Let \(f \in \mathscr{f}^{(m)}(E)\), where \(E\) is an open subset of \(R^{n}\). Fix
a \(\in E\), and suppose \(x \in R^{n}\) is so close to 0 that the points
$$
\mathbf{p}(t)=\mathbf{a}+\boldsymbol{r} \mathbf{x}
$$
lie in \(E\) whenever \(0 \leq t \leq 1\). Define
$$
h(t)=f(\mathrm{p}(t))
$$
for all \(t \in R^{1}\) for which \(\mathrm{p}(t) \in E\).
(a) For \(1 \leq k \leq m\), show (by repeated application of the chain rule)
that
$$
h^{(k)}(t)=\sum\left(D_{i_{1} \ldots i_{k}} f\right)(\mathrm{p}(t)) x_{i_{1}
\ldots} x_{i_{k}} .
$$
The sum extends over all ordered \(k\) -tuples \(\left(i_{1}, \ldots,
i_{k}\right)\) in which each \(i\), is one of the integers \(1, \ldots, n\).
(b) By Taylor's theorem (5.15),
$$
h(1)=\sum_{k=0}^{m-1} \frac{h^{(k)}(0)}{k !}+\frac{h^{(m)}(t)}{m !}
$$
for some \(t \in(0,1) .\) Use this to prove Taylor's theorem in \(n\) variables by
showing that the formula
$$
f(\mathbf{a}+\mathbf{x})=\sum_{k=0}^{m-1} \frac{1}{k !} \sum\left(D_{i 1}
\ldots, f\right)(\mathbf{a}) x_{i_{1}} \ldots x_{1,}+r(\mathbf{x})
$$
represents \(f(\mathrm{a}+\mathbf{x})\) as the sum of its so-called "Taylor
polynomial of degree \(m-1, "\) plus a remainder that satisfies
$$
\lim _{x \rightarrow 0} \frac{r(\mathbf{x})}{|x|=-1}=0
$$
Each of the inner sums extends over all ordered \(k\) -tuples \(\left(i_{1},
\ldots, i_{k}\right)\), 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 a is \(f(\mathbf{a})\).
(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 corresponding three terms can be
written in the form
$$
3\left(D_{1}^{2} D_{3} f\right)(a) x_{1}^{2} x_{3}
$$
Prove (by calculating how often each derivative occurs) that the Taylor
polynomial in \((b)\) can be written in the form
$$
\sum \frac{\left(D_{1}^{21} \cdots D_{n}^{s n} f\right)(a)}{s_{1} ! \cdots
s_{n} !} x_{1}^{21} \cdots x_{n}^{s_{n}}
$$
Here the summation extends over all ordered \(n\) -tuples \(\left(s_{1}, \ldots,
s_{n}\right)\) such that each \(s_{1}\) is a nonnegative integer, and
\(s_{1}+\cdots+s_{n} \leq m-1\).
