Symmetry of second derivatives

[1][2] 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.

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,[3] although already in 1721 Bernoulli had implicitly assumed the result with no formal justification.

[4] 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

[5] Other attempts were made by P. Blanchet (1841), Duhamel (1856), Sturm (1857), Schlömilch (1862), and Bertrand (1864).

Finally in 1867 Lindelöf systematically analyzed all the earlier flawed proofs and was able to exhibit a specific counterexample where mixed derivatives failed to be equal.

Minor variants of earlier proofs were published by Laurent (1885), Peano (1889 and 1893), J. Edwards (1892), P. Haag (1893), J. K. Whittemore (1898), Vivanti (1899) and Pierpont (1905).

Further progress was made in 1907-1909 when E. W. Hobson and W. H. Young found proofs with weaker conditions than those of Schwarz and Dini.

, which readily entails the result in general) is by applying Green's theorem to the gradient of

now imply that This account is a straightforward classical method found in many text books, for example in Burkill, Apostol and Rudin.

[10][11][12] Although the derivation above is elementary, the approach can also be viewed from a more conceptual perspective so that the result becomes more apparent.

Recall that the elementary discussion on maxima or minima for real-valued functions implies that if

with its dual norm, yields the following inequality: These versions of the mean valued theorem are discussed in Rudin, Hörmander and elsewhere.

The properties of repeated Riemann integrals of a continuous function F on a compact rectangle [a,b] × [c,d] are easily established.

[28] The equality above is a simple case of Fubini's theorem, involving no measure theory.

To prove Clairaut's theorem, assume f is a differentiable function on an open set U, for which the mixed second partial derivatives fyx and fxy exist and are continuous.

Using the fundamental theorem of calculus twice, Similarly The two iterated integrals are therefore equal.

[34] Another strengthening of the theorem, in which existence of the permuted mixed partial is asserted, was provided by Peano in a short 1890 note on Mathesis: The theory of distributions (generalized functions) eliminates analytic problems with the symmetry.

An example of non-symmetry is the function (due to Peano)[36][37] This can be visualized by the polar form

Intuitively, therefore, the local behavior of the function at (0, 0) cannot be described as a quadratic form, and the Hessian matrix thus fails to be symmetric.

This kind of example belongs to the theory of real analysis where the pointwise value of functions matters.

When viewed as a distribution the second partial derivative's values can be changed at an arbitrary set of points as long as this has Lebesgue measure 0.

That is, Di in a sense generates the one-parameter group of translations parallel to the xi-axis.

These groups commute with each other, and therefore the infinitesimal generators do also; the Lie bracket is this property's reflection.

[38] In the middle of the 18th century, the theory of differential forms was first studied in the simplest case of 1-forms in the plane, i.e.

At that stage his investigations were interpreted as ways of solving ordinary differential equations.