Symmetry of second derivatives


In mathematics, the symmetry of second derivatives refers to the possibility under certain conditions 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 Schwarz's theorem, Clairaut's theorem, or Young's theorem.
In the context of partial differential equations it is called the
Schwarz integrability condition.

Hessian matrix

This matrix of second-order partial derivatives of f is called the Hessian matrix of f. The entries in it off the main diagonal are the mixed derivatives; that is, successive partial derivatives with respect to different variables.
In most "real-life" circumstances the Hessian matrix is symmetric, although there are many functions that do not have this property. Mathematical analysis reveals that symmetry requires a hypothesis on f that goes further than simply stating the existence of the second derivatives at a particular point. The theorem of Schwarz gives a sufficient condition on f for this to occur.

Formal expressions of symmetry

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.

Theorem of Schwarz

In mathematical analysis, Schwarz's theorem named after Alexis Clairaut and Hermann Schwarz, states that if,, some neighborhood of is contained in,
and has continuous second partial derivatives at the point in, then
The partial derivatives of this function commute at that point. One easy way to establish this theorem is by applying Green's theorem to the gradient of
An elementary proof for functions on open subsets of the plane is as follows. Let be a differentiable function on an open rectangle containing and suppose that is continuous with and both continuous. Define
These functions are defined for, where and.
By the mean value theorem, intermediate values can be found in with
Since, the first equality below can be divided by :
Letting tend to zero in the last equality, the continuity assumptions on and now imply that
This account is a straightforward classical method found in many text books, for example in Burkill, Apostol and Rudin.
Although the derivation above is elementary, the approach can also be viewed from a more conceptual perspective so that the result becomes more apparent. Indeed the difference operators commute and tend to as tends to 0, with a similar statement for second order operators. Here, for a vector in the plane and a directional vector, the difference operator is defined by
By the fundamental theorem of calculus for functions on an open interval with
Hence
This is a generalized version of the mean value theorem. Recall that the elementary discussion on maxima or minima for real-valued functions implies that if is continuous on and differentiable on, then there is a point in such that
For vector-valued functions with a finite-dimensional normed space, there is no analogue of the equality above, indeed it fails. But since, the inequality above is a useful substitute. Moreover, using the pairing of the dual of with its dual norm, yields the following equality:
These versions of the mean valued theorem are discussed in Rudin, Hörmander and elsewhere.
For a function on an open set in the plane, define and. Furthermore for set
Then for in the open set, the generalized mean value theorem can be applied twice:
Thus tends to as tends to 0. The same argument shows that tends to. Hence, since the difference operators commute, so do the partial differential operators and, as claimed.
Remark. By two applications of the classical mean value theorem,
for some and in. Thus the first elementary proof can be reinterpreted using difference operators. Conversely, instead of using the generalized mean value theorem in the second proof, the classical mean valued theorem could be used. Since it was first established in 1883, Camille Jordan's 19th-century proof of symmetry of second mixed derivatives has been described as "perfect".

Theorems of Fubini and Clairaut

Let and be closed intervals in the real line and let be a continuous function on. Thus is uniformly continuous on. One of the main steps in establishing Fubini's theorem is to show that for any such
The first task is to prove Dieudonné's theorem: that any continuous function on can be approximated uniformly by a finite sum. Approximations of this kind were known to be consequences of the Stone-Weierstrass theorem established in the 1940s. Earlier Jean Dieudonné succeeded in proving these directly in 1937 using simpler methods.
Given, there are triangular functions with such that
and each is supported in intervals of length less than. The family is thus an example of a continuous "partition of unity": these were first defined by Dieudonné for solving exactly these kinds of problems. By uniform continuity of, given, there is a constant such that whenever and. Here can chosen sufficiently large that every interval is less than. Choose for each and set
Then for all
so that tends to uniformly.
In fact to check the estimate above, note that
observing that vanishes off.
Fubini's theorem can now be established by formally defining iterated Riemann integrals for simple functions
and then passing to the limit by continuity. This follows the framework developed for the Daniell integral by, but in a very much simplified setting. Real-valued functions will be used here, but—as with the Riemann integration in one variable—the passage to complex-valued functions is routine.
Let be denote Riemann integration on for intervals. Then for in,
For, if is real, then. The form is positive since and, if then. Thus if.
Similarly for integration on and in,
with if.
For functions given by finite sums in
It will be established that also defines a positive form; and that if, then.
These inequalities can be extended to in by approximately such finite sums uniformly. Indeed if,
Thus tends uniformly to, tends to uniformly to
, which must therefore be in.
Finally, on integrating over, it follows that tends to with. Similarly, taking the limits over and then, the integrals tend to. But for, it was seen that integrating in the different order gives the same answer.
Summarising, for in, the following equality for iterated integral holds:
This is Fubini's theorem in the special case of.
To prove Clairaut's theorem, let be differentiable on the open set in the plane with both and continuous. Taking a rectangle the integral can be computed using the fundamental theorem of calculus
Similarly
The hypotheses of continuity and Fubini's theorem implies that for any rectangle. Since is a positive form, if or its negative became strictly positive on the rectangle, this would give a contradiction.

