Editing Symmetry of second derivatives

{{Short description|Mathematical theorem}}
{{redirect|Clairaut's theorem (calculus)|other Clairaut's results|Clairaut's formula (disambiguation)}}
{{use dmy dates|cs1-dates=ly|date=August 2021}}
In [[mathematics]], the '''symmetry of second derivatives''' (also called the '''equality of mixed partials''') is the fact that exchanging the order of [[partial derivative]]s of a [[multivariate function]]
:<math>f\left(x_1,\, x_2,\, \ldots,\, x_n\right)</math>
does not change the result if some [[continuous function|continuity]] conditions are satisfied (see below); that is, the second-order partial derivatives satisfy the [[Identity (mathematics)|identities]]
:<math>\frac {\partial}{\partial x_i} \left( \frac{\partial f}{\partial x_j} \right) \ = \  
       \frac {\partial}{\partial x_j} \left( \frac{\partial f}{\partial x_i} \right).
</math>
In other words, the matrix of the second-order partial derivatives, known as the [[Hessian matrix]], is a [[symmetric matrix]]. 

Sufficient conditions for the symmetry to hold are given by '''Schwarz's theorem''', also called '''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 |publisher=University of California Berkeley}}</ref>{{sfn|Allen|1964|pages=[https://books.google.com/books?id=fgm9O6reUcsC&pg=PA300 300–305]}}

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 cutin the complex <math>\left(x,\, y\right)</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.
-->

== Formal expressions of symmetry ==
In symbols, the symmetry may be expressed as:

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

\qquad\text{or}\qquad

\frac {\partial^2\! f} {\partial x\,\partial y} \ =\ \frac{\partial^2\! f} {\partial y\,\partial x}.

</math>

Another notation is:

:<math>\partial_x\partial_y f = \partial_y\partial_x f \qquad\text{or}\qquad f_{yx} = f_{xy}.</math>

In terms of [[Function composition|composition]] of the [[differential operator]] {{math|''D''<sub>''i''</sub>}} which takes the partial derivative with respect to {{math|''x''<sub>''i''</sub>}}:

:<math>D_i \circ D_j = D_j \circ D_i</math>.

From this relation it follows that the [[ring (mathematics)|ring]] of differential operators with [[constant coefficients]], generated by the {{math|''D''<sub>''i''</sub>}}, is [[commutative]]; but this is only true as operators over a domain of sufficiently differentiable functions. It is easy to check the symmetry as applied to [[monomial]]s, so that one can take [[polynomial]]s in the {{math|''x''<sub>''i''</sub>}} as a domain. In fact [[smooth function]]s 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 [[Leonard Euler|Euler]]'s, published in 1740,{{sfn|Euler|1740}} although already in 1721 [[Nicolas Bernoulli|Bernoulli]] had implicitly assumed the result with no formal justification.{{sfn|Sandifer|2007|pages=[https://books.google.com/books?id=3-DyDwAAQBAJ&pg=PA142 142–147] |loc=footnote: Comm. Acad. Sci. Imp. Petropol. '''7''' (1734/1735) '''1740''', 174-189, 180-183; ''Opera Omnia'', 1.22, 34-56.}} [[Alexis Clairaut|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]] (1797) was improved by [[Cauchy]] (1823), but assumed the existence and continuity of the partial derivatives <math>\tfrac{\partial^2 f}{\partial x^2}</math> and <math>\tfrac{\partial^2 f}{\partial y^2}</math>.{{sfn|Minguzzi|2015}} Other attempts were made by P. Blanchet (1841), [[Jean-Marie Duhamel|Duhamel]] (1856), [[Jacques Charles François Sturm|Sturm]] (1857), [[Schlömilch]] (1862), and [[Joseph Bertrand|Bertrand]] (1864). Finally in 1867 [[Lorenz Leonard Lindelöf|Lindelöf]] systematically analyzed all the earlier flawed proofs and was able to exhibit a specific counterexample where mixed derivatives failed to be equal.{{sfn|Lindelöf|1867}}{{sfn|Higgins|1940}}

