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.^[1][2]

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