Symmetry of second derivatives

In mathematics, the symmetry of second derivatives (also called the equality of mixed partials) is the fact that exchanging the order of partial derivatives of a multivariate function

${\displaystyle f\left(x_{1},\,x_{2},\,\ldots ,\,x_{n}\right)}$

does not change the result if some continuity conditions are satisfied (see below); that is, the second-order partial derivatives satisfy the identities

${\displaystyle {\frac {\partial }{\partial x_{i}}}\left({\frac {\partial f}{\partial x_{j}}}\right)\ =\ {\frac {\partial }{\partial x_{j}}}\left({\frac {\partial f}{\partial x_{i}}}\right).}$

In other words, the matrix of the second-order partial derivatives, known as the Hessian matrix, is a symmetric matrix.

Sufficient conditions for the symmetry to hold are given by Schwarz's theorem, also called Clairaut's theorem or Young's theorem.

In the context of partial differential equations, it is called the Schwarz integrability condition.