Question

Real Mathematical Analysis, Pugh, 5.29 : Prove Corollary18 that r^th-order differentiability implies symmetry of D^rf, r ≥ 3. Use induction to show that (D^rf)_p (v₁,.....,v_r) is symmetric with respect to permutations of v₁,...,v_r−1 and of v₂,...,v_r. Then take advantage of the fact that r is strictly greater than 2. (Please provide a formal proof. Thanks)

Corollary 18: The r^th derivative, if it exists, is symmetric: Permutation of the vectors v₁,...,v_r does not affect the value of (D^rf)_p(v₁,...,v_r). Corresponding mixed higher-order partials are equal.

Answer 1

Proof: Statement: The -th order mixed derivative is symmetric in it's arguments.

Base case: For , we know that:

and hence it's true for .

Induction hypothesis: Let the statement be true for and arbitrary.

Then, for and for some , we have:

by the induction hypothesis.

Also, for some and identifying as during the permutation and vice versa, we have:

again by the induction hypothesis.

Now, note that using just these cases, it's possible to build all possible permutations of , because (it's already true for , so we leave that case out). Thus, the statement is true for .

Since was arbitrary, hence the statement is true for all .