1. Clairaut's Theorem Clarification

Hi

I'm trying to learn this specific proof on the Equality of Mixed Partials and I pretty much understand the proof, but I need one clarification.

The proof is posted here:
http://homepage.smc.edu/nestler_andr...11clairaut.pdf

I am confused on what $\displaystyle \Delta(h)$ means. At first I thought it was the change in h, but the parenthesis lead me to believe that it stands for a function of h. It would make sense to me if this was actually just a function of h, but I am not one-hundred percent sure.

Could someone tell me what $\displaystyle \Delta(h)$ represents?

Thank you.

2. Originally Posted by Anthonny
Hi

I'm trying to learn this specific proof on the Equality of Mixed Partials and I pretty much understand the proof, but I need one clarification.

The proof is posted here:
http://homepage.smc.edu/nestler_andr...11clairaut.pdf

I am confused on what $\displaystyle \Delta(h)$ means. At first I thought it was the change in h, but the parenthesis lead me to believe that it stands for a function of h. It would make sense to me if this was actually just a function of h, but I am not one-hundred percent sure.

Could someone tell me what $\displaystyle \Delta(h)$ represents?

Thank you.
In the second line of the proof, we're told that $\displaystyle \Delta(h)=(f(a+h,b+h)-f(a+h,b))-(f(a,b+h)-f(a,b))$...

3. $\displaystyle \Delta(h)$ is not the change in h. $\displaystyle \Delta(h)$ is a function of h. So, yes, you are right.