Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?


share|cite|improve this question
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. – Bill Cook Mar 19 '12 at 18:13
Thank you! Very helpful. – docjay Mar 19 '12 at 18:15
Yes, that is correct. You only need second order partials to be continuous. See – Robert Israel Mar 19 '12 at 18:15

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.