Six years after that, [[H. A. Schwarz|Schwarz]] succeeded in giving the first rigorous proof.{{sfn|Schwarz|1873}} [[Ulisse Dini|Dini]] later contributed by finding more general conditions than those of Schwarz. Eventually a clean and more general version was found by [[Camille Jordan|Jordan]] in 1883 that is still the proof found in most textbooks. Minor variants of earlier proofs were published by [[Paul Matthieu Hermann Laurent|Laurent]] (1885), [[Peano]] (1889 and 1893), J. Edwards (1892), P. Haag (1893), J. K. Whittemore (1898), [[Giulio Vivanti|Vivanti]] (1899) and [[James Pierpont (mathematician)|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]].{{sfn|Higgins|1940}}
== Schwarz's theorem<!--'Schwarz's theorem' and 'Clairaut's theorem on equality of mixed partials' redirect here--> ==
{{redirect|Schwarz's theorem|the result in complex analysis|Schwarz lemma}}
In [[mathematical analysis]], '''Schwarz's theorem'''<!--boldface per WP:R#PLA--> (or '''Clairaut's theorem on equality of mixed partials'''<!--boldface per WP:R#PLA-->){{sfn|James|1966|p={{pn|date=August 2021}}}} named after [[Alexis Clairaut]] and [[Hermann Schwarz]], states that for a function <math>f \colon \Omega \to \mathbb{R}</math> defined on a set <math>\Omega \subset \mathbb{R}^n</math>, if <math>\mathbf{p}\in \mathbb{R}^n</math> is a point such that some [[Neighbourhood (mathematics)|neighborhood]] of <math>\mathbf{p}</math> is contained in <math>\Omega</math> and <math>f</math> has [[continuous function|continuous]] second [[partial derivatives]] on that neighborhood of <math>\mathbf{p}</math>, then for all {{mvar|i}} and {{mvar|j}} in <math>\{1, 2 \ldots,\, n\},</math>

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

The partial derivatives of this function commute at that point.

[[#Sufficiency of twice-differentiability|There exists]] a version of this theorem where <math>f</math> is only required to be twice differentiable at the point <math>\mathbf{p}</math>.

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 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).<ref name=burkhill>{{harvnb|Burkill|1962|pages=154–155}}</ref> Let <math>f(x,y)</math> be a [[differentiable function]] on an open rectangle <math>\Omega</math> containing a point <math>(a,b)</math> and suppose that <math>df</math> is continuous with continuous <math>\partial_x \partial _y f</math> and <math>\partial_y\partial_x f</math> over <math>\Omega.</math> Define

:<math>\begin{align}
  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{align}</math>

These functions are defined for <math>\left|h\right|,\, \left|k\right| < \varepsilon</math>, where <math>\varepsilon > 0 </math> and <math>\left[a - \varepsilon,\, a + \varepsilon\right] \times \left[b - \varepsilon,\, b + \varepsilon\right]</math> is contained in <math>\Omega.</math>

By the [[mean value theorem]], for fixed {{mvar|h}} and {{mvar|k}} non-zero,  <math>\theta, \theta',  \phi, \phi'</math> can be found in the open interval <math> (0,1)</math> with

:<math>\begin{align}
     w\left(h,\, k\right)
  &= u\left(h,\, k\right) - u\left(0,\, k\right) = h\, \partial_x u\left(\theta h,\, k\right) \\
  &= h\,\left[\partial_x f\left(a + \theta h,\, b + k\right) - \partial_x f\left(a + \theta h,\, b\right)\right] \\
  &= hk \, \partial_y \partial_x f\left(a + \theta h,\, b + \theta^\prime k\right) \\

     w\left(h,\, k\right)
  &= v\left(h,\, k\right) - v\left(h,\, 0\right) = k\,\partial_y v\left(h,\, \phi k\right) \\
  &= k\left[\partial_y f\left(a + h,\, b + \phi k\right) - \partial_y f\left(a,\, b + \phi k\right)\right] \\
  &= hk\, \partial_x\partial_y f \left(a + \phi^\prime h,\, b + \phi k\right).
\end{align}</math>

Since <math>h,\,k \neq 0</math>, the first equality below can be divided by <math>hk</math>:

:<math>\begin{align}
  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{align}</math>

Letting <math>h,\,k</math> tend to zero in the last equality, the continuity assumptions on <math>\partial_y\partial_x f</math> and <math>\partial_x\partial_y f</math> now imply that

:<math>
  \frac{\partial^2}{\partial x\partial y}f\left(a,\, b\right) =
  \frac{\partial^2}{\partial y\partial x}f\left(a,\, b\right).
</math>

