View source for 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 interchanging the order of taking [[partial derivative]]s of a function

:<math>f(x_{1},x_{2}, \dots ,x_{n})</math>

of ''n'' variables. If the partial derivative with respect to <math>x_{i}</math> is denoted with a subscript <math>i</math>, then the symmetry is the assertion that the second-order partial derivatives <math>f_{ij}</math> satisfy the identity

:<math>f_{ij}=f_{ji}</math>

so that they form an ''n'' × ''n'' [[symmetric matrix]]. This is sometimes known as '''Schwarz's theorem''', '''Clairaut's theorem''', or '''Young's theorem'''.<ref>{{cite web |url=http://are.berkeley.edu/courses/ARE210/fall2005/lecture_notes/Young%27s-Theorem.pdf |title=Young's Theorem |access-date=2015-01-02 |url-status=dead |archive-url=https://web.archive.org/web/20060518134739/http://are.berkeley.edu/courses/ARE210/fall2005/lecture_notes/Young%27s-Theorem.pdf |archive-date=May 18, 2006 |df= }}</ref><ref>{{cite book |last=Allen |first=R. G. D. |title=Mathematical Analysis for Economists |location=New York |publisher=St. Martin's Press |year=1964 |isbn= |pages=300–305 |url=https://books.google.com/books?id=fgm9O6reUcsC&pg=PA300 }}</ref>

In the context of [[partial differential equation]]s it is called the
'''Schwarz [[integrability conditions for differential systems|integrability]] condition'''. <!-- In physics, however, it is important
for the understanding of many phenomena in nature to remove this restrictions and allow functions to violate the
Schwarz integrability criterion, which makes them multivalued.
The simplest example is the function
<math>\arctan\; y/x</math>. At first one defines this with a cut
in the complex
<math>(x,y)</math>-plane running from 0 to infinity.
The cut makes the function single-valued. In complex analysis, however, one thinks of this function as having several 'sheets' (forming a [[Riemann surface]]).
 It is useless until they explain where and how the function violates 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 matrix|symmetric]], although there are a great number of 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. [[#Schwarz's theorem|Schwarz' theorem]] gives a sufficient condition on ''f'' for this to occur.

== Formal expressions of symmetry ==
In symbols, the symmetry says that, for example,

:<math>\frac {\partial}{\partial x} \left( \frac { \partial f }{ \partial y} \right) =
       \frac {\partial}{\partial y} \left( \frac { \partial f }{ \partial x} \right).</math>

This equality can also be written as

:<math>\partial_{xy} f = \partial_{yx} f.</math>

Alternatively, the symmetry can be written as an algebraic statement involving the [[differential operator]] ''D''<sub>''i''</sub> which  takes the partial derivative with respect to ''x''<sub>''i''</sub>:

:''D''<sub>''i''</sub> . ''D''<sub>''j''</sub> = ''D''<sub>''j''</sub> . ''D''<sub>''i''</sub>.

From this relation it follows that the [[ring (mathematics)|ring]] of differential operators with [[constant coefficients]], generated by the ''D''<sub>''i''</sub>, is [[commutative]]. But one should naturally specify some domain for these operators. It is easy to check the symmetry as applied to [[monomial]]s, so that one can take [[polynomial]]s in the ''x''<sub>''i''</sub> as a domain. In fact [[smooth function]]s are possible.

== Schwarz's theorem<!--'Schwarz's theorem' and 'Clairaut's theorem on equality of mixed partials' redirect here--> ==
In [[mathematical]] [[mathematical analysis|analysis]], '''Schwarz's theorem'''<!--boldface per WP:R#PLA--> (or '''Clairaut's theorem on equality of mixed partials'''<!--boldface per WP:R#PLA-->)<ref>{{cite book |last=James |first=R. C. |year=1966 |title=Advanced Calculus |location=Belmont, CA |publisher=Wadsworth |isbn= |url=https://books.google.com/books?id=d5kpAQAAMAAJ }}</ref> named after [[Alexis Clairaut]] and [[Hermann Schwarz]], states that if <math>(a_1, \dots, a_n)\in\mathbb{R}^n</math>, <math>\Omega\subseteq\mathbb{R}^n</math>, some neighborhood of <math>(a_1, \dots, a_n)</math> is contained in <math>\Omega</math>,

:<math>f \colon \Omega \to \mathbb{R}</math>

and <math>f</math> has [[continuous function|continuous]] second [[partial derivatives]] at the point in <math>(a_1, \dots, a_n)</math>, then <math>\forall i, j \in \{ 1,2,\ldots, n\},</math>

:<math>\frac{\partial^2 f}{\partial x_i\, \partial x_j}(a_1, \dots, a_n) = \frac{\partial^2 f}{\partial x_j\, \partial x_i}(a_1, \dots, a_n).</math>

The partial derivatives of this function are [[commutative operation|commutative]] at that point. One easy way to establish this theorem (in the case where <math>n = 2</math>, <math>i = 1</math>, and <math>j = 2</math>, which readily entails the result in general) is by applying [[Green's theorem]] to the [[gradient]] of <math>f.</math> An elementary proof is as follows. Let

