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

In summary, Browder is discussing differentiable maps in Chapter 8 of his book "Mathematical Analysis: An Introduction". In particular, he focuses on Section 8.2 where he defines differentials and provides some remarks in Definition 8.9. These remarks state that for any fixed vector k and a positive value t, we have an equation involving L(tk) and M(tk). In order to understand these remarks better, the reader asks two questions. First, why does Browder put h = tk and let t approach 0? The answer is that by doing this, he is able to make a distinction between fixed and variable vectors and show that the equation holds for all t approaching 0. Second, how does
  • #1
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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 \(\displaystyle k \neq 0\) 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
 

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  • #2
Peter said:
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 $Lk-Mk = 0$.
 
  • #3
Opalg said:
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
 

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

1. What is the meaning of "differentials in R^n" in the context of Browder's work?

"Differentials in R^n" refers to the study of differential equations in n-dimensional space, as discussed in Section 8.2 of Browder's work. This involves analyzing how functions change over time in multiple dimensions.

2. How does Browder define differentials in R^n?

Browder defines differentials in R^n as the study of differential equations in n-dimensional space, which involves the use of partial derivatives and vector calculus to understand how functions change over time in multiple dimensions.

3. What is the significance of studying differentials in R^n?

Studying differentials in R^n is important because it allows us to understand and predict how systems in multiple dimensions will change over time. This has applications in various fields such as physics, engineering, and economics.

4. What is the main focus of Section 8.2 in Browder's work?

The main focus of Section 8.2 in Browder's work is on the application of differentials in R^n to solve differential equations in multiple dimensions. It covers topics such as partial derivatives, gradient vectors, and the chain rule.

5. Are there any real-world examples of differentials in R^n?

Yes, there are many real-world examples of differentials in R^n, such as predicting the movement of celestial bodies in space, analyzing the flow of fluids in pipes, and modeling the spread of diseases in a population. These examples all involve understanding how systems change over time in multiple dimensions.

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