Symmetry of second derivatives#Schwarz.27s theorem

In symbols, the symmetry may be expressed as:

:$\backslash frac\; \{\backslash partial\}\{\backslash partial\; x\}\; \backslash left(\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\; y\}\; \backslash right)\; \backslash \; =\; \backslash$

\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}.

Another notation is:

:$\backslash partial\_x\backslash partial\_y\; f\; =\; \backslash partial\_y\backslash partial\_x\; f\; \backslash qquad\backslash text\{or\}\backslash qquad\; f\_\{yx\}\; =\; f\_\{xy\}.$

In terms of composition of the differential operator {{math|D_i}} which takes the partial derivative with respect to {{math|x_i}}:

:$D\_i\; \backslash circ\; D\_j\; =\; D\_j\; \backslash circ\; D\_i$.

From this relation it follows that the ring of differential operators with constant coefficients, generated by the {{math|D_i}}, 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 monomials, so that one can take polynomials in the {{math|x_i}} as a domain. In fact smooth functions are another valid domain.

The result on the equality of mixed partial derivatives under certain conditions has a long history. The list of unsuccessful proposed proofs started with Euler's, published in 1740,{{sfn|Euler|1740}} although already in 1721 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.}} 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 $\backslash tfrac\{\backslash partial^2\; f\}\{\backslash partial\; x^2\}$ and $\backslash tfrac\{\backslash partial^2\; f\}\{\backslash partial\; y^2\}$.{{sfn|Minguzzi|2015}} Other attempts were made by P. Blanchet (1841), Duhamel (1856), Sturm (1857), Schlömilch (1862), and Bertrand (1864). Finally in 1867 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, Schwarz succeeded in giving the first rigorous proof.{{sfn|Schwarz|1873}} Dini later contributed by finding more general conditions than those of Schwarz. Eventually a clean and more general version was found by Jordan in 1883 that is still the proof found in most textbooks. Minor variants of earlier proofs were published by Laurent (1885), Peano (1889 and 1893), J. Edwards (1892), P. Haag (1893), J. K. Whittemore (1898), Vivanti (1899) and 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}}

{{redirect|Schwarz's theorem|the result in complex analysis|Schwarz lemma}}

In mathematical analysis, Schwarz's theorem (or Clairaut's theorem on equality of mixed partials){{sfn|James|1966|p={{pn|date=August 2021}}}} named after Alexis Clairaut and Hermann Schwarz, states that for a function $f\; \backslash colon\; \backslash Omega\; \backslash to\; \backslash mathbb\{R\}$ defined on a set $\backslash Omega\; \backslash subset\; \backslash mathbb\{R\}^n$, if $\backslash mathbf\{p\}\backslash in\; \backslash mathbb\{R\}^n$ is a point such that some neighborhood of $\backslash mathbf\{p\}$ is contained in $\backslash Omega$ and $f$ has continuous second partial derivatives on that neighborhood of $\backslash mathbf\{p\}$, then for all {{mvar|i}} and {{mvar|j}} in $\backslash \{1,\; 2\; \backslash ldots,\backslash ,\; n\backslash \},$

:$$

\frac{\partial^2}{\partial x_i\, \partial x_j}f(\mathbf{p}) =

\frac{\partial^2}{\partial x_j\, \partial x_i}f(\mathbf{p}).

The partial derivatives of this function commute at that point.

There exists a version of this theorem where $f$ is only required to be twice differentiable at the point $\backslash mathbf\{p\}$.

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 $f.$

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).{{harvnb|Burkill|1962|pages=154–155}} Let $f(x,y)$ be a differentiable function on an open rectangle $\backslash Omega$ containing a point $(a,b)$ and suppose that $df$ is continuous with continuous $\backslash partial\_x\; \backslash partial\; \_y\; f$ and $\backslash partial\_y\backslash partial\_x\; f$ over $\backslash Omega.$ Define

:$\backslash begin\{align\}$

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}

These functions are defined for , where $\backslash varepsilon\; >\; 0$ and $\backslash left[a\; -\; \backslash varepsilon,\backslash ,\; a\; +\; \backslash varepsilon\backslash right]\; \backslash times\; \backslash left[b\; -\; \backslash varepsilon,\backslash ,\; b\; +\; \backslash varepsilon\backslash right]$ is contained in $\backslash Omega.$

