\magnification=\magstep1
\centerline{Mathematics 173 -- Applied Mathematics 1}
\centerline{Review Sheet for Midterm 1}
\centerline{September 29, 1999}
\bigskip
\noindent
{\it General Information}
\bigskip
As announced in the course syllabus, the first midterm exam will
be a closed book exam
given in class on Thursday, October 7.  The exam will take the
entire class period that day.  The exam will have three or four
questions similar to questions from the problem sets.
\bigskip
\noindent
{\it Topics to Be Covered}
\bigskip
The exam will cover the material from class, starting from the 
beginning of the semester through class on Thursday, September 30.
This includes Chapter 1 and Chapter 2 sections 1,2,3 in the text, plus
the extra material on vector calculus -- see below.  Specifically,
this means (not in the order we talked about them in class):
\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.  They will not be given.  Also be aware that 
it may ``pay'' to use polar, cylindrical, or spherical coordinates to 
evaluate some of the double and triple integrals that come up.)
\item{2)}  The 1D, 2D, and 3D heat equations.  
Know the derivation of these equations from the conservation of energy 
principle as we saw in class for the 1D case and on Discussion 2 for 
the 2D and 3D cases. (Note: The 2D and 3D cases use a number of ideas from 
1 above). Know how to verify that given functions are solutions.
\item{3)}  Boundary conditions -- know how to express various
boundary conditions for the 1D heat equation such as
\itemitem{a)} ``the ends of the rod are maintained at constant temperature''
\itemitem{b)} ``the ends of the rod are insulated''
\itemitem{c)} ``the heat flux at the ends equals ... ''
\item{} as equations.
\item{4)}  Equilibrium solutions of boundary value problems 
for the heat equation (know how to derive them, and be able to interpret 
what they mean)
\item{5)}  Laplace's Equation for equilibrium solutions of the
2D heat equation.  Know the form of the Laplacian in rectangular
and polar coordinates and how to use them.  
(But I will not ask you for the derivation of the polar coordinate form.)
\item{6)} Separation of variables for the 1D Heat Equation
(ends maintained at zero temperature), solutions
of the ordinary differential equation
${d^2 \phi \over dx^2} = -\lambda \phi$.  Superposition of solutions.
\vfill\eject
\noindent
{\it Review Session}
\bigskip
I would be happy to hold an evening review session before the 
exam next week.  Is Tuesday evening good?
\bigskip
\noindent
{\it Some Review Problems}
\bigskip
\noindent
I.  Let ${\bf F} = (x^2 - y^2, xy)$.  Let $D$ be the region in the 
plane bounded by the line $y = x$ and the parabola $y = x^2$, 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.
\bigskip
\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 $\alpha$.)
\bigskip
\noindent
III.  Let ${\bf F} = (y^2, x + y^2 + z, x - z)$, let $\Phi$
be the surface formed by the upper 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.
\bigskip
\noindent
IV.  (``Convection'') Assume we have a long thin wire or rod with constant 
thermal properties ($c,\rho,K_0$ all constant).  Assume the wire is
{\it not insulated along its lateral surface\/}, so that the rate of change 
(per unit time) of heat energy per unit surface area
along the lateral surface is
$C(u(x,t) - T)$, where $T$ is the temperature of the medium 
surrounding the wire, and $C$ is a positive constant.  
\item{A)} Show that
the heat equation in this situation takes the form
$$c\rho u_t = K_0 u_{xx} + Q - {C P \over A}(u - T)$$
($A$ is the cross-section area of the wire; $P$ is the perimeter of a
cross-section.)   Hint: Proceed as in our derivation of the 1D heat 
equation from the first day of class.
\item{B)} Suppose
the surrounding temperature is $T = 0$, there are heat sources in the 
wire, and that the temperature in the rod is uniform ($u(x,t) = u(t)$,
independent of $x$.)  Determine $u(t)$, if $u(0) = u_0$ is given.
\bigskip
\noindent
From the text: 1.4.1, 1.4.2, 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!), 1.5.12, 1.5.13, 1.5.14 (Extra Credit problem), 2.3.2
\bye
