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| 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
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| :<math>f(x_{1},x_{2}, \dots ,x_{n})</math>
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| 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
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| :<math>f_{ij}=f_{ji}</math>
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| so that they form an ''n'' × ''n'' [[symmetric matrix]]. This is sometimes known as '''Young's theorem'''.<ref>http://are.berkeley.edu/courses/ARE210/fall2005/lecture_notes/Young%27s-Theorem.pdf</ref>
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| In the context of [[partial differential equation]]s it is called the
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| '''Schwarz [[integrability conditions for differential systems|integrability]] condition'''. <!-- In physics, however, it is important
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| for the understanding of many phenomena in nature to remove this restrictions and allow functions to violate the
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| Schwarz integrability criterion, which makes them multivalued.
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| The simplest example is the function
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| <math>\arctan\; y/x</math>. At first one defines this with a cut
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| in the complex
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| <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]]).
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| It is useless until they explain where and how the function violates Schwarz integrability condition -->
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| ==Hessian matrix==
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| 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.
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| In most "real-life" circumstances the Hessian matrix is [[symmetric matrix|symmetric]], although there are a far greater 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' theorem|Schwarz' theorem]] gives a sufficient condition on ''f'' for this to occur.
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| ==Formal expressions of symmetry==
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| In symbols, the symmetry says that, for example,
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| :<math>\frac {\partial}{\partial x} \left( \frac { \partial f }{ \partial y} \right) =
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| \frac {\partial}{\partial y} \left( \frac { \partial f }{ \partial x} \right).</math>
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| This equality can also be written as
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| :<math>\partial_{xy} f = \partial_{yx} f.</math>
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| 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>:
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| :''D''<sub>''i''</sub> . ''D''<sub>''j''</sub> = ''D''<sub>''j''</sub> . ''D''<sub>''i''</sub>.
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| 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.
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| ==Schwarz' theorem==
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| In [[mathematical]] [[mathematical analysis|analysis]], ''Schwarz' theorem'' (or ''Clairaut's theorem''<ref>James, R.C. (1966) ''Advanced Calculus''. Belmont, CA, Wadsworth.</ref>) named after [[Alexis Clairaut]] and [[Hermann Schwarz]], states that if
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| :<math>f \colon \mathbb{R}^n \to \mathbb{R}</math>
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| has [[continuous function|continuous]] second [[partial derivatives]] at any given point in <math> \mathbb{R}^n </math>, say, <math> (a_1, \dots, a_n),</math> then <math>\forall i, j \in \mathbb{N} \backslash \{0\}: i,j \leq n,</math>
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| :<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>
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| The partial derivations of this function are [[commutative operation|commutative]] at that point. 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.
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| ==Distribution theory formulation==
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| The theory of [[distribution (mathematics)|distribution]]s eliminates analytic problems with the symmetry. The derivative of any [[integrable]] function can be defined as a distribution. The use of [[integration by parts]] puts the symmetry question back onto the test functions{{Clarify|date=July 2010}}, which are smooth and certainly satisfy the symmetry. In the sense of distributions, symmetry always holds.
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| 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.
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| ==Requirement of continuity==
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| The symmetry may be broken if the function fails to satisfy the premises of Clairaut's theorem, such as if the derivatives are not [[Continuous function|continuous]].
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| [[File:Graph001.png|thumb|right|This function, ''f(x,y)'', does not have symmetric second derivatives at its origin.]]
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| An example of non-symmetry is the function:
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| :<math>f(x,y) = \begin{cases}
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| \frac{xy(x^2 - y^2)}{x^2+y^2} & \mbox{ for } (x, y) \ne (0, 0)\\
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| 0 & \mbox{ for } (x, y) = (0, 0).
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| \end{cases}</math>
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| Although this function is everywhere continuous, its algebraic derivatives are [[Undefined (mathematics)|undefined]] at the [[Origin (mathematics)|origin]] point.
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| Elsewhere along the ''x''-axis the ''y''-derivative <math>\partial_y f|_{(x,0)}=x</math>, and so:
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| :<math>\partial_x\partial_y f|_{(0,0)} =
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| \lim_{\epsilon\rightarrow 0} \frac { \partial_y f|_{(\epsilon,0)}-\partial_y f|_{(0,0)} } \epsilon = 1.</math>
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| Vice versa, along the ''y''-axis the ''x''-derivative <math>\partial_x f|_{(0,y)}=-y</math>,
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| and so <math>\partial_y\partial_x f|_{(0,0)} = -1</math>.
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| That is, <math>\partial_{xy}f\ne\partial_{yx}f</math> at (0, 0), although the mixed partial derivates of this function do exist and are symmetric at every other point.
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| In general, the [[interchange of limiting operations]] need not [[Commutative property|commute]]. Given two variables near (0, 0) and two limiting processes on
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| :<math>f(h,k) - f(h,0) - f(0,k) + f(0,0)</math>
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| 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 <math>0</math>. Since in the example the Hessian is symmetric everywhere except <math>(0,0)</math>, there is no contradiction with the fact that the Hessian, viewed as a [[Schwartz distribution]], is symmetric.
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| ==In Lie theory==
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| 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 generator]]s do also; the [[Lie bracket]]
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| :[''D''<sub>''i''</sub>, ''D''<sub>''j''</sub>] = 0
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| is this property's reflection. In other words, the Lie derivative of one coordinate with respect to another is zero.
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| ==References==
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| {{Reflist}}
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| == Books ==
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| *{{Springer|id=P/p071620|title=Partial derivative}}
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| *{{cite book
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| |last=Kleinert |first=H.
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| |year=2008
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| |title=Multivalued Fields in Condensed Matter, Electrodynamics, and Gravitation
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| |url=http://users.physik.fu-berlin.de/~kleinert/public_html/kleiner_reb11/psfiles/mvf.pdf
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| |publisher=[[World Scientific]]
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| |isbn=978-981-279-170-2
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| }}
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| {{DEFAULTSORT:Symmetry Of Second Derivatives}}
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| [[Category:Multivariable calculus]]
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| [[Category:Generalized functions]]
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| [[Category:Symmetry]]
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| [[Category:Theorems in analysis]]
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