By the mean value theorem, for fixed {{mvar|h}} and {{mvar|k}} non-zero, $\backslash theta,\; \backslash theta\text{'},\; \backslash phi,\; \backslash phi\text{'}$ can be found in the open interval $(0,1)$ with

:$\backslash begin\{align\}$

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}

Since $h,\backslash ,k\; \backslash neq\; 0$, the first equality below can be divided by $hk$:

:$\backslash begin\{align\}$

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}

Letting $h,\backslash ,k$ tend to zero in the last equality, the continuity assumptions on $\backslash partial\_y\backslash partial\_x\; f$ and $\backslash partial\_x\backslash partial\_y\; f$ now imply that

:$$

\frac{\partial^2}{\partial x\partial y}f\left(a,\, b\right) =

\frac{\partial^2}{\partial y\partial x}f\left(a,\, b\right).

This account is a straightforward classical method found in many text books, for example in Burkill, Apostol and Rudin.{{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 operators $\backslash Delta^t\_x,\backslash ,\backslash ,\backslash Delta^t\_y$ commute and $\backslash Delta^t\_x\; f,\backslash ,\backslash ,\backslash Delta^t\_y\; f$ tend to $\backslash partial\_x\; f,\backslash ,\backslash ,\; \backslash partial\_y\; f$ as $t$ 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 functions on the plane. Under Fourier transform, the difference and differential operators are just multiplication operators.{{sfn|Hörmander|2015|loc=Chapter VII}}}} Here, for $z$ a vector in the plane and $u$ a directional vector $\backslash tbinom\{1\}\{0\}$ or $\backslash tbinom\{0\}\{1\}$, the difference operator is defined by

:$\backslash Delta^t\_u\; f(z)=\; \{f(z+tu)\; -\; f(z)\backslash over\; t\}.$

By the fundamental theorem of calculus for $C^1$ functions $f$ on an open interval $I$ with $(a,b)\; \backslash subset\; I$

:$\backslash int\_a^b\; f^\backslash prime\; (x)\; \backslash ,\; dx\; =\; f(b)\; -\; f(a).$

Hence

:$|f(b)\; -\; f(a)|\; \backslash le\; (b-a)\backslash ,\; \backslash sup\_\{c\backslash in\; (a,b)\}\; |f^\backslash prime(c)|$.

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 $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then there is a point $c$ in $(a,b)$ such that

:$\{f(b)\; -\; f(a)\; \backslash over\; b\; -\; a\}\; =\; f^\backslash prime(c).$

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

:$\backslash |f(b)\; -\; f(a)\backslash |\; \backslash le\; (b-a)\backslash ,\; \backslash sup\_\{c\backslash in\; (a,b)\}\; \backslash |f^\backslash prime(c)\backslash |$.

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 $f$ a $C^2$ function on an open set in the plane, define $D\_1\; =\; \backslash partial\_x$ and $D\_2\; =\; \backslash partial\_y$. Furthermore for $t\; \backslash ne\; 0$ set

:$\backslash Delta\_1^t\; f(x,y)\; =\; [f(x+t,y)-f(x,y)]/t,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash Delta^t\_2f(x,y)=[f(x,y+t)\; -f(x,y)]/t$.

Then for $(x\_0,y\_0)$ in the open set, the generalized mean value theorem can be applied twice:

:$\backslash left|\backslash Delta\_1^t\backslash Delta\_2^t\; f(x\_0,y\_0)\; -\; D\_1\; D\_2f(x\_0,y\_0)\backslash right|\backslash le\; \backslash sup\_\{0\backslash le\; s\; \backslash le\; 1\}\; \backslash left|\backslash Delta\_1^t\; D\_2\; f(x\_0,y\_0\; +\; ts)\; -D\_1D\_2\; f(x\_0,y\_0)\backslash right|\backslash le\; \backslash sup\_\{0\backslash le\; r,s\backslash le\; 1\}\; \backslash left|D\_1D\_2f(x\_0+tr,y\_0+ts)\; -\; D\_1D\_2f(x\_0,y\_0)\backslash right|.$

