Multivariable Analysis .... Directional and Partial Derivatives .... D&K Propostion 2.3.3 ....

[Math Amateur]

I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ...

I am focused on Chapter 2: Differentiation ... ...

I need help with an aspect of the proof of Proposition 2.3.2 ... ...

Duistermaat and Kolk's Proposition 2.3.2 and its proof read as follows:
View attachment 7847
https://www.physicsforums.com/attachments/7848
In the above proof by D&K we read the following:

" ... ... Assertion (i) follows from Formula (2.11). ... ..."Can someone please demonstrate (formally and rigorously) that this is the case ... that is that assertion (i) follows from Formula (2.11). ... ...Help will be appreciated ...

Peter==========================================================================================***NOTE***

It may help readers of the above post to have access to the start of Section "2.3: Directional and Partial Derivatives" ... in order to understand the context and notation of the post ... so I am providing the same ... as follows:https://www.physicsforums.com/attachments/7849Hope that the above helps readers of the post understand the context and notation of the post ...

Peter

 

[GJA]

Hi, Peter.

" ... ... Assertion (i) follows from Formula (2.11). ... ..."

Can someone please demonstrate (formally and rigorously) that this is the case ... that is that assertion (i) follows from Formula (2.11). ... ...

Try dividing the first line of (2.11) through by $t$. Now take the limit as $t\rightarrow 0$, use the definition of directional derivative and the second part of (2.11) to obtain the desired result.