This account is a straightforward classical method found in many text books, for example in Burkill, Apostol and Rudin.<ref name=burkhill/>{{sfn|Apostol|1965}}{{sfn|Rudin|1976}}
 
Although the derivation above is elementary, the approach can also be viewed from a more conceptual perspective so that the result becomes more apparent.{{sfn|Hörmander|2015|pages=7, 11|ps=. This condensed account is possibly the shortest.}}{{sfn|Dieudonné|1960|pages=179–180}}{{sfn|Godement|1998b|pages=287–289}}{{sfn|Lang|1969|pages=108–111}}{{sfn|Cartan|1971|pages=64–67}} Indeed the [[difference operator]]s <math>\Delta^t_x,\,\,\Delta^t_y</math> commute and <math>\Delta^t_x f,\,\,\Delta^t_y f</math> tend to <math>\partial_x f,\,\, \partial_y f</math> as <math>t</math> tends to 0, with a similar statement for second order operators.{{efn|name="Schwartz"|1=These can also be rephrased in terms of the action of operators on [[Schwartz function]]s on the plane. Under [[Fourier transform]], the difference and differential operators are just multiplication operators.{{sfn|Hörmander|2015|loc=Chapter VII}}}} Here, for <math>z</math> a vector in the plane and <math>u</math> a directional vector <math>\tbinom{1}{0}</math> or <math>\tbinom{0}{1}</math>, the difference operator is defined by

:<math>\Delta^t_u f(z)= {f(z+tu) - f(z)\over t}.</math>

By the [[fundamental theorem of calculus]] for <math>C^1</math> functions <math>f</math> on an open interval <math>I</math> with <math> (a,b) \subset I</math>

:<math>\int_a^b f^\prime (x) \, dx = f(b) - f(a).</math>

Hence

:<math>|f(b) - f(a)| \le (b-a)\, \sup_{c\in (a,b)} |f^\prime(c)|</math>.

This is a generalized version of the [[mean value theorem]]. Recall that the elementary discussion on maxima or minima for real-valued functions implies that if <math>f</math> is continuous on <math>[a,b]</math> and differentiable on <math>(a,b)</math>, then there is a point <math>c</math> in <math>(a,b)</math> such that

:<math> {f(b) - f(a) \over b - a} = f^\prime(c).</math>

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

:<math>\|f(b) - f(a)\| \le (b-a)\, \sup_{c\in (a,b)} \|f^\prime(c)\|</math>.

These versions of the mean valued theorem are discussed in Rudin, Hörmander and elsewhere.{{sfn|Hörmander|2015|page=6}}{{sfn|Rudin|1976|page={{pn|date=August 2021}}}}

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

:<math>\Delta_1^t f(x,y) = [f(x+t,y)-f(x,y)]/t,\,\,\,\,\,\,\Delta^t_2f(x,y)=[f(x,y+t) -f(x,y)]/t</math>.

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

:<math> \left|\Delta_1^t\Delta_2^t f(x_0,y_0) - D_1 D_2f(x_0,y_0)\right|\le \sup_{0\le s \le 1} \left|\Delta_1^t D_2 f(x_0,y_0 + ts)  -D_1D_2 f(x_0,y_0)\right|\le \sup_{0\le r,s\le 1} \left|D_1D_2f(x_0+tr,y_0+ts) - D_1D_2f(x_0,y_0)\right|.</math>

Thus <math>\Delta_1^t\Delta_2^t f(x_0,y_0)</math> tends to <math>D_1 D_2f(x_0,y_0)</math> as <math>t</math> tends to 0. The same argument shows that <math>\Delta_2^t\Delta_1^t f(x_0,y_0)</math> tends to <math>D_2 D_1f(x_0,y_0)</math>. Hence, since the difference operators commute, so do the partial differential operators <math>D_1</math> and <math>D_2</math>, as claimed.{{sfn|Hörmander|2015|page=11}}{{sfn|Dieudonné|1960}}{{sfn|Godement|1998a}}{{sfn|Lang|1969}}{{sfn|Cartan|1971}}

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

:<math>\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)</math>

for some <math>\theta</math> and <math>\theta^\prime</math> in <math>(0,1)</math>. 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 {{mvar|F}} on a compact rectangle {{math|[''a'',''b''] × [''c'',''d'']}} are easily established.{{sfn|Titchmarsh|1939|p={{pn|date=August 2021}}}} The [[uniformly continuous|uniform continuity]] of {{mvar|F}} implies immediately that the functions <math>g(x)=\int_c^d F(x,y)\, dy</math> and <math>h(y)=\int_a^b F(x,y)\, dx</math> are continuous.{{sfn|Titchmarsh|1939|pages=23–25}} It follows that

