Symmetry Of Second Derivatives 113u1j

Symmetry of second derivatives In mathematics, the symmetry of second derivatives (also called the equality of mixed partials) refers to the possibility under certain conditions (see below) of inter- ∂xy f = ∂yx f. changing the order of taking partial derivatives of a func- Alternatively, the symmetry can be written as an algebraic tion statement involving the differential operator Di which takes the partial derivative with respect to xi: f (x1 , x2 , . . . , xn )

Di . Dj = Dj . Di.

of n variables. If the partial derivative with respect to From this relation it follows that the ring of differenxi is denoted with a subscript i , then the symmetry is tial operators with constant coefficients, generated by the the assertion that the second-order partial derivatives fij Di, is commutative. But one should naturally specify satisfy the identity some domain for these operators. 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 funcfij = fji tions are possible. so that they form an n × n symmetric matrix. This is sometimes known as Schwarz' theorem or Young’s theorem.[1][2]

3 Schwarz’s theorem

In the context of partial differential equations it is called In mathematical analysis, Schwarz’s theorem (or Clairaut’s the Schwarz integrability condition. theorem[3] ) named after Alexis Clairaut and Hermann Schwarz, states that if

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Hessian matrix f : Rn → R

This matrix of second-order partial derivatives of f is has continuous second partial derivatives at any given called the Hessian matrix of f. The entries in it off the point in Rn , say, (a1 , . . . , an ), then ∀i, j ∈ main diagonal are the mixed derivatives; that is, succes{1, 2, . . . , n}, sive partial derivatives with respect to different variables. In most “real-life” circumstances the Hessian matrix is ∂2f ∂2f symmetric, although there are a great number of func(a1 , . . . , an ) = (a1 , . . . , an ). tions that do not have this property. Mathematical analy- ∂xi ∂xj ∂xj ∂xi sis reveals that symmetry requires a hypothesis on f that goes further than simply stating the existence of the sec- The partial derivations of this function are commutative ond derivatives at a particular point. Schwarz' theorem at that point. One easy way to establish this theorem (in the case where n = 2 , i = 1 , and j = 2 , which gives a sufficient condition on f for this to occur. readily entails the result in general) is by applying Green’s theorem to the gradient of f .

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Formal expressions of symmetry 4 Sufficiency of differentiability

In symbols, the symmetry says that, for example,

∂ ∂x

(

∂f ∂y

)

∂ = ∂y

(

∂f ∂x

)

twice-

A weaker condition than the continuity of second partial derivatives (which is implied by the latter) which nevertheless suffices to ensure symmetry is that all partial derivatives are themselves differentiable.[4]

.

This equality can also be written as 1

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REFERENCES

Distribution theory formulation

exist and are everywhere continuous. However, the second partial derivatives are not continuous at (0,0), and the The theory of distributions (generalized functions) elim- symmetry fails. In fact, along the x-axis the y-derivative inates analytic problems with the symmetry. The deriva- is ∂y f |(x,0) = x , and so: tive of an integrable function can always be defined as a distribution, and symmetry of mixed partial derivatives ∂ f| − ∂y f |(0,0) always holds as an equality of distributions. The use of ∂x ∂y f |(0,0) = lim y (ϵ,0) = 1. ϵ→0 ϵ formal integration by parts to define differentiation of distributions puts the symmetry question back onto the Vice versa, along the y-axis the x-derivative ∂x f |(0,y) = test functions, which are smooth and certainly satisfy this −y , and so ∂y ∂x f |(0,0) = −1 . That is, ∂xy f ̸= ∂yx f symmetry. In more detail (where f is a distribution, writ- at (0, 0), although the mixed partial derivatives do exist, ten as an operator on test functions, and φ is a test func- and at every other point the symmetry does hold. tion), In general, the interchange of limiting operations need not commute. Given two variables near (0, 0) and two limiting processes on (D1 D2 f )[ϕ] = −(D2 f )[D1 ϕ] = f [D2 D1 ϕ] = f [D1 D2 ϕ] = −(D1 f )[D2 ϕ] = (D2 D1 f )[ϕ]. Another approach, which defines the Fourier transform of a function, is to note that on such transforms par- f (h, k) − f (h, 0) − f (0, k) + f (0, 0) tial derivatives become multiplication operators that com- corresponding to making h → 0 first, and to making k mute much more obviously. → 0 first. It can matter, looking at the first-order , which is applied first. This leads to the construction of pathological examples in which second derivatives are 6 Requirement of continuity non-symmetric. This kind of example belongs to the theory of real analysis where the pointwise value of functions The symmetry may be broken if the function fails to matters. When viewed as a distribution the second parhave differentiable partial derivatives, which is possible tial derivative’s values can be changed at an arbitrary set if Clairaut’s theorem is not satisfied (the second partial of points as long as this has Lebesgue measure 0. Since in the example the Hessian is symmetric everywhere exderivatives are not continuous). cept (0,0), there is no contradiction with the fact that the Hessian, viewed as a Schwartz distribution, is symmetric.

7 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 [Di, Dj] = 0

The function f(x,y), as shown in equation (1), does not have symmetric second derivatives at its origin.

is this property’s reflection. In other words, the Lie derivative of one coordinate with respect to another is zero.

An example of non-symmetry is the function:

8 References [1] Archived May 18, 2006, at the Wayback Machine.

This function is everywhere continuous, but its derivatives at (0,0) cannot be computed algebraically. Rather, the limit of difference quotients shows that ∂x f |(0,0) = ∂y f |(0,0) = 0 , so the graph z = f(x,y) has a horizontal tangent plane at (0,0), and the partial derivatives ∂x f, ∂y f

[2] Allen, R. G. D. (1964). Mathematical Analysis for Economists. New York: St. Martin’s Press. pp. 300–305. [3] James, R. C. (1966). Advanced Calculus. Belmont, CA: Wadsworth.

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[4] Hubbard, John; Hubbard, Barbara. Vector Calculus, Linear Algebra and Differential Forms (5th ed.). Matrix Editions. pp. 732–733.

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Further reading • Hazewinkel, Michiel, ed. (2001), “Partial derivative”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

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