Using differentiation under integral sign to proof equality of mixed partials

In summary, Castilla is asking for help understanding how to use differentiation under the integral sign to proof the equality of mixed partials in real functions of two real variables. They have read a proof in a book by an English author, David Stirling, and are hoping for clarification from the community. They mention a related thread on the topic and express their difficulty in understanding the proof, as they are not a mathematician or engineer.
  • #1
Castilla
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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.
 
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  • #2
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 theorm (equality of mixed partial dervatives), which has a very elementary proof that doesn't require integration.


PS: Welcome to the Physics Forums! :cool:
 
  • #3
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.
 

What is the concept of using differentiation under integral sign to proof equality of mixed partials?

The concept involves using the property of differentiation under the integral sign to prove that the mixed partial derivatives of a function are equal. This property states that if a function f(x,y) is continuous and has continuous partial derivatives with respect to both x and y, then the order of differentiation does not matter and the mixed partial derivatives are equal.

Why is it useful to use differentiation under integral sign for proving equality of mixed partials?

Using this property simplifies the process of proving equality of mixed partials, as it eliminates the need for complicated algebraic manipulations. It also allows for a more elegant and concise proof compared to other methods.

What are the assumptions required for using differentiation under integral sign to proof equality of mixed partials?

The main assumption is that the function must be continuous and have continuous partial derivatives with respect to both x and y. This is also known as the continuity and differentiability criteria for using this property.

Can differentiation under integral sign be used for all functions to prove equality of mixed partials?

No, this property can only be applied to functions that meet the assumptions mentioned earlier. If the function is not continuous or does not have continuous partial derivatives, then this method cannot be used to prove equality of mixed partials.

Are there any limitations to using differentiation under integral sign for proving equality of mixed partials?

One limitation is that this method can only be applied to functions with two independent variables. It also does not work for functions with discontinuous partial derivatives or functions with more than two independent variables.

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