mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the symmetry of second derivatives (also called the equality of mixed partials) is the fact that exchanging the order of

partial derivative In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). P ...

s of a

multivariate function In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words ''map'', ''mapping'', ''transformation'', ''correspondence'', and ''operator'' are sometimes used synonymously. The set is called ...

f\left(x_1,\, x_2,\, \ldots,\, x_n\right)

does not change the result if some continuity conditions are satisfied (see below); that is, the second-order partial derivatives satisfy the identities :

\frac  \left( \frac \right) \ = \  
       \frac  \left( \frac \right).

In other words, the matrix of the second-order partial derivatives, known as the Hessian matrix, is a

symmetric matrix In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with ...

. Sufficient conditions for the symmetry to hold are given by Schwarz's theorem, also called Clairaut's theorem or Young's theorem. In the context of

partial differential equation In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to ho ...

s, it is called the Schwarz integrability condition.

Formal expressions of symmetry

In symbols, the symmetry may be expressed as: :

\frac  \left( \frac \right) \ = \  
       \frac  \left( \frac \right)

\qquad\text\qquad

\frac   \ =\ \frac .

Another notation is: :

\partial_x\partial_y f = \partial_y\partial_x f \qquad\text\qquad f_ = f_.

In terms of composition of the differential operator which takes the partial derivative with respect to : :

D_i \circ D_j = D_j \circ D_i

. From this relation it follows that the ring of differential operators with constant coefficients, generated by the , is

commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a pr ...

; 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

polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...

s in the as a domain. In fact smooth functions are another valid domain.

History

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 Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaCauchy (1823), but assumed the existence and continuity of the partial derivatives

\tfrac

and

\tfrac

. 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. Six years after that, Schwarz succeeded in giving the first rigorous proof. Dini later contributed by finding more general conditions than those of Schwarz. Eventually a clean and more general version was found by

Jordan Jordan, officially the Hashemite Kingdom of Jordan, is a country in the Southern Levant region of West Asia. Jordan is bordered by Syria to the north, Iraq to the east, Saudi Arabia to the south, and Israel and the occupied Palestinian ter ...

in 1883 that is still the proof found in most textbooks. 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. In 1918, Carathéodory gave a different proof based on the

Lebesgue integral In mathematics, the integral of a non-negative Function (mathematics), function of a single variable can be regarded, in the simplest case, as the area between the Graph of a function, graph of that function and the axis. The Lebesgue integral, ...

Schwarz's theorem

mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ...

, Schwarz's theorem (or Clairaut's theorem on equality of mixed partials) named after Alexis Clairaut and Hermann Schwarz, states that for a function

f \colon \Omega \to \mathbb

defined on a set

\Omega \subset \mathbb^n

, if

\mathbf\in \mathbb^n

is a point such that some

neighborhood A neighbourhood (Commonwealth English) or neighborhood (American English) is a geographically localized community within a larger town, city, suburb or rural area, sometimes consisting of a single street and the buildings lining it. Neigh ...

\mathbf

is contained in

\Omega

and

f

has continuous second partial derivatives on that neighborhood of

\mathbf

, then for all and in

\,

\fracf(\mathbf) =
  \fracf(\mathbf).

The partial derivatives of this function commute at that point. There exists a version of this theorem where

f

is only required to be twice differentiable at the point

\mathbf

. One easy way to establish this theorem (in the case where

n = 2

i = 1

, and

j = 2

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

f.

An elementary proof for functions on open subsets of the plane is as follows (by a simple reduction, the general case for the theorem of Schwarz easily reduces to the planar case). Let

f(x,y)

be a differentiable function on an open rectangle

\Omega

containing a point

(a,b)

and suppose that

df

is continuous with continuous

\partial_x \partial _y f

and

\partial_y\partial_x f

over

\Omega.

Define :

\begin
  u\left(h,\, k\right) &= f\left(a + h,\, b + k\right) - f\left(a + h,\, b\right), \\
  v\left(h,\, k\right) &= f\left(a + h,\, b + k\right) - f\left(a,\, b + k\right), \\
  w\left(h,\, k\right) &= f\left(a + h,\, b + k\right) - f\left(a + h,\, b\right) - f\left(a,\, b + k\right) + f\left(a,\, b\right).
\end

These functions are defined for

\left, h\,\, \left, k\ < \varepsilon

, where

\varepsilon > 0

and

\left - \varepsilon,\, a + \varepsilon\right \times \left - \varepsilon,\, b + \varepsilon\right /math> is contained in \Omega. By the

mean value theorem In mathematics, the mean value theorem (or Lagrange's mean value theorem) states, roughly, that for a given planar arc (geometry), arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant lin ...

