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

of n variables. If the partial derivative with respect to is denoted with a subscript, then the symmetry is the assertion that the second-order partial derivatives satisfy the identity

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

This property may also be considered as a condition for the function to be single-valued. Then it is called the Schwarz integrability condition. In physics, however, it is important for the understanding of many phenomena in nature to remove this restrictions and allow functions to violate the Schwarz integrability criterion, which makes them multivalued. The simplest example is the function . At first one defines this with a cut in the complex -plane running from 0 to infinity. The cut makes the function single-valued. In complex analysis, however, one thinks of this function as having several 'sheets' (forming a Riemann surface).