:<math>\int_a^b \int_c^d F(x,y) \, dy\, dx = \int_c^d \int_a^b F(x,y) \, dx \, dy</math>;

moreover it is immediate that the [[iterated integral]] is positive if {{mvar|F}} is positive.{{sfn|Titchmarsh|1939|pages=49–50}} The equality above is a simple case of [[Fubini's theorem]], involving no [[measure theory]]. {{harvtxt|Titchmarsh|1939}} proves it in a straightforward way using [[Riemann sums#Higher dimensions|Riemann approximating sums]] corresponding to subdivisions of a rectangle into smaller rectangles.

To prove Clairaut's theorem, assume {{mvar|f}} is a differentiable function on an open set {{mvar|U}}, for which the mixed second partial derivatives {{math|''f''<sub>''yx''</sub>}} and {{math|''f''<sub>''xy''</sub>}} exist and are continuous. Using the [[fundamental theorem of calculus]] twice,

:<math>\int_c^d \int_a^b f_{yx}(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).</math>

Similarly

:<math>\int_a^b \int_c^d f_{xy}(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).</math>

The two iterated integrals are therefore equal. On the other hand, since {{math|''f''<sub>''xy''</sub>(''x'',''y'')}} is continuous, the second iterated integral can be performed by first integrating over {{mvar|x}} and then afterwards over {{mvar|y}}. But then the iterated integral of {{math|''f''<sub>''yx''</sub> − ''f''<sub>''xy''</sub>}} on {{math|[''a'',''b''] × [''c'',''d'']}} must vanish. However, if the iterated integral of a continuous function function {{mvar|F}} vanishes for all rectangles, then {{mvar|F}} must be identically zero; for otherwise {{mvar|F}} or {{math|−''F''}} would be strictly positive at some point and therefore by continuity on a rectangle, which is not possible. Hence {{math|''f''<sub>''yx''</sub> − ''f''<sub>''xy''</sub>}} must vanish identically, so that {{math|1=''f''<sub>''yx''</sub> = ''f''<sub>''xy''</sub>}} everywhere.{{sfn|Spivak|1965|page=61}}{{sfn|McGrath|2014}}{{sfn|Aksoy|Martelli|2002}}{{sfn|Axler|2020|pages=142–143}}<ref>{{citation|first=Donald E.|last=Marshall|title=Theorems of Fubini and Clairaut |publisher=University of Washington |url=https://sites.math.washington.edu/~marshall/math_136/FubiniClairaut.pdf}}</ref>

== 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]].{{sfn|Hubbard|Hubbard|2015|pages=732–733}} 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_1 f(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>\left(x_0,\, y_0\right) \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>\left(x_0,\, y_0\right)</math> and <math>\partial_{1,2}f\left(x_0,\, y_0\right) = \partial_{2,1}f\left(x_0,\, y_0\right)</math>.''{{sfn|Rudin|1976|pages=235–236}}

== 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>\left(D_1 D_2 f\right)[\phi] = -\left(D_2f\right)\left[D_1\phi\right] = f\left[D_2 D_1\phi\right] = f\left[D_1 D_2\phi\right] = -\left(D_1 f\right)\left[D_2\phi\right] = \left(D_2 D_1 f\right)[\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.{{efn|name="Schwartz"}}

== 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'',&thinsp;''y''), as shown in equation ({{EquationNote|1}}), does not have symmetric second derivatives at its origin.]]
An example of non-symmetry is the function (due to [[Peano]]){{sfn|Hobson|1921|pages=403–404}}{{sfn|Apostol|1974|pages=358–359}}
{{NumBlk|:
| <math>f(x,\, y) = \begin{cases}
  \frac{xy\left(x^2 - y^2\right)}{x^2 + y^2} & \mbox{ for } (x,\, y) \ne (0,\, 0),\\
                                           0 & \mbox{ for } (x,\, y) = (0,\, 0).
\end{cases}</math>
| {{EquationRef|1}}
}}

