Exercise 9.30

Exercise 30: Let $f\in C^{(m)}(E)$, where $E$ is an open subset of $n$. Fix $ǎ\in E$, and suppose $\check{x}\in n$ is so close to $\check{0}$ that the points $\check{p}(t)=ǎ+t\check{x}$ lie in $E$ whenever $0\le t\le 1$. Define

for all $t\in 1$ for which $\check{p}(t)\in E$.

(a) For $1\le k\le m$, show (by repeated application of the chain rule) that

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),

for some $t\in (0,1)$. Use this to prove Taylor’s theorem in $n$ variables showing that the formula

represents $f(ǎ+\check{x})$ as the sum of its so-called “Taylor polynomial of degree $m-1$,” plus a remainder that satisfies

Each of the inner sums extends over all ordered $k$-tuples $(i_1,\dots ,i_k)$, 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 $ǎ$ is $f(ǎ)$.

(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

Prove (by calculating how often each derivative occurs) that the Taylor polynomial in (b) can be written in the form

Here the summation extends over all ordered $n$-tuples $(s_1,\dots ,s_n)$ such that each $s_i$ is a nonnegative integer, and $s_1+\cdots +s_n\le m-1$.