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\begin{document}

\title{Partial Differential Equations}
\author{Homework 3}
\date{due October 7, 2002}
\maketitle

\begin{enumerate}
\item Prove that the Taylor series with integral remainder for a
function $u \in C^{N}( \mathbb{R}^n)$ is
\[
  u(x) = \sum_{|\alpha| \leq N-1}
         \frac{D^\alpha u(x_0)}{\alpha!} \, (x-x_0)^\alpha
         + R_N ,
\]
where
\[
  R_N = N \sum_{|\alpha|=N} \frac{(x-x_0)^\alpha}{\alpha!}
        \int_0^1 (1-t)^{N-1} \, 
        D^\alpha u(x_0 + t(x-x_0)) \, dt .
\]
Hint: Apply the one-dimensional Taylor formula with integral remainder
to the function $f(s) = u(x_0 + s(x-x_0))$.



\item Evans, p. 86/87 problems 5 and 6, 8.

\end{enumerate}


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