In mathematics, the symmetry of second derivatives refers to the possibility of interchanging the order of taking partial derivatives of a function Mathematics, often abbreviated maths in Commonwealth English and math in American English, is the study of abstraction. ... 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. ...

f(x₁, x₂, ..., x_n)

of n variables. If the partial derivative with respect to x_i is denoted with a subscript i, then the symmetry is the assertion that

f_ij

is an n × n symmetric matrix. This matrix 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 normal circumstances the Hessian matrix is indeed symmetric; but from the point of view of mathematical analysis this isn't a safe statement, without some hypothesis on f that goes further than simply stating the existence of the second derivatives at a particular point. Clairaut's theorem gives a sufficient condition on f for this to occur. In linear algebra, a symmetric matrix is a matrix that is its own transpose. ... In mathematics, the Hessian matrix is the square matrix of second partial derivatives of a scalar-valued function. ... In linear algebra, the main diagonal of a square matrix is the diagonal which runs from the top left corner to the bottom right corner. ... Analysis is that branch of mathematics which deals with the real numbers, complex numbers, and their functions. ... In mathematical analysis, Clairauts theorem states that if has continuous second partial derivatives at then for In words, the partial derivatives of this function commute. ...

In symbols, the symmetry says that, for example,

This equality can also be written as

Alternatively, the symmetry can be written as an algebraic statement involving the differential operator D_i which takes the partial derivative with respect to x_i: In mathematics, a differential operator is a linear operator defined as a function of the differentiation operator. ...

D_i . D_j = D_j . D_i.

From this relation it follows that the ring of differential operators with constant coefficients, generated by the D_i, is commutative. But one should naturally specify some domain for these operators. It is easy to check the symmetry as applied to monomials, so that one can take polynomials in the x_i as a domain. In fact smooth functions is possible. In ring theory, a branch of abstract algebra, a ring is an algebraic structure in which addition and multiplication are defined and have similar properties to those familiar from the integers. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... In mathematics, a monomial is a particular kind of polynomial, having just one term. ... In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ... In mathematics, a smooth function is one that is infinitely differentiable, i. ...

One can also apply the theory of distributions to get round any analytic problems with the symmetry. Firstly the derivative of any function can be defined (provided it is integrable), as a distribution. Secondly the use of integration by parts throws the symmetry question back onto the test functions, which are smooth and certainly satisfy the symmetry. One concludes that, in the sense of distributions, the symmetry always holds. (Another approach, where the Fourier transform of a function is defined, is to note that on transforms the partial derivatives become multiplication operators that commute much more obviously). This page deals with mathematical distributions. ... Integrability is a mathematical concept used in different areas. ... In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, possibly simpler, integrals. ... The Fourier transform, named after Jean Baptiste Joseph Fourier, is an integral transform that re-expresses a function in terms of sinusoidal basis functions, i. ...

The fact remains that in the worst case there are counterexamples. In the case of two variables, near (0, 0) one can consider two limiting processes on In logic, and especially in its applications to mathematics and philosophy, a counterexample is an exception to a proposed general rule, i. ...

f(h, k) − f(h, 0) − f(0, k) + f(0, 0)

corresponding to making h → 0 first, and to making k → 0 first. These processes need not commute: it can matter, looking at the first-order terms, which is applied first. This leads to the construction of pathological examples in which the symmetry of second derivatives is not true. [Could someone write up such an example here or in its own article, and add that to the list of mathematical examples?] Given that the derivatives as Schwartz distributions are symmetric, this kind of example belongs in the 'fine' theory of real analysis. In mathematics, a pathological example is one whose properties are (or should be considered) untypically bad. ... This page will attempt to list examples in mathematics. ... Real analysis is that branch of mathematical analysis dealing with the set of real numbers and functions of real numbers. ...

A more sophisticated argument is this: consider the first-order differential operators D_i to be infinitesimal operators on Euclidean space. That is, D_i in a sense generates the one-parameter group of translations parallel to the x_i-axis. These groups certainly all commute with each other, and therefore we expect that the infinitesimal generators do also; the Lie bracket In mathematics, an infinitesimal transformation is a limiting form of small transformation. ... In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphism φ : R → G from the real line R (as an additive group) to some other topological group G. That means that it is not in fact a group, strictly speaking; if φ is injective... Translation is an activity comprising the interpretation of the meaning of a text in one language—the called the source text—and the production of a new, equivalent text in another language—called the target text, or the translation. ... In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ... A lie bracket can refer to: Lie algebra Lie derivative This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...

[D_i, D_j] = 0

is the way that is reflected. When one comes to ask whether this is try as applied to spaces of functions on Euclidean space, this question is an elementary part of the theory of differentiable vectors in representation theory; that is, a suitable subspace of a function space on which the Lie algebra representation deriving from a representation of a Lie group has desirable and transparent properties. In mathematics Representation theory is the name given to the study of standard representations of abstract mathematical structures. ... In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in most applications, it is a topological space or/and a vector space. ... In mathematics, a Lie algebra (named after Sophus Lie, pronounced lee) is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds. ... In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ...

Counter Example

Consider the function

with f(0,0) = 0. Then the mixed partial derivatives of f exist, and are continuous everywhere except at (0,0). Moreover

at (0,0).