Thus $\backslash Delta\_1^t\backslash Delta\_2^t\; f(x\_0,y\_0)$ tends to $D\_1\; D\_2f(x\_0,y\_0)$ as $t$ tends to 0. The same argument shows that $\backslash Delta\_2^t\backslash Delta\_1^t\; f(x\_0,y\_0)$ tends to $D\_2\; D\_1f(x\_0,y\_0)$. Hence, since the difference operators commute, so do the partial differential operators $D\_1$ and $D\_2$, 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,

:$\backslash Delta\_1^t\backslash Delta\_2^t\; f(x\_0,y\_0)=\; D\_1\; D\_2\; f(x\_0+t\backslash theta,y\_0\; +t\backslash theta^\backslash prime)$

for some $\backslash theta$ and $\backslash theta^\backslash prime$ in $(0,1)$. 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.

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 uniform continuity of {{mvar|F}} implies immediately that the functions $g(x)=\backslash int\_c^d\; F(x,y)\backslash ,\; dy$ and $h(y)=\backslash int\_a^b\; F(x,y)\backslash ,\; dx$ are continuous.{{sfn|Titchmarsh|1939|pages=23–25}} It follows that

:$\backslash int\_a^b\; \backslash int\_c^d\; F(x,y)\; \backslash ,\; dy\backslash ,\; dx\; =\; \backslash int\_c^d\; \backslash int\_a^b\; F(x,y)\; \backslash ,\; dx\; \backslash ,\; dy$;

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 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_yx}} and {{math|f_xy}} exist and are continuous. Using the fundamental theorem of calculus twice,

:$\backslash int\_c^d\; \backslash int\_a^b\; f\_\{yx\}(x,y)\; \backslash ,\; dx\; \backslash ,\; dy\; =\; \backslash int\_c^d\; f\_y(b,y)\; -\; f\_y(a,y)\; \backslash ,\; dy\; =\; f(b,d)-f(a,d)-f(b,c)+f(a,c).$

Similarly

:$\backslash int\_a^b\; \backslash int\_c^d\; f\_\{xy\}(x,y)\; \backslash ,\; dy\; \backslash ,\; dx\; =\; \backslash int\_a^b\; f\_x(x,d)\; -\; f\_x(x,c)\; \backslash ,\; dx\; =\; f(b,d)-f(a,d)-f(b,c)+f(a,c).$

The two iterated integrals are therefore equal. On the other hand, since {{math|f_xy(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_yx − f_xy}} 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_yx − f_xy}} must vanish identically, so that {{math|1=f_yx = f_xy}} everywhere.{{sfn|Spivak|1965|page=61}}{{sfn|McGrath|2014}}{{sfn|Aksoy|Martelli|2002}}{{sfn|Axler|2020|pages=142–143}}{{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}}

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.{{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:

: If $f:E\; \backslash to\; \backslash mathbb\{R\}$ is defined on an open set $E\; \backslash subset\; \backslash R^2$; $\backslash partial\_1\; f(x,\backslash ,\; y)$ and $\backslash partial\_\{2,1\}f(x,\backslash ,\; y)$ exist everywhere on $E$; $\backslash partial\_\{2,1\}f$ is continuous at $\backslash left(x\_0,\backslash ,\; y\_0\backslash right)\; \backslash in\; E$, and if $\backslash partial\_\{2\}f(x,\backslash ,\; y\_0)$ exists in a neighborhood of $x\; =\; x\_0$, then $\backslash partial\_\{1,2\}f$ exists at $\backslash left(x\_0,\backslash ,\; y\_0\backslash right)$ and $\backslash partial\_\{1,2\}f\backslash left(x\_0,\backslash ,\; y\_0\backslash right)\; =\; \backslash partial\_\{2,1\}f\backslash left(x\_0,\backslash ,\; y\_0\backslash right)$.{{sfn|Rudin|1976|pages=235–236}}

The theory of 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 functions, 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),

: $\backslash left(D\_1\; D\_2\; f\backslash right)[\backslash phi]\; =\; -\backslash left(D\_2f\backslash right)\backslash left[D\_1\backslash phi\backslash right]\; =\; f\backslash left[D\_2\; D\_1\backslash phi\backslash right]\; =\; f\backslash left[D\_1\; D\_2\backslash phi\backslash right]\; =\; -\backslash left(D\_1\; f\backslash right)\backslash left[D\_2\backslash phi\backslash right]\; =\; \backslash left(D\_2\; D\_1\; f\backslash right)[\backslash phi].$

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"}}

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).

