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So, my advanced calculus book (Folland) has this theorem...

If f is of class [tex]C^k[/tex] on an open set S, then...

[tex]\partial i_1 \partial i_2 ... \partial i_k f = \partial j_1 \partial j_2 \partial j_k f[/tex] on an open set S whenever the sequence [tex]{j_1 ,..., j_k}[/tex] is a reordering of the sequence [tex]{i_1 ,..., i_k}[/tex], which defines a smooth function when [tex]k = \infty[/tex]

So my question is, is Clairaut's Theorem a special case of this? (when k = 2?). Also, is the existence of derivatives in the nxn Hessian matrix logically equivalent to the conclusion of Clairaut's Theorem? Does this theorem even have a name? (I can't find it on Wikipedia or Mathworld anywhere).

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# Clairaut's theorem and smooth functions

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