This can be visualized by the polar form <math>f(r \cos(\theta), r\sin(\theta)) = \frac{r^2 \sin(4\theta)}{4}</math>; it is everywhere continuous, but its derivatives at {{nowrap|(0,&thinsp;0)}} cannot be computed algebraically. Rather, the limit of difference quotients shows that <math>f_x(0,0) = f_y(0,0) = 0</math>, so the graph <math>z = f(x, y)</math> has a horizontal tangent plane at {{nowrap|(0,&thinsp;0)}}, and the partial derivatives <math>f_x, f_y</math> exist and are everywhere continuous. However, the second partial derivatives are not continuous at {{nowrap|(0,&thinsp;0)}}, and the symmetry fails. In fact, along the ''x''-axis the ''y''-derivative is <math>f_y(x,0) = x</math>, and so:
:<math>
  f_{yx}(0,0) =
  \lim_{\varepsilon \to 0} \frac{f_y(\varepsilon,0) - f_y(0,0)}{\varepsilon} =
  1.
</math>

In contrast, along the ''y''-axis the ''x''-derivative <math>f_x(0,y) = -y</math>, and so <math>f_{xy}(0,0) = -1</math>. That is, <math>f_{yx} \ne f_{xy}</math> at {{nowrap|(0,&thinsp;0)}}, 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
:<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 (0,&thinsp;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 [[commutative property|commute]]. Given two variables near {{nowrap|(0,&thinsp;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 {{nowrap|(0,&thinsp;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 (geometry)|translations]] 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>.{{sfn|Tu|2010}}

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. <math>A\,dx + B\,dy</math>, where <math>A</math> and <math>B</math> 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]]s. Formally Clairaut showed that a 1-form <math>\omega = A \, dx + B \, dy</math> on an open rectangle is closed, i.e. <math>d\omega=0</math>, if and only <math>\omega</math> has the form <math>df</math> for some function <math>f</math> in the disk. The solution for <math>f</math> can be written by Cauchy's integral formula

:<math>f(x,y)=\int_{x_0}^x A(x,y)\, dx + \int_{y_0} ^y B(x,y)\, dy;</math>

while if <math> \omega= df</math>, the closed property <math> d\omega=0</math> is the identity <math>\partial_x\partial_y f = \partial_y\partial_x f</math>. (In modern language this is one version of the [[Poincaré lemma]].){{sfn|Katz|1981}}

