Editing Symmetry of second derivatives (section)

== 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>