Exercise 9.29

Question

Exercise 29: Let $E$ be an open set in $\R n$. The classes $\mathscr C'(E)$ and $\mathscr C''(E)$ are defined in the text. By induction $\mathscr C^{(k)}(E)$ can be defined as follows, for all
    positive integers $k$: To say that $f\in\mathscr C^{(k)}(E)$ means that the partial derivatives $D_1f,\ldots,D_nf$ belong to $\mathscr C^{(k-1)}(E)$.

Assume $f\in\mathscr C^{(k)}(E)$, and show (by repeated application of Theorem 9.41) that the $k$th-order derivative
    $$
        D_{i_1i_2\cdots i_k}f=D_{i_1}D_{i_2}\cdots D_{i_k}f
    $$
    is unchanged if the subscripts $i_1,\ldots,i_k$ are permuted. For instance, if $n\ge3$, then $D_{1213}f=D_{3112}f$ for every $f\in\mathscr C^{(4)}$.

analambanomenos · Accepted Answer

If we let $g=D_{i_{n+1}\cdots i_k}f$, then by Theorem 9.41 we have 
    \begin{align*}
        D_{i_1\cdots i_k}f &= D_{i_1\cdots i_{n-2}}(D_{i_{n-1}i_n}g) \
        &= D_{i_1\cdots i_{n-2}}(D_{i_ni_{n-1}}g) \
        &= D_{i_1\cdots i_{n-2}i_ni_{n-1}\cdots i_k}f,
    \end{align*}
    that is, we can swap any two adjacent indices and the derivative remains unchanged. We can apply this result to show that we can swap any two indices:
    \begin{align*}
        D_{i_1\cdots i_m\cdots i_n\cdots i_k}f &= D_{i_1\cdots i_m\cdots i_ni_{n-1}i_{n+1}\cdots i_k}f \
        &= \hbox{(keep swapping adjacent indices until $i_n$ comes before $i_m$)} \
        &= D_{i_1\cdots i_ni_m\cdots i_{n-1}i_{n+1}\cdots i_k}f \
        &= D_{i_1\cdots i_ni_{m+1}i_m\cdots i_{n-1}i_{n+1}\cdots i_k}f \
        &= \hbox{(keep swapping adjacent indices until $i_m$ comes after $i_{n-1}$)} \
        &= D_{i_1\cdots i_{m-1}i_ni_{m+1}\cdots i_{n-1}i_mi_{n+1}\cdots i_k}f
    \end{align*}
    Since any permutation is the result of pairwise swaps (this is usually shown in elementary abstract algebra courses when discussing the permutation group), we see that in the case of $f\in
    \mathscr C^{(k)}$, we can permute the order of partial differentiation without changing the derivative.

Exercise 9.29

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Exercise 9.29

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