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 \centerline{\large Math.\ 401, Sec.\
500\hfill    Spring, 2005}
 \bigskip
\centerline{\bfseries\large Homework 13, due April 29}
\bigskip

\begin{enumerate}


\item {\it [Schaum's, p.\ 64, Ex.\ 3.22]\/}\quad
Let $\mathbf{r}$ be any three-dimensional vector, and 
$\{\mathbf i,\mathbf j,\mathbf k\}$ the standard unit vectors 
along the 
coordinate axes.  Prove the following finite-dimensional 
analogues of the theorems about orthonormal bases of functions: 
\begin{enumerate}
\item {\bf (Parseval's equation)} \quad $(\mathbf 
r\cdot\mathbf i)^2 + 
(\mathbf r\cdot\mathbf j)^2 +(\mathbf r\cdot\mathbf k)^2 = 
\mathbf r\cdot \mathbf r$.
 \item {\bf (Bessel's inequality)} \quad
$(\mathbf r\cdot\mathbf i)^2 + 
(\mathbf r\cdot\mathbf j)^2 \le \mathbf r\cdot  \mathbf r$.
\item {\bf (least squares approximation)} \quad
$ \mathbf p \equiv (\mathbf r\cdot \mathbf i)\mathbf i + 
(\mathbf r\cdot \mathbf j)\mathbf j $ is, of all vectors in the 
$x$--$y$ plane, the 
closest to $\mathbf r$; furthermore, $\mathbf p$ is perpendicular 
to 
$\mathbf r-\mathbf p$.
\end{enumerate}

\item {\it [Schaum's, p.\ 64, Ex.\ 3.23]\/}\quad 
Suppose that one term in any orthonormal basis (i.e., a complete 
sequence of orthonormal functions) is omitted.
(For example, leave $\,\sin(10x)$ out of the basis 
functions for the Fourier series on $(-\pi,\pi)$.)
Call the resulting (amputated) sequence $\{\psi_n(x)\}$.

\begin{enumerate}
\item Can we expand an arbitrary function $f(x)$  
as a series $\sum_n c_n \psi_n(x)\,$?
\item Is Parseval's identity always satisfied?
Is it ever satisfied?
\item Is Bessel's inequality always satisfied?
\end{enumerate}
 Justify your answers.



 \item {\it [Logan, p.\ 195, Ex.\ 3.1(b)]\/} \quad
Find the eigenvalues and eigenfunctions for the problem
$$y''+\lambda y=0, \quad 0<x<1,$$
$$y(0)=0, \qquad y(1) -y'(1) =0.$$


 
\item {\it [Schaum's, p.\ 64, Ex.\ 3.43]\/} \quad
Show that the solution of the boundary value problem
$${\partial u \over \partial t} = {\partial ^2u\over \partial 
x^2} 
\qquad (0<x<L, \ 0<t),$$
$${\partial u\over \partial x} (t,0) = h u(t,0) , \qquad
{\partial u\over \partial x}(t,L) = -h u(t,L), \qquad u(0,x) = 
f(x),$$
where $h$ and $L$ are constants, is
$$u(t,x) = \sum_{n=1}^\infty
e^{-\omega_n{}\!^2 t} \,
{\omega_n \cos (\omega_n x) + h \sin (\omega_n x) \over 
(\omega_n{}\!^2 + h^2)L + 2h}\, c_n$$
where
$$c_n = \int_0^L f(x) (\omega_n \cos (\omega_n x) + h \sin 
(\omega_n x)) \,dx$$
and the $\omega_n$ are solutions of
$$\tan (\omega L) = {2h\omega\over \omega^2 - h^2}\,.$$




\item  Let $r$ and $\theta$ be polar coordinates, defined in the 
usual way:
$$x= r \cos \theta, \quad y = r\sin\theta.$$
Show that
$$\nabla^2u \equiv {\partial^2u\over \partial x^2} + 
{\partial ^2u\over \partial y^2} =
{\partial ^2u\over\partial r^2} +\frac1r\,
{\partial  u\over\partial r}
+ \frac1{r^2}\,{\partial ^2u\over \partial \theta^2}\,.$$
{\sl Hint:\/} First show that (acting on any function)
$${\partial \over\partial x} = \cos \theta\, {\partial 
\over\partial r} -{\sin \theta\over r}\, {\partial 
\over\partial \theta }\,, $$
$${ \partial\over\partial y} = \sin \theta\, {\partial \over 
\partial r} +{\cos \theta\over r}\, {\partial\over\partial\theta}
\,.$$
{\sl Hint for hint:\/} The calculations are easier if you use some kind
of implicit differentiation.  For example, a quick way to show that
$${\partial\theta \over \partial  y} = {\cos \theta \over r}$$
is to differentiate the equation
$$\tan \theta = \frac yx$$
with respect to $y$ (with $x$ fixed) and perform some necessary algebra on the
result.  Remember that (for example) a $y$ derivative with $x$ fixed is not
the same as a $y$ derivative with $r$ fixed!


 \item {\it [Logan, p.\ 304, Ex.\ 3.13]\/} \quad
In three dimensions the wave equation is
$${\partial ^2u\over \partial t^2} -c^2 \nabla^2 u =0$$
where $\nabla^2$ is the Laplacian.
For waves with spherical symmetry,
$u=u(r,t)$ and 
$$\nabla^2 u = {\partial ^2u\over \partial r^2} + \frac2r \, 
{\partial u\over \partial r}\,.$$
By introducing the variable $U = ru$,
show that the general solution for the spherically symmetric
wave equation is
$$ u = \frac1r \, f(r-ct) + \frac1r \, g(r+ct).$$




\item  {\it [Logan, p.\ 159,  Ex.\ 1.3]\/}\quad
Determine in which regions each equation is hyperbolic, 
elliptic, or parabolic. 
Subscripts indicate partial differentiation with respect to the 
indicated variables.
 [Read ahead in the notes if necessary 
(the section ``Classification \dots''). {\sl Everybody\/}
should be prepared to answer questions like this on the final 
exam.]

\begin{enumerate}
\item\quad $t\,u_{tt} + u_{xx} =0$
\item\quad $u_{tt} - u_{xx}=0$
\item\quad $u_{tt} + (1+x^2) u_x -u_t = e^t$
\item\quad $u_{xx} + u_{yy} = f(x,y)$
\end{enumerate}

\end{enumerate}
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