:<math>u(h_1,h_2)=f(a_1+h_1,a_2+h_2)-f(a_1+h_1,a_2),</math>
:<math>v(h_1,h_2)=f(a_1+h_1,a_2+h_2)-f(a_1,a_2+h_2),</math>
:<math>w(h_1,h_2)=f(a_1+h_1,a_2+h_2)-f(a_1+h_1,a_2)-f(a_1,a_2+h_2)+f(a_1,a_2).</math>

These functions are defined for <math>|h_1|,|h_2|<\varepsilon</math>, where <math>\varepsilon</math> is chosen so that holds <math>\varepsilon>0</math> and <math>[a_1-\varepsilon,a_1+\varepsilon]\times[a_2-\varepsilon,a_2+\varepsilon]\subseteq\Omega</math>.

By [[mean value theorem]], for <math>h_1,h_2\neq 0</math> holds

:<math>w(h_1,h_2)=u(h_1,h_2)-u(0,h_2)=h_1\frac{\partial u}{\partial x}(\theta_1(h_1,h_2)h_1,h_2)
=h_1\left(\frac{\partial f}{\partial x}(a_1+\theta_1(h_1,h_2)h_1,a_2+h_2)-\frac{\partial f}{\partial x}(a_1+\theta_1(h_1,h_2)h_1,a_2)\right)
=h_1h_2\frac{\partial^2f}{\partial x\partial y}(a_1+\theta_1(h_1,h_2)h_1,a_2+\theta_2(h_1,h_2)h_2)</math>,

:<math>w(h_1,h_2)=v(h_1,h_2)-v(h_1,0)=h_2\frac{\partial v}{\partial y}(h_1,\theta_3(h_1,h_2)h_2)
=h_2\left(\frac{\partial f}{\partial y}(a_1+h_1,a_2+\theta_3(h_1,h_2)h_2)-\frac{\partial f}{\partial x}(a_1,a_2+\theta_3(h_1,h_2)h_2)\right)
=h_1h_2\frac{\partial^2f}{\partial y\partial x}(a_1+\theta_4(h_1,h_2)h_1,a_2+\theta_3(h_1,h_2)h_2)</math>,

for some functions <math>\theta_i(h_1,h_2)</math> such that <math>0<\theta_i(h_1,h_2)<1</math>. The special case is that for <math>h\neq 0</math> holds

:<math>h^2\frac{\partial^2f}{\partial x\partial y}(a_1+\theta_1(h,h)h,a_2+\theta_2(h,h)h)=h^2\frac{\partial^2f}{\partial y\partial x}(a_1+\theta_4(h,h)h,a_2+\theta_3(h,h)h),</math>
:<math>\frac{\partial^2f}{\partial x\partial y}(a_1+\theta_1(h,h)h,a_2+\theta_2(h,h)h)=\frac{\partial^2f}{\partial y\partial x}(a_1+\theta_4(h,h)h,a_2+\theta_3(h,h)h).</math>

Letting that <math>h</math> tends to zero in the last equality, we conclude that

:<math>\frac{\partial^2f}{\partial x\partial y}(a_1,a_2)=\frac{\partial^2f}{\partial y\partial x}(a_1,a_2).</math>

== 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 function#Differentiability in higher dimensions|differentiable]].<ref>{{cite book|last1=Hubbard|first1=John|last2=Hubbard|first2=Barbara|title=Vector Calculus, Linear Algebra and Differential Forms|publisher=Matrix Editions|pages=732–733|edition=5th}}</ref> 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_(journal)|Mathesis]]:

''If <math>f:E\to \mathbb{R}</math> is defined on an open set <math>E\subset \R^2</math>; <math>\partial_1f(x,y)</math> and <math> \partial_{2,1}f(x,y)</math> exist everywhere on <math>E</math>; <math>\partial_{2,1}f</math> is continuous at <math>(x_0,y_0)\in E</math>, and if <math>\partial_{2}f(x,y_0)</math> exists in a neighborhood of <math>x=x_0</math>, then <math>\partial_{1,2}f</math> exists at <math>(x_0,y_0)</math> and <math>\partial_{1,2}f(x_0,y_0)=\partial_{2,1}f(x_0,y_0)</math>.''<ref>{{Cite book|url=https://archive.org/details/1979RudinW|title=Principles of Mathematical Analysis|last=Rudin|first=Walter|publisher=McGraw-Hill|year=1976|isbn=0-07-054235-X|location=New York|pages=235–236}}</ref>