== Notes ==
{{notelist}}
{{Reflist|24em}}

== References ==
*{{citation|first1=A.|last1= Aksoy|first2= M.|last2= Martelli|title=Mixed Partial Derivatives and Fubini's Theorem|journal= College Mathematics Journal of MAA|volume=33|year= 2002|issue= 2| pages=126–130|doi= 10.1080/07468342.2002.11921930 |s2cid= 124561972|url=https://scholarship.claremont.edu/cgi/viewcontent.cgi?referer=https://www.google.com/&httpsredir=1&article=1580&context=cmc_fac_pub}}
*{{cite book |last=Allen |first=R. G. D. |title=Mathematical Analysis for Economists |location=New York |publisher=St. Martin's Press |year=1964 |isbn=9781443725224 |url=https://books.google.com/books?id=fgm9O6reUcsC}}
*{{citation|title=Mathematical analysis: a modern approach to advanced calculus|first=Tom M.|last=Apostol |author-link=Tom M. Apostol|place=London|publisher=Addison-Wesley|year=1965|oclc=901554874}}
*{{citation|title=Mathematical Analysis|first=Tom M.|last= Apostol|author-link=Tom M. Apostol|publisher=Addison-Wesley|year= 1974| isbn=9780201002881}}
*{{citation|last=Axler|first=Sheldon|title=Measure, integration & real analysis|series=Graduate Texts in Mathematics|volume=282|publisher=Springer|year=2020|isbn=9783030331436}}
*{{citation|last=Bourbaki|first= Nicolas|author-link= Nicolas Bourbaki|title=Eléments de mathématique, Livre VI: Intégration| chapter=Chapitre III: Mesures sur les espaces localement compacts|language=fr|publisher= Hermann et Cie|year= 1952}}
*{{citation|title=A First Course in Mathematical Analysis|first=J. C.|last= Burkill|author-link=J. C. Burkill|publisher=[[Cambridge University Press]]|year= 1962|isbn=9780521294683}} (reprinted 1978)
*{{citation|title=Calcul Differentiel|language=fr|first=Henri|last= Cartan|author-link= Henri Cartan|publisher=[[Éditions Hermann|Hermann]]|year= 1971|isbn=9780395120330}}
*{{citation|journal=Mémoires de l'Académie Royale des Sciences|author-link=Alexis Clairaut|first= A. C.|last=Clairaut|year=1739|title= Recherches générales sur le calcul intégral|pages=425–436|url=https://www.biodiversitylibrary.org/item/87741#page/563/mode/1up}}
*{{citation|last=Clairaut|first= A. C.|author-link=Alexis Clairaut|year=1740|title=Sur l'integration ou la construction des equations différentielles du premier ordre|journal=Mémoires de l'Académie Royale des Sciences|volume=2 |pages=293–323|url=https://www.biodiversitylibrary.org/item/87743#page/447/mode/1up}}
*{{citation|last=Dieudonné|first= J. |author-link=Jean Dieudonné|year=1937|title= Sur les fonctions continues numérique définies dans une produit de deux espaces compacts|journal= Comptes Rendus de l'Académie des Sciences de Paris|volume= 205|pages= 593–595}}
*{{citation|first=J.|last= Dieudonné|author-link=Jean Dieudonné|title= Foundations of Modern Analysis |series=Pure and Applied Mathematics|volume=10|year=1960|publisher= Academic Press|isbn=9780122155505}}
*{{citation|last=Dieudonné|first=J.|author-link=Jean Dieudonné|title=Treatise on analysis. Vol. II. |translator= I. G. Macdonald|translator-link=I. G. Macdonald|series= Pure and Applied Mathematics|volume= 10-II |publisher=Academic Press |year= 1976|isbn=9780122155024}}
*{{cite journal |last=Euler |first=Leonhard |year=1740|title=De infinitis curvis eiusdem generis seu methodus inveniendi aequationes pro infinitis curvis eiusdem generis |language=la |trans-title=On infinite(ly many) curves of the same type, that is, a method of finding equations for infinite(ly many) curves of the same type |journal=Commentarii Academiae Scientiarum Petropolitanae |volume=7 |pages=174-189, 180-183 |url=https://scholarlycommons.pacific.edu/euler-works/44/ |via=The Euler Archive, maintained by the University of the Pacific}}
*{{citation|last1=Gilkey|first1=Peter|last2= Park|first2=JeongHyeong|last3= Vázquez-Lorenzo|first3= Ramón |title=Aspects of differential geometry I|series=Synthesis Lectures on Mathematics and Statistics|volume= 15|publisher=Morgan & Claypool |year=2015|isbn=9781627056632}}
*{{citation|title=Analyse mathématique I|first=Roger|last=Godement|author-link=Roger Godement|publisher=Springer|year=1998a|url=https://archive.org/details/ananlyse-mathematique-tome-i-ii-iii-iv-roger-godemet}} 
*{{citation|title=Analyse mathématique II|first=Roger|last=Godement|author-link=Roger Godement|publisher=Springer|year=1998b|url=https://archive.org/details/ananlyse-mathematique-tome-i-ii-iii-iv-roger-godemet}}
*{{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|access-date=2017-04-19|archive-url=https://web.archive.org/web/20170419195440/http://mathforum.org/kb/thread.jspa?forumID=13&threadID=1580215&messageID=5765194|archive-date=2017-04-19}}
