Question

Taylor Polynomials Let $f \in \mathscr{C}^{(m)}(E),$ where $E$ is an open subset of $R^{n}$. Fix \textbf{a} $\in E$, and suppose \textbf{x} $\in R^{n}$ is so close to \textbf{0} that the points
\begin{equation*}
\textbf{p}(t)=\textbf{a}+t\textbf{x}
\end{equation*}
lie in $E$ whenever $0 \leq t \leq 1$. Define
\begin{equation*}
h(t)=f(\textbf{p}(t))
\end{equation*}
for all t $\in R^{1}$ for which \textbf{p}(t) $\in E$.
(a) For $1 \leq k \leq m$, show (by repeated application of the chain rule) that
\begin{equation*}
h^{(k)}(t)=\sum(D_{i_{1} \dots i_{k}}f)(\textbf{p}(t))x_{i_{1} \dots

Answer 1

Let $f \in \mathscr{C}^{(m)}(E),$ where $E$ is an open subset of $R^{n}$. Fix \textbf{a} $\in E$, and suppose \textbf{x} $\in R^{n}$ is so close to \textbf{0} that the points
\begin{equation*}
\textbf{p}(t)=\textbf{a}+t\textbf{x}
\end{equation*}
lie in $E$ whenever $0 \leq t \leq 1$. Define
\begin{equation*}
h(t)=f(\textbf{p}(t))
\end{equation*}
for all t $\in R^{1}$ for which \textbf{p}(t) $\in E$.
(a) For $1 \leq k \leq m$, show (by repeated application of the chain rule) that
\begin{equation*}
h^{(k)}(t)=\sum(D_{i_{1} \dots i_{k}}f)(\textbf{p}(t))x_{i_{1} \dots}x_{i_{k}}.
\end{equation*}
The sum extends over all ordered $k$-tuples $(i_{1, \dots ,}i_{k}$ in which each $i_{j}$ is one of the integers $1,\dots , n$.\\
($b$) By Taylor's theorem (5.15),
\begin{equation*}
h(1)=\sum_{k=0}^{m-1}\frac{h^{(k)}(0)}{k!}+\frac{h^{(m)}(t)}{m!}
\end{equation*}
for some $t \in (0,1)$. Use this to prove Taylor's theorem in $n$ variables by showing that the formula
\begin{equation*}
f(\textbf{a}+\textbf{x})=\sum_{k=0}^{m-1}\frac{1}{k!}\sum(D_{i_{1}...i_{k}}f)(\textbf{a})x_{i_{1}...}x_{i_{k}}+r(\textbf{x})
\end{equation*}
represents $f$(\textbf{a}+\textbf{x}) as the sum of its so-called ``Taylor polynomial of degree $m-1$", plus a remainder that satisfies
\begin{equation*}
\lim_{x \rightarrow 0}\frac{r(\textbf{x})}{|\textbf{x}|^{m-1}}=0.
\end{equation*}
Each of the inner sums extends over all ordered $k$-tuples $(i_{i,...,}i_{k})$, as in part $(a)$; as usual, the zero-order derivative of $f$ is simply $f$, so that the constant term of the Taylor polynomial of $f$ at \textbf{a} is $f$(\textbf{a}).\\
$(c)$ Exercise 29 shows that repetition occurs in the Taylor polynomial as written in part $(b)$. For instance, $D_{113}$ occurs three times, as $D_{113}$, $D_{131}$, $D_{311}$. The sum of the correspondin gthree terms can be written in the form
\begin{equation*}
3(D_{1}^{2}D_{3}f)(\textbf{a})x_{1}^{2}x_{3}.
\end{equation*}
Prove (by calculating how often each derivative occurs) that the Taylor polynomial in $(b)$ can be written in the form
\begin{equation*}
\sum\frac{(D_{1}^{s_{1}} \dots D_{n}^{s_{n}}f)(\textbf{a})}{s_{1}! \dots s_{n}!}x_{1}^{s_{1}} \dots x_{n}^{s_{n}}
\end{equation*}
Here the summation extends over all ordered $n$-tuples $(s_{1,...,}s_{n})$ such that each $s_{i}$ is a nonnegative integer, and $s_{1}+ \dots + s_{n} \leq m-1$.