This matrix of second-order partial derivatives of f is called the Hessian matrix of f. The entries in it off the main diagonal are the mixed derivatives; that is, successive partial derivatives with respect to different variables.
In most circumstances the Hessian matrix is symmetric. Mathematical analysis reveals that symmetry requires a hypothesis on f that goes further than simply stating the existence of the second derivatives at a particular point. Clairaut's theorem gives a sufficient condition on f for this to occur.
Read more about this topic: Symmetry Of Second Derivatives
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