*{{citation|last=Hobson|first= E. W.|author-link=E. W. Hobson|title=The theory of functions of a real variable and the theory of Fourier's series. Vol. I.|publisher=[[Cambridge University Press]]|year=1921}}
*{{citation|title=The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis|series=Classics in Mathematics|first=Lars|last=Hörmander|author-link=Lars Hörmander|year=2015 |publisher=Springer |edition=2nd|isbn=9783642614972}}
*{{cite book|last1=Hubbard|first1=John|last2=Hubbard|first2=Barbara|author1-link=John H. Hubbard|author2-link=Barbara Burke Hubbard|year=2015|title=Vector Calculus, Linear Algebra and Differential Forms|publisher=Matrix Editions|edition=5th|isbn=9780971576681 |url=http://matrixeditions.com/5thUnifiedApproach.html}}
*{{cite book |last=James |first=R. C. |year=1966 |title=Advanced Calculus |location=Belmont, CA |publisher=Wadsworth |url=https://books.google.com/books?id=d5kpAQAAMAAJ}}
*{{citation|last=Jordan|first= Camille|author-link=Camille Jordan|title= Cours d'analyse de l'École polytechnique. Tome I. Calcul différentiel (Les Grands Classiques Gauthier-Villars) |publisher= Éditions Jacques Gaba]| year=1893}}
*{{citation|journal=[[Historia Mathematica]]|volume= 8|year=1981|pages =161–188|title=The history of differential forms from Clairaut to Poincaré|first=Victor J.|last=Katz|issue= 2|doi= 10.1016/0315-0860(81)90027-6|doi-access=}}
*{{citation|title=Real Analysis|publisher=[[Addison-Wesley]]|first=Serge|last= Lang|year=1969 |isbn= 0201041790|author-link=Serge Lang}}
*{{citation|last=Lindelöf|first=L. L.|author-link=Lorenz Leonard Lindelöf|title=Remarques sur les différentes manières d'établir la formule d<sup>2</sup> z/dx dy = d<sup>2</sup> z/dy dx|journal=Acta Societatis Scientiarum Fennicae| volume= 8|year=1867|pages= 205–213|url=https://www.biodiversitylibrary.org/item/48435#page/225/mode/1up}}
*{{citation|last= Loomis|first= Lynn H.|author-link=Lynn Loomis| title=An introduction to abstract harmonic analysis|publisher= D. Van Nostrand |year= 1953|hdl= 2027/uc1.b4250788|hdl-access= free}} 
*{{citation|last=McGrath|first= Peter J.|title=Another proof of Clairaut's theorem|journal=[[Amer. Math. Monthly]]|volume= 121|year=2014|issue= 2|pages= 165–166|doi= 10.4169/amer.math.monthly.121.02.165|s2cid= 12698408}}
*{{cite journal|first1=E.|last1=Minguzzi|title=The equality of mixed partial derivatives under weak differentiability conditions|journal=Real Analysis Exchange|year=2015|volume=40|pages=81–98 |doi=10.14321/realanalexch.40.1.0081 |s2cid=119315951|arxiv=1309.5841}}
*{{citation|last=Nachbin|first= Leopoldo|author-link=Leopoldo Nachbin|title= Elements of approximation theory|series= Notas de Matemática|volume = 33 |publisher=Fascículo publicado pelo Instituto de Matemática Pura e Aplicada do Conselho Nacional de Pesquisas|location= Rio de Janeiro|year= 1965}}
*{{citation|title=Principles of Mathematical Analysis|first=Walter|last=Rudin|author-link=Walter Rudin |publisher=McGraw-Hill|year=1976|series=International Series in Pure & Applied Mathematics|isbn=0-07-054235-X |url=https://archive.org/details/1979RudinW}}
*{{citation|last=Sandifer|first=C. Edward|year=2007|chapter=Mixed partial derivatives are equal |chapter-url=https://books.google.com/books?id=3-DyDwAAQBAJ&pg=PA142 |title=The Early Mathematics of Leonhard Euler, Vol. 1 |publisher=Mathematics Association of America|isbn=9780883855591}}
*{{citation|last=Schwarz|first= H. A.|author-link=Hermann Schwarz|title=Communication|journal=Archives des Sciences Physiques et Naturelles|volume=  48 |year=1873|pages= 38–44 |url=https://www.biodiversitylibrary.org/item/27383#page/37/mode/1up}}
*{{citation|last= Spivak|first= Michael|author-link=Michael Spivak|title=  Calculus on manifolds. A modern approach to classical theorems of advanced calculus|publisher= W. A. Benjamin|year=1965}}
*{{citation|last=Tao|first= Terence|author-link=Terence Tao|title=Analysis II|series=Texts and Readings in Mathematics|volume= 38| publisher=Hindustan Book Agency|year= 2006|doi= 10.1007/978-981-10-1804-6|isbn=  8185931631 |url=https://link.springer.com/content/pdf/10.1007%2F978-981-10-1804-6.pdf}}
*{{citation|last=Titchmarsh|first=E. C.|author-link=E. C. Titchmarsh|title=The Theory of Functions |edition=2nd|publisher=[[Oxford University Press]]|year=1939}}
*{{citation|last=Tu|first=Loring W.|year=2010|title=An Introduction to Manifolds|publisher=Springer |isbn=978-1-4419-7399-3|edition=2nd|place=New York |url=https://archive.org/details/TuL.W.AnIntroductionToManifolds2e2010Springer}}

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

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