Symmetry of second derivatives

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

of n variables. If the partial derivative with respect to ${\displaystyle x_{i}}$ is denoted with a subscript ${\displaystyle i}$, then the symmetry is the assertion that the second-order partial derivatives ${\displaystyle f_{ij}}$ satisfy the identity

so that they form an n × n symmetric matrix. This is sometimes known as Schwarz's theorem or Young's theorem.

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

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