Differentials in R^n .... Another Remark by Browder, Section 8.2 .... ....

[Math Amateur]

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 \(\displaystyle k \neq 0\) and \(\displaystyle t \gt 0\), we have \(\displaystyle \frac{1}{ |tk| }( L(tk) - M(tk) ) = \frac{1}{|k|},(Lk - Mk )\) ... ... ... "
My questions are as follows:Question 1

Browder puts \(\displaystyle h = tk\) and then let's \(\displaystyle t \to 0\) ... why is Browder doing this ... what is the logic behind this ... what do we gain by putting \(\displaystyle h = tk\) ... both \(\displaystyle h\) and \(\displaystyle k \in \mathbb{R}^n \) and also isn't \(\displaystyle h\) just as arbitrary as \(\displaystyle k\) ... ?
Question 2

How exactly (and in detail) does letting \(\displaystyle t \to 0\) allow us to conclude that \(\displaystyle Lk = Mk\) ...
Help will be much appreciated ...

Peter

 

[Opalg]

In the above Remark by Browder we read the following:

"for any fixed \(\displaystyle k \neq 0\) and \(\displaystyle t \gt 0\), we have \(\displaystyle \frac{1}{ |tk| }( L(tk) - M(tk) ) = \frac{1}{|k|},(Lk - Mk )\) ... ... ... "
My questions are as follows:Question 1

Browder puts \(\displaystyle h = tk\) and then let's \(\displaystyle t \to 0\) ... why is Browder doing this ... what is the logic behind this ... what do we gain by putting \(\displaystyle h = tk\) ... both \(\displaystyle h\) and \(\displaystyle k \in \mathbb{R}^n \) and also isn't \(\displaystyle h\) just as arbitrary as \(\displaystyle k\) ... ?
Question 2

How exactly (and in detail) does letting \(\displaystyle t \to 0\) allow us to conclude that \(\displaystyle Lk = Mk\) ...

You need to make a clear distinction between fixed and variable vectors. In the equation \(\displaystyle \lim_{h\to0}\frac1{|h|}(Lh - Mh) = 0\), $h$ has to be a variable vector that converges to $0$. But $k$ is a fixed (nonzero) vector. if $h=tk$ then $h$ varies as $t$ varies. And as $t$ goes to $0$, $h$ also goes to $0$. So you can conclude that \(\displaystyle \lim_{t\to0}\frac1{|tk|}(L(tk) - M(tk)) = 0\). But \(\displaystyle \frac1{|tk|}(L(tk) - M(tk)) = \frac1{|k|}(Lk-Mk)\), which is constant. It follows that\(\displaystyle \frac1{|k|}(Lk-Mk) = 0\), and since $|k|$ is nonzero the conclusion is that $Lk-Mk = 0$.

 

[Math Amateur]

You need to make a clear distinction between fixed and variable vectors. In the equation \(\displaystyle \lim_{h\to0}\frac1{|h|}(Lh - Mh) = 0\), $h$ has to be a variable vector that converges to $0$. But $k$ is a fixed (nonzero) vector. if $h=tk$ then $h$ varies as $t$ varies. And as $t$ goes to $0$, $h$ also goes to $0$. So you can conclude that \(\displaystyle \lim_{t\to0}\frac1{|tk|}(L(tk) - M(tk)) = 0\). But \(\displaystyle \frac1{|tk|}(L(tk) - M(tk)) = \frac1{|k|}(Lk-Mk)\), which is constant. It follows that\(\displaystyle \frac1{|k|}(Lk-Mk) = 0\), and since $|k|$ is nonzero the conclusion is that $Lk-Mk = 0$.

Thanks Opalg ...

Appreciate your help ...

Peter