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Concept
Symmetry of second derivatives
Summary
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 without changing the result under certain conditions (see below). The symmetry is the assertion that the second-order partial derivatives satisfy the identity
so that they form an n × 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.
In the context of partial differential equations it is called the Schwarz integrability condition.
In symbols, the symmetry may be expressed as:
Another notation is:
In terms of composition of the differential operator Di which takes the partial derivative with respect to xi:
From this relation it follows that the ring of differential operators with constant coefficients, generated by the Di, is commutative; but this is only true as operators over a domain of sufficiently differentiable functions. It is easy to check the symmetry as applied to monomials, so that one can take polynomials in the xi as a domain. In fact smooth functions are another valid domain.
The result on the equality of mixed partial derivatives under certain conditions has a long history. The list of unsuccessful proposed proofs started with Euler's, published in 1740, although already in 1721 Bernoulli had implicitly assumed the result with no formal justification. Clairaut also published a proposed proof in 1740, with no other attempts until the end of the 18th century. Starting then, for a period of 70 years, a number of incomplete proofs were proposed. The proof of Lagrange (1797) was improved by Cauchy (1823), but assumed the existence and continuity of the partial derivatives and . Other attempts were made by P. Blanchet (1841), Duhamel (1856), Sturm (1857), Schlömilch (1862), and Bertrand (1864).
Official source
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