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\centerline{Mathematics 373 -- Applied Mathematics (PDE)}
\centerline{Problem Set 4}
\centerline{March 27, 2001}
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I.  For each of the following vector fields and surfaces, verify
the Divergence Theorem by direct computation of the surface integral
and the triple integral:
\item{A)}  ${\bf F} = (z,y,x)$, $S$ the sphere of radius 1 centered at 
$(0,0,0)$.
\item{B)}  ${\bf F} = (x+y,y+z,x+z)$, $S$ the surface of the tetrahedron
bounded by the coordinate planes and the plane $x + y + z = 1$.  (For
the surface integral, you will need to parametrize each of the four
faces of the tetrahedron separately, and add the corresponding surface
integrals.)
{\it Answer: 1/2 (both values)}.
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\noindent
II.  Recall the definitions of the hyperbolic cosine and sine functions:
$$\cosh x = {e^x + e^{-x}\over 2} \qquad \sinh x = {e^x - e^{-x}\over 2}$$
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\item{A)}  Show that for all $x \in {\bf R}$, $\cosh^2 x - \sinh^2 x = 1$.
Consider the parametric curve $\alpha(t) = (\cosh t, \sinh t)$ in the 
plane.
How does this curve help explain the names {\it hyperbolic cosine\/} and 
{\it sine\/}?
\item{B)}  Express the derivatives of the hyperbolic functions
$\cosh x, \sinh x$, and the functions $\tanh x = {\sinh x\over \cosh x}, 
\coth x = {\cosh x\over \sinh x}, {\rm sech} x = {1\over \cosh x}, 
{\rm csch} x = {1 \over \sinh x}$ analogous to the other trig functions
in terms of the same hyperbolic functions.
\item{C)}  Show that any function $g(x) = Ae^x + Be^{-x}$ can
be written as a linear combination of $\cosh(x)$ and $\sinh(x)$.
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\noindent
II.  Consider the heat equation 
$$u_t = k(u_{xx} + u_{yy})$$
on the rectangle $[0,a]\times [0,b]$ in the $(x,y)$-plane.  (Think
of a thin metal plate in this shape and heat conduction within the 
plate.)  An {\it equilibrium temperature distribution} is a solution
$u(x,y,t)$ of the heat equation that {\it does not depend on $t$} -- 
in other words, a solution that has ``settled down'' and is not
changing any longer.  When $u$ is independent of $t$, the heat equation
reduces to 
$$u_{xx} + u_{yy} = 0\leqno(1)$$
which is called {\it Laplace's Equation}.  In this problem, we will apply
separation of variables and Fourier series techniques to solve a
boundary value problem for Laplace's Equation on the rectangle.
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\item{A)}  Say $u(x,y) = X(x)Y(y)$ solves (1).  What 2 ODE's
are satisfied by $X$ and $Y$?
\item{B)}  A typical problem here would be to solve (1) together
with boundary conditions such as 
$$u_y(x,0) = u_y(x,b) = 0,$$ 
all $x$ (horizontal
edges of the plate insulated), and 
$$u(0,y) = 0 \qquad u(a,y) = g(y)$$ 
for
some given function $g$ (vertical edges held at given temperatures).  
What are the eigenfunctions and eigenvalues for the boundary value
problem you get for $Y(y)$?  
\item{C)}  Show how to develop a Fourier series solution of Laplace's
Equation satisfying the boundary conditions from B.  (Note: you will probably
want to use the hyperbolic functions from problem II here to deal
with the factors $X(x)$ in the product solutions.)
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IV.  Suppose we have a thin metal ring with constant
$c, \rho, K_0$ in the shape of 
$D = \{(r, \theta) : a \le r \le b, 0 \le \theta \le 2\pi\}$
in the polar coordinate plane.  The ring has a radially symmetric 
equilibrium temperature function $u = u(r)$ that satisfies
$${\partial u\over \partial r}(a) = \beta, \quad {\partial u\over \partial r}(b) = 
1.$$
Show that $\beta = b/a$ and interpret this result physically.  
(Note: As for the case of rectangular coordinates, 
${\partial u\over \partial r}(a)$ is proportional to the heat flux
across the inner boundary of the ring.  Similarly at $r = b$.)
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{\it Assignment}
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Individual writeups due in class, Tuesday, April 3.
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