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Real Mathematical Analysis, Pugh, 5.29 : Prove Corollary18 that rth-order di?erentiability implies symmetry of Drf, r ?...

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Real Mathematical Analysis, Pugh, 5.29 : Prove Corollary18 thatrth-order di?erentiability implies symmetry ofDrf, r ? 3. Use induction to show that(Drf)p(v1,.....,vr) is symmetric with respect topermutations of v1,...,vr?1 and ofv2,...,vr. Then take advantage of the factthat r is strictly greater than 2. (Please provide a formal proof.Thanks)

Corollary 18: The rth derivative, if it exists, issymmetric: Permutation of the vectorsv1,...,vr does not a?ect the value of(Drf)p(v1,...,vr).Corresponding mixed higher-order partials are equal.

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Proof Statement The th order mixedderivative is symmetric in its argumentsBase case For we know thatand hence its    See Answer
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