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I was wondering if anyone knows of an extension of Clairaut's theorem for interchanging the order of partial differentiation. For example, just recently I noticed that for a lot of functions

$$f_{xyy} = f_{yyx}$$.

Tdonut
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1 Answers1

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I would not really call that an extension. If the first order derivatives of $f$ also satisfy the conditions of Clairaut's theorem, then by two consecutive applications of that theorem

$$f_{xyy}=(f_{xy})_y=(f_{yx})_y=(f_y)_{xy}=(f_y)_{yx}=f_{yyx}$$

Justpassingby
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  • But is there an extension of Clairaut's theorem? That is my question. – Tdonut Nov 28 '15 at 03:02
  • My point was: such an extension can be formulated but the proof is so obvious that nobody bothers to give it a special name other than "repeated application of Clairaut's theorem". It's like commutativity in groups: the definition mentions exchanging the order of only 2 group elements but it is easy to conclude that any number of elements can change places. – Justpassingby Nov 28 '15 at 09:56