Proving a theorem: Changing the Order of Differentiation (large question)
This question is right out of Taylor & Mann Advanced Calculus, Third Edition, and is a REALLY tricky one. Due to how context-heavy this is, I can't write it word-for-word.
We're to prove the following theorem:
Let and its first partial derivatives be defined in a neighborhood of the point , and suppose that and are differentiable at that point. Then .
This theorem uses part of another theorem's proof to start things off, before getting to the parts that I can't figure out.
Let be a number different from zero such that the point is inside a square having its center at . We then consider the following expression:
If we introduce the function
we can express in the form
Now has the derivative
Hence is continuous, and we may apply the mean-value theorem for derivatives to (*), obtaining the following:
That ends the part of the separate proof; now to parts which I'm in the dark about (these are from the actual question).
From the fact that is differentiable at , one can write
, where as .
Explain why this is so.
Next, go on to explain how to obtain the following expression:
where as .
Explain the derivation of the similar expression
where as ,
using the fact that is differentiable at .
With all this, complete the proof of the theorem.
This is a lot of info to work with, I know. But I can't summarize it any better than this because the question has such a high context requirement.