi$, superposition of solutions.
\item{6)} Fourier Series, the Pointwise Convergence Theorem and
its consequences, Fourier Sine
and Cosine series on $[0,L]$, and their relation with odd and even
periodic extensions. 
\item{7)}  The 1D and 2D Wave Equation, boundary value problems,
normal modes of vibration for strings and rectangular membranes with 
zero boundary conditions on the vertical displacement function.
Fourier series solutions.
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\noindent
{\it Review Session}
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I would be happy to hold an evening review session before the 
exam.
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\noindent
{\it Some Review Problems}
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\noindent
I.  Let ${\bf F} = (xy - y, xy^2)$.  Let $D$ be the region in the 
plane bounded by the line $y = -x$ and the parabola $y = 1 - x^2$, and 
let $\alpha$ be the positively oriented boundary of $D$.  Show 
that the conclusion of Green's Theorem is valid by computing both
sides of the equation.
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\noindent
II. Show using the ``alternate form'' of Green's Theorem that
if $f,g$ are any two functions with continuous second order 
partial derivatives on the region $D$ bounded by a simple closed
curve $\alpha$, then
$$\int_\alpha f \nabla g \cdot {\bf N}\ ds = 
\int\!\!\!\int_D f {\rm div}(\nabla g) + \nabla f \cdot \nabla g\ dA$$
(Here ${\bf N}$ is the outward unit normal to $\alpha$.)
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\noindent
III.  Let ${\bf F} = (y^2, x + 4 y + z, x^2 - z)$, let $\Phi$
be the surface formed by the lower hemisphere of the sphere of 
radius 1 centered at (0,0,0), and the disk in the $xy$-plane inside
$x^2 + y^2 = 1$.  Say why the Divergence Theorem applies in this
case, and compute both sides in the equality, showing they are the same.

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\noindent
Consult the review sheet for the first midterm and the calculus
textbooks on reserve in the Science Library for 
additional problems of these kinds.
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\noindent
From the text: 
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\item{} 1.4.1, 1.4.6, 1.4.7, 1.5.9 (Assume constant
thermal properties, assume the temperature
is radially symmetric, and that there are no heat sources. Use 
polar coordinates!)
\item{} 1.5.12, 1.5.13,
\item{} 2.3.1-8,
\item{} 2.4.1,
\item{} 2.5.1,
\item{} 3.2.1-4,
\item{} 3.3.1-3,3.3.18,
\item{} 4.4.1,4.4.7,4.4.8
\bye