File:Graph001.png

An example of non-symmetry is the function (due to Peano){{sfn|Hobson|1921|pages=403–404}}{{sfn|Apostol|1974|pages=358–359}}

{{NumBlk|:

| $f(x,\backslash ,\; y)\; =\; \backslash begin\{cases\}$

\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}

| {{EquationRef|1}}

}}

This can be visualized by the polar form $f(r\; \backslash cos(\backslash theta),\; r\backslash sin(\backslash theta))\; =\; \backslash frac\{r^2\; \backslash sin(4\backslash theta)\}\{4\}$; it is everywhere continuous, but its derivatives at {{nowrap|(0, 0)}} cannot be computed algebraically. Rather, the limit of difference quotients shows that $f\_x(0,0)\; =\; f\_y(0,0)\; =\; 0$, so the graph $z\; =\; f(x,\; y)$ has a horizontal tangent plane at {{nowrap|(0, 0)}}, and the partial derivatives $f\_x,\; f\_y$ exist and are everywhere continuous. However, the second partial derivatives are not continuous at {{nowrap|(0, 0)}}, and the symmetry fails. In fact, along the x-axis the y-derivative is $f\_y(x,0)\; =\; x$, and so:

:$$

f_{yx}(0,0) =

\lim_{\varepsilon \to 0} \frac{f_y(\varepsilon,0) - f_y(0,0)}{\varepsilon} =

1.

In contrast, along the y-axis the x-derivative $f\_x(0,y)\; =\; -y$, and so $f\_\{xy\}(0,0)\; =\; -1$. That is, $f\_\{yx\}\; \backslash ne\; f\_\{xy\}$ at {{nowrap|(0, 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

:$f(r,\backslash ,\; \backslash theta)\; =\; \backslash frac\{r^2\; \backslash sin\{4\backslash theta\}\}\{4\},$

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, 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 commute. Given two variables near {{nowrap|(0, 0)}} and two limiting processes on

:$f(h,\backslash ,\; k)\; -\; f(h,\backslash ,\; 0)\; -\; f(0,\backslash ,\; k)\; +\; f(0,\backslash ,\; 0)$

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 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, 0)}}, there is no contradiction with the fact that the Hessian, viewed as a Schwartz distribution, is symmetric.

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 commute with each other, and therefore the infinitesimal generators do also; the Lie bracket

: [D_i, D_j] = 0

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

The Clairaut-Schwarz theorem is the key fact needed to prove that for every $C^\backslash infty$ (or at least twice differentiable) differential form $\backslash omega\backslash in\backslash Omega^k(M)$, the second exterior derivative vanishes: $d^2\backslash omega\; :=\; d(d\backslash omega)\; =\; 0$. This implies that every differentiable exact form (i.e., a form $\backslash alpha$ such that $\backslash alpha\; =\; d\backslash omega$ for some form $\backslash omega$) is closed (i.e., $d\backslash alpha\; =\; 0$), since $d\backslash alpha\; =\; d(d\backslash omega)\; =\; 0$.{{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. $A\backslash ,dx\; +\; B\backslash ,dy$, where $A$ and $B$ 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 equations. Formally Clairaut showed that a 1-form $\backslash omega\; =\; A\; \backslash ,\; dx\; +\; B\; \backslash ,\; dy$ on an open rectangle is closed, i.e. $d\backslash omega=0$, if and only $\backslash omega$ has the form $df$ for some function $f$ in the disk. The solution for $f$ can be written by Cauchy's integral formula

:$f(x,y)=\backslash int\_\{x\_0\}^x\; A(x,y)\backslash ,\; dx\; +\; \backslash int\_\{y\_0\}\; ^y\; B(x,y)\backslash ,\; dy;$

while if $\backslash omega=\; df$, the closed property $d\backslash omega=0$ is the identity $\backslash partial\_x\backslash partial\_y\; f\; =\; \backslash partial\_y\backslash partial\_x\; f$. (In modern language this is one version of the Poincaré lemma.){{sfn|Katz|1981}}

{{notelist}}

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Category:Multivariable calculus

Category:Generalized functions

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Category:Theorems in mathematical analysis