, for fixed and non-zero,

\theta, \theta',  \phi, \phi'

can be found in the open interval

(0,1)

with :

\\ &= hk\, \partial_x\partial_y f \left(a + \phi^\prime h,\, b + \phi k\right). \end

Since

h,\,k \neq 0

, the first equality below can be divided by

hk

: :

\begin
  hk\,\partial_y\partial_x f\left(a + \theta h,\, b + \theta^\prime k\right) &=
    hk \, \partial_x\partial_y f\left(a + \phi^\prime h,\, b + \phi k\right), \\
 \partial_y\partial_x f\left(a + \theta h,\, b + \theta^\prime k\right) &=
    \partial_x\partial_y f\left(a + \phi^\prime h,\, b + \phi k\right).
\end

Letting

h,\,k

tend to zero in the last equality, the continuity assumptions on

\partial_y\partial_x f

and

\partial_x\partial_y f

now imply that :

\fracf\left(a,\, b\right) =
  \fracf\left(a,\, b\right).

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

\Delta^t_x,\,\,\Delta^t_y

commute and

\Delta^t_x f,\,\,\Delta^t_y f

tend to

\partial_x f,\,\, \partial_y f

t

tends to 0, with a similar statement for second order operators. Here, for

z

a vector in the plane and

u

a directional vector

\tbinom

\tbinom

, the difference operator is defined by :

\Delta^t_u f(z)= .

By the fundamental theorem of calculus for

C^1

functions

f

on an open interval

I

with

(a,b) \subset I

\int_a^b f^\prime (x) \, dx = f(b) - f(a).

Hence :

, f(b) - f(a),  \le (b-a)\, \sup_ , f^\prime(c),

. This is a generalized version of the

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

f

is continuous on

,b /math> and differentiable on (a,b), then there is a point c in (a,b) such that

: = f^\prime(c). For vector-valued functions with V a finite-dimensional normed space, there is no analogue of the equality above, indeed it fails. But since \inf f^\prime \le f^\prime(c) \le \sup f^\prime, the inequality above is a useful substitute. Moreover, using the pairing of the dual of V with its dual norm, yields the following inequality:

: \, f(b) - f(a)\,  \le (b-a)\, \sup_ \, f^\prime(c)\, .

These versions of the mean valued theorem are discussed in Rudin, Hörmander and elsewhere.

For f a C^2 function on an open set in the plane, define D_1 = \partial_x and D_2 = \partial_y . Furthermore for t \ne 0 set

: \Delta_1^t f(x,y) = (x+t,y)-f(x,y) t,\,\,\,\,\,\,\Delta^t_2f(x,y)= (x,y+t) -f(x,y) t .

Then for (x_0,y_0) in the open set, the generalized mean value theorem can be applied twice:

: \left, \Delta_1^t\Delta_2^t f(x_0,y_0) - D_1 D_2f(x_0,y_0)\\le \sup_ \left, \Delta_1^t D_2 f(x_0,y_0 + ts)  -D_1D_2 f(x_0,y_0)\\le \sup_ \left, D_1D_2f(x_0+tr,y_0+ts) - D_1D_2f(x_0,y_0)\. Thus \Delta_1^t\Delta_2^t f(x_0,y_0) tends to D_1 D_2f(x_0,y_0) as t tends to 0. The same argument shows that \Delta_2^t\Delta_1^t f(x_0,y_0) tends to D_2 D_1f(x_0,y_0) . Hence, since the difference operators commute, so do the partial differential operators D_1 and D_2, as claimed.

Remark. By two applications of the classical mean value theorem,

: \Delta_1^t\Delta_2^t f(x_0,y_0)= D_1 D_2 f(x_0+t\theta,y_0 +t\theta^\prime) for some \theta and \theta^\prime in (0,1) . 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.

Proof of Clairaut's theorem using iterated integrals

The properties of repeated Riemann integrals of a continuous function on a compact rectangle are easily established. The

uniform continuity In mathematics, a real function f of real numbers is said to be uniformly continuous if there is a positive real number \delta such that function values over any function domain interval of the size \delta are as close to each other as we want. In ...

of implies immediately that the functions

g(x)=\int_c^d F(x,y)\, dy

and

h(y)=\int_a^b F(x,y)\, dx

are continuous. It follows that :

\int_a^b \int_c^d F(x,y) \, dy\, dx = \int_c^d \int_a^b F(x,y) \, dx \, dy

; moreover it is immediate that the iterated integral is positive if is positive. The equality above is a simple case of Fubini's theorem, involving no

measure theory In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude (mathematics), magnitude, mass, and probability of events. These seemingl ...

. proves it in a straightforward way using Riemann approximating sums corresponding to subdivisions of a rectangle into smaller rectangles. To prove Clairaut's theorem, assume is a differentiable function on an open set , for which the mixed second partial derivatives and exist and are continuous. Using the fundamental theorem of calculus twice, :

\int_c^d \int_a^b f_(x,y) \, dx \, dy = \int_c^d f_y(b,y) - f_y(a,y) \, dy  = f(b,d)-f(a,d)-f(b,c)+f(a,c).

