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I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ...
I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.2 Differentials ... ...
I need some further help in fully understanding some remarks by Browder made after Definition 8.9 ...
Definition 8.9 and the following remark read as follows:
View attachment 9408
In the above Remark by Browder we read the following:
"for any fixed $$k \neq 0$$ and $$t \gt 0$$, we have $$\frac{1}{ |tk| }( L(tk) - M(tk) ) = \frac{1}{|k|},(Lk - Mk )$$ ... ... ... "
My questions are as follows:Question 1
Browder puts $$h = tk$$ and then let's $$t \to 0$$ ... why is Browder doing this ... what is the logic behind this ... what do we gain by putting $$h = tk$$ ... both $$h$$ and $$k \in \mathbb{R}^n $$ and also isn't $$h$$ just as arbitrary as $$k$$ ... ?
Question 2
How exactly (and in detail) does letting $$t \to 0$$ allow us to conclude that $$Lk = Mk$$ ...
Help will be much appreciated ...
Peter
I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.2 Differentials ... ...
I need some further help in fully understanding some remarks by Browder made after Definition 8.9 ...
Definition 8.9 and the following remark read as follows:
View attachment 9408
In the above Remark by Browder we read the following:
"for any fixed $$k \neq 0$$ and $$t \gt 0$$, we have $$\frac{1}{ |tk| }( L(tk) - M(tk) ) = \frac{1}{|k|},(Lk - Mk )$$ ... ... ... "
My questions are as follows:Question 1
Browder puts $$h = tk$$ and then let's $$t \to 0$$ ... why is Browder doing this ... what is the logic behind this ... what do we gain by putting $$h = tk$$ ... both $$h$$ and $$k \in \mathbb{R}^n $$ and also isn't $$h$$ just as arbitrary as $$k$$ ... ?
Question 2
How exactly (and in detail) does letting $$t \to 0$$ allow us to conclude that $$Lk = Mk$$ ...
Help will be much appreciated ...
Peter