A function $f$ that has continuous third order partial derivatives in $\mathbb{R}^n$. I'm just wondering that since the partial derivatives are continuous then the Hessian matrix is symmetric. Is that correct?
Thanks.
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- $\begingroup$ Yes. If the third order partials are continuous, then the second order partials are and so Clairaut's theorem applies -- mixed partials are equal and thus the Hessian is symmetric. $\endgroup$Bill Cook– Bill Cook2012-03-19 18:13:36 +00:00Commented Mar 19, 2012 at 18:13
- $\begingroup$ Yes, that is correct. You only need second order partials to be continuous. See en.wikipedia.org/wiki/Symmetry_of_second_derivatives $\endgroup$Robert Israel– Robert Israel2012-03-19 18:15:43 +00:00Commented Mar 19, 2012 at 18:15
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1 Answer
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The mere existence of third order partial derivatives implies that the second order derivatives are continuous. As noted by Robert Israel in comments, the continuity of second order partial derivatives is a sufficient condition for the symmetry of Hessian.