Using differentiation under integral sign to proof equality of mixed partials

[Castilla]

Hello. Excuse my english, it is not my natal tongue. I have a question about real functions of two real variables.

Does someone knows how to use differentiation under the integral sign to proof the equality of mixed partials ?

I have read said proof in a book of an english author, David Stirling (Mathematical analysis: a fundamental and straightforward approach), and it would be great if someone here could throw some light on it...

Regards,

Castilla.

 

[Galileo]

Hi Castilla,

I remember starting a somewhat related thread about a month ago, you may want to read through it as well.
https://www.physicsforums.com/showthread.php?t=36033

I can be shown easily if you may use Fubini's theorem (interchanging the order of integration), but it seems silly to use that, since it is harder to prove than Clairaut's theorem (equality of mixed partial dervatives), which has a very elementary proof that doesn't require integration.


PS: Welcome to the Physics Forums!

 

[Castilla]

Hi, Galileo, I had already read the thread. You used the equality of mixed partials as a tool to proof the Leibniz Integral Rule (which I first called "differentiation under the integral sign").

But what I am talking about is rather the inverse. How to use the Leibniz Integral Rule to proof the equality of mixed partials. I have found the proof in a book ("Mathematical Analysis: a fundamental and straightforward approach", by David Stirling) but I can not understand every detail of it. Well, I am neither a mathematician nor an engineer.

Regards,

Castilla.