== 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 [[Alexis Clairaut|Clairaut]] (1740), [[Lagrange]] (1797), [[Cauchy]] (1823) and many others in the 19th century.  None of these proofs were without fault however (for example, Clairaut assumed all definite integrals could be differentiated under the integral sign).  In 1867 [[Ernst Leonard Lindelöf]] published a paper<ref>{{cite journal|last1=Lindelöf|first1=Ernst Leonard|title=Remarques sur les différentes manières d'établir la formule <math>\frac{d^2 z}{d xd y} = \frac{d^2 z}{d yd x}</math>|journal=Acta Societatis Scientiarum Fennicae|date=1867|volume=8, part 1|pages=205–213|url=https://archive.org/stream/actasocietatissc81suom#page/205/mode/2up}}</ref> criticizing in detail all the proofs he was familiar with.  Finally, six years later [[Hermann Schwarz]] (1873) 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 [[Ulisse Dini|Dini]], [[Camille Jordan|Jordan]], [[Peano]], [[E. W. Hobson]], [[W. H. Young]]. For a good historical account, see <ref>{{cite journal|last1=Higgins|first1=Thomas James|title=A note on the history of mixed partial derivatives|journal=Scripta Mathematica|date=1940|volume=7|pages=59–62|url=http://mathforum.org/kb/thread.jspa?forumID=13&threadID=1580215&messageID=5765194|accessdate=19 April 2017}}</ref>.

== Distribution theory formulation ==

The theory of [[distribution (mathematics)|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]] to define differentiation of distributions puts the symmetry question back onto the [[test function]]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),
: <math> (D_1D_2f)[\phi] = -(D_2f)[D_1\phi] = f[D_2D_1\phi] = f[D_1D_2\phi] = -(D_1f)[D_2\phi] = (D_2D_1f)[\phi] .</math>

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 (the second partial derivatives are not [[Continuous function|continuous]]).

[[File:Graph001.png|thumb|right|The function ''f(x,y)'', as shown in equation ({{EquationNote|1}}), does not have symmetric second derivatives at its origin.]]
An example of non-symmetry is the function
{{NumBlk|:|<math>f(x,y) = \begin{cases}
                     \frac{xy(x^2 - y^2)}{x^2+y^2} & \mbox{ for } (x, y) \ne (0, 0)\\
                      0                            & \mbox{ for } (x, y) = (0, 0).
                \end{cases}</math>|{{EquationRef|1}}}}
This function is everywhere continuous, but its derivatives at (0,0) cannot be computed algebraically. Rather, the limit of difference quotients shows that <math>\partial_x f|_{(0,0)}=\partial_y f|_{(0,0)} = 0</math>, so the graph z = f(x,y) has a horizontal tangent plane at (0,0), and the partial derivatives <math>\partial_x f, \partial_y f</math> exist and are everywhere continuous. However, the second partial derivatives are not continuous at (0,0), and the symmetry fails. In fact, along the ''x''-axis the ''y''-derivative is <math>\partial_y f|_{(x,0)}=x</math>, and so:
:<math>\partial_x\partial_y f|_{(0,0)} =
\lim_{\epsilon\rightarrow 0} \frac { \partial_y f|_{(\epsilon,0)}-\partial_y f|_{(0,0)} } \epsilon = 1.</math>
In contrast, along the ''y''-axis the ''x''-derivative <math>\partial_x f|_{(0,y)}=-y</math>,
and so <math>\partial_y\partial_x f|_{(0,0)} = -1</math>.
That is, <math>\partial_{xy}f\ne\partial_{yx}f</math> at (0,&nbsp;0), 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
:<math>f(r, \theta) = \frac{r^2 \sin{4\theta}}{4},</math>
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 <math>(0, 0)</math> 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 [[Commutative property|commute]]. Given two variables near (0,&nbsp;0) and two limiting processes on
:<math>f(h,k) - f(h,0) - f(0,k) + f(0,0)</math>
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 (mathematics)|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 (0,0), 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''<sub>''i''</sub> to be [[infinitesimal operator]]s on [[Euclidean space]]. That is, ''D''<sub>''i''</sub> in a sense generates the [[one-parameter group]] of [[translation]]s parallel to the ''x''<sub>''i''</sub>-axis. These groups commute with each other, and therefore the [[Lie group#The Lie algebra associated to a Lie group|infinitesimal generators]] do also; the [[Lie bracket]]

:[''D''<sub>''i''</sub>, ''D''<sub>''j''</sub>] = 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 <math>C^\infty</math> (or at least twice differentiable) [[differential form]] <math>\omega\in\Omega^k(M)</math>, the second exterior derivative vanishes: <math>d^2\omega:=d(d\omega)=0</math>.  This implies that every differentiable [[Exact differential form|exact]] form (i.e., a form <math>\alpha</math> such that <math>\alpha=d\omega</math> for some form <math>\omega</math>) is [[Closed differential form|closed]] (i.e., <math>d\alpha=0</math>), since <math>d\alpha=d(d\omega)=0</math>.<ref>{{Cite book|url=https://archive.org/details/TuL.W.AnIntroductionToManifolds2e2010Springer|title=An Introduction to Manifolds|last=Tu|first=Loring W.|publisher=Springer|year=2010|isbn=978-1-4419-7399-3|edition=2nd|location=New York|pages=}}</ref>

== References ==
{{Reflist}}

== Further reading ==
* {{Springer|id=P/p071620|title=Partial derivative}}

[[Category:Multivariable calculus]]
[[Category:Generalized functions]]
[[Category:Symmetry]]
[[Category:Theorems in analysis]]