Symmetry of second derivatives

In mathematics, the symmetry of second derivatives (also called the equality of mixed partials) refers to the possibility of interchanging the order of taking partial derivatives of a function

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

of ${\displaystyle n}$ variables without changing the result under certain conditions (see below). The symmetry is the assertion that the second-order partial derivatives satisfy the identity

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

so that they form an ${\displaystyle n\times n}$ symmetric matrix, known as the function's Hessian matrix. Sufficient conditions for the above symmetry to hold are established by a result known as Schwarz's theorem, Clairaut's theorem, or Young's theorem.^[1][2]

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