Similarly :

\int_a^b \int_c^d f_(x,y) \, dy \, dx = \int_a^b f_x(x,d) - f_x(x,c) \, dx = f(b,d)-f(a,d)-f(b,c)+f(a,c).

The two iterated integrals are therefore equal. On the other hand, since is continuous, the second iterated integral can be performed by first integrating over and then afterwards over . But then the iterated integral of on must vanish. However, if the iterated integral of a continuous function function vanishes for all rectangles, then must be identically zero; for otherwise or would be strictly positive at some point and therefore by continuity on a rectangle, which is not possible. Hence must vanish identically, so that everywhere.

Sufficiency of twice-differentiability

A weaker condition than the continuity of second partial derivatives (which is implied by the latter) 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: : ''If

f:E \to \mathbb

is defined on an open set

E \subset \R^2

;

\partial_1 f(x,\, y)

and

\partial_f(x,\, y)

exist everywhere on

E

;

\partial_f

is continuous at

\left(x_0,\, y_0\right) \in E

, and if

\partial_f(x,\, y_0)

exists in a neighborhood of

x = x_0

, then

\partial_f

exists at

\left(x_0,\, y_0\right)

and

\partial_f\left(x_0,\, y_0\right) = \partial_f\left(x_0,\, y_0\right)

.''

Distribution theory formulation

The theory of distributions (generalized functions) 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 In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivati ...

to define differentiation of distributions puts the symmetry question back onto the

test function In mathematical analysis, a bump function (also called a test function) is a function f : \Reals^n \to \Reals on a Euclidean space \Reals^n which is both smooth (in the sense of having continuous derivatives of all orders) and compactly suppor ...

s, which are smooth and certainly satisfy this symmetry. In more detail (where ''f'' is a distribution, written as an operator on test functions, and ''φ'' is a test function), :

\left(D_1 D_2 f\right) phi = -\left(D_2f\right)\left_1\phi\right = f\left_2 D_1\phi\right = f\left_1 D_2\phi\right = -\left(D_1 f\right)\left_2\phi\right = \left(D_2 D_1 f\right) phi

Another approach, which defines the

Fourier transform In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...

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 (the second partial derivatives are not continuous). Graph001

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

f(r \cos(\theta), r\sin(\theta)) = \frac

; it is everywhere continuous, but its derivatives at cannot be computed algebraically. Rather, the limit of difference quotients shows that

f_x(0,0) = f_y(0,0) = 0

, so the graph

z = f(x, y)

has a horizontal tangent plane at , and the partial derivatives

f_x, f_y

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

f_y(x,0) = x

, and so: :

f_(0,0) =
  \lim_ \frac =
  1.

In contrast, along the ''y''-axis the ''x''-derivative

f_x(0,y) = -y

, and so

f_(0,0) = -1

. That is,

f_ \ne f_

at , although the mixed partial derivatives do exist, and at every other point the symmetry does hold. The above function, written in polar coordinates, can be expressed as :

f(r,\, \theta) = \frac,

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 (0, 0) 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 :

f(h,\, k) - f(h,\, 0) - f(0,\, k) + f(0,\, 0)

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 In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include co ...

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 In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it c ...

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 ''D''_''i'' to be infinitesimal operators on

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...

. That is, ''D''_''i'' in a sense generates the one-parameter group of translations parallel to the ''x''_''i''-axis. These groups commute with each other, and therefore the infinitesimal generators do also; the Lie bracket : 'D''_''i'', ''D''_''j''= 0 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

C^\infty

(or at least twice differentiable) differential form

\omega\in\Omega^k(M)

, the second exterior derivative vanishes:

d^2\omega := d(d\omega) = 0

. This implies that every differentiable exact form (i.e., a form

\alpha

such that

\alpha = d\omega

for some form

\omega

) is closed (i.e.,

d\alpha = 0

), since

d\alpha = d(d\omega) = 0

. 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.

A\,dx + B\,dy

, where

A

and

B

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 equation In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable (mathematics), variable. As with any other DE, its unknown(s) consists of one (or more) Function (mathematic ...

s. Formally Clairaut showed that a 1-form

\omega = A \, dx + B \, dy

on an open rectangle is closed, i.e.

d\omega=0

, if and only

\omega

has the form

df

for some function

f

in the disk. The solution for

f

can be written by Cauchy's integral formula :

f(x,y)=\int_^x A(x,y)\, dx + \int_ ^y B(x,y)\, dy;

while if

\omega= df

, the closed property

d\omega=0

is the identity

\partial_x\partial_y f = \partial_y\partial_x f

. (In modern language this is one version of the Poincaré lemma.)

Notes

References

* * * * * * * (reprinted 1978) * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *