Editing Symmetry of second derivatives (section)

{{Short description|Mathematical theorem}}
{{redirect|Clairaut's theorem (calculus)|other Clairaut's results|Clairaut's formula (disambiguation)}}
{{use dmy dates|cs1-dates=ly|date=August 2021}}
In [[mathematics]], the '''symmetry of second derivatives''' (also called the '''equality of mixed partials''') is the fact that exchanging the order of [[partial derivative]]s of a [[multivariate function]]
:<math>f\left(x_1,\, x_2,\, \ldots,\, x_n\right)</math>
does not change the result if some [[continuous function|continuity]] conditions are satisfied (see below); that is, the second-order partial derivatives satisfy the [[Identity (mathematics)|identities]]
:<math>\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).
</math>
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'''.<ref>{{cite web |url=http://are.berkeley.edu/courses/ARE210/fall2005/lecture_notes/Young%27s-Theorem.pdf |title=Young's Theorem |access-date=2015-01-02 |url-status=dead |archive-url=https://web.archive.org/web/20060518134739/http://are.berkeley.edu/courses/ARE210/fall2005/lecture_notes/Young%27s-Theorem.pdf |archive-date=May 18, 2006 |publisher=University of California Berkeley}}</ref>{{sfn|Allen|1964|pages=[https://books.google.com/books?id=fgm9O6reUcsC&pg=PA300 300–305]}}

In the context of [[partial differential equation]]s, it is called the '''Schwarz [[integrability conditions for differential systems|integrability]] condition'''.
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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 <math>\arctan\; y/x</math>. At first one defines this with a cutin the complex <math>\left(x,\, y\right)</math>-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]]). It is useless until they explain where and how the function violates Schwarz integrability condition.
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