Sufficiency of twice-differentiability

A weaker condition than the continuity of second partial derivatives which suffices to ensure symmetry is that all partial derivatives are themselves differentiable. 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:

History

The result of the equality of the mixed partial derivatives under certain conditions has a long history. Nicolaus I Bernoulli implicitly assumed the result as early as 1721, but Euler was the first to provide a proof. Other proofs followed by Clairaut, Lagrange, Cauchy and many others in the 19th century. None of these proofs were without fault however. In 1867 Ernst Leonard Lindelöf published a paper criticizing in detail all the proofs he was familiar with. Finally, six years later Hermann Schwarz gave the first satisfactory proof. This was followed by successive refinements that relaxed the hypotheses in Schwarz's theorem in various ways, among others by Dini, Jordan, Peano, E. W. Hobson, W. H. Young. For a good historical account, see..
Most advanced calculus texts contain sufficient conditions and proof for the equality of second mixed partial derivatives. Hence this is something that should interest those involved in teaching and learning that part of analysis. The topic can be separated into be divided into two distinct line of attack. The first came in 1867 when, following many announcements of incomplete proofs, the Finnish mathematician Lindelöf found a counter-example. The second in 1873 was the success by the German analyst H. A. Schwarz in discovering a first rigorous proof of sufficient conditions.
In 1898 Moritz Cantor outlined the historical status of second mixed derivatives before 1800. In 1740 Leonard Euler was the first to publish a proposed proof. However, already in 1721, the works of Nicolas Bernoulli had tacitly assumed the property without any formal proof. At the same time as Euler, Clairaut proposed a proof, unchallenged for most of the century. Then successively Lagrange, Cauchy, P. Blanchet, Duhamel, Sturm, Schlömilch, and Bertrand published incomplete proofs. All of the proposed proofs had been criticized, particularly when subtle points on limiting procedures arose. It was as a result of a detailed study of the deficiencies that Lindelöf could explicitly exhibit a counter-example, thus ending the stage of "primitive" investigations.
Six years after Lindelöf, Schwarz published the first satisfactory proof, thus starting the next stage of investigations.
Mathematicians tried to relax some of the assumptions of Schwarz. After an unsuccessful attempt by Thomae in 1875, the Italian mathematician Dini made an improved on Schwarz by introducing the more general "Dini-Schwarz conditions". Following another fruitless effort by Harnack in 1881, Jordan in 1882 was able to make headway. Assuming less than Dini, he published in 1883 the proof that can now be found in most text books. Along with this popular account, there are other versions by Laurent, Peano, J. Edwards, P. Haag, J. K. Whittemore, Vivanti, and Pierpont. Some of these expositions were perfect, some not, but essentially apart from changing some points of view in a minor way, Jordan's proof was adopted.
Further advances were made by E. W. Hobson in 1907 when he introduced successive differentiation, further relaxing the Dini-Schwarz conditions. Later in 1909, W. H. Young independently found less restrictions than the
Dini-Schwarz conditions. Just at that time Young published a theorem which he referred to as him as "the fundamental theorem
of the theory of differentials of two variables" stating that "if
and each
have differentials of the first order, the function
possesses a differential of the second order." In his proof, he showed that in those particular circumstances the function had
equal mixed second derivatives. Finally, in 1918, Carathéodory gave an original and unique contribution in this context using Lebesgue integration.

Distribution theory formulation

The theory of distributions eliminates analytic problems with the symmetry. The derivative of an integrable function can always be defined as a distribution, and symmetry of mixed partial derivatives always holds as an equality of distributions. The use of formal integration by parts to define differentiation of distributions puts the symmetry question back onto the test functions, which are smooth and certainly satisfy this symmetry. In more detail,
Another approach, which defines the Fourier transform of a function, is to note that on such transforms partial derivatives become multiplication operators that commute much more obviously.

Requirement of continuity

The symmetry may be broken if the function fails to have differentiable partial derivatives, which is possible if Clairaut's theorem is not satisfied.
An example of non-symmetry is the function
This can be visualized by the polar form ; it is everywhere continuous, but its derivatives at cannot be computed algebraically. Rather, the limit of difference quotients shows that, so the graph z = f has a horizontal tangent plane at, and the partial derivatives exist and are everywhere continuous. However, the second partial derivatives are not continuous at, and the symmetry fails. In fact, along the x-axis the y-derivative is, and so:
In contrast, along the y-axis the x-derivative, and so. That is, at, although the mixed partial derivatives do exist, and at every other point the symmetry does hold.
The above function, written in a cylindrical coordinate system, can be expressed as
showing that the function oscillates four times when traveling once around an arbitrarily small loop containing the origin. Intuitively, therefore, the local behavior of the function at cannot be described as a quadratic form, and the Hessian matrix thus fails to be symmetric.
In general, the interchange of limiting operations need not commute. Given two variables near and two limiting processes on
corresponding to making h → 0 first, and to making k → 0 first. It can matter, looking at the first-order terms, which is applied first. This leads to the construction of pathological examples in which second derivatives are non-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. Since in the example the Hessian is symmetric everywhere except, there is no contradiction with the fact that the Hessian, viewed as a Schwartz distribution, is symmetric.

In Lie theory

Consider the first-order differential operators Di to be infinitesimal operators on Euclidean space. 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. In other words, the Lie derivative of one coordinate with respect to another is zero.

Application to differential forms

The Clairaut-Schwarz theorem is the key fact needed to prove that for every differential form, the second exterior derivative vanishes:. This implies that every differentiable exact form is closed, since.
In the middle of the 18th century, the theory of differential forms first studied in the simplest case of 1-forms in the plane, i.e., where and are functions in the plane. The study of 1-forms and the differentials of functions began with Clairaut's papers in 1739 and 1740. At that stage his investigations were interpreted as ways of solving ordinary differential equations. Formally Clairaut showed that a 1-form on an open rectangle is closed, i.e., if and only has the form for some function in the disk. The solution for can be written by Cauchy's integral formula
while if, the closed property is the identity.