Differentials/Total Derivatives in R^n .... Browder, Proposition 8.12 ....

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In summary: This means that the function f can be approximated by a series of differentiable functions, each of which is better at representing the function at a particular point.
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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 help in formulating a proof of Proposition 8.12 ...

Proposition 8.12 reads as follows:

View attachment 9428

Can someone please help me to demonstrate a formal and rigorous proof of Proposition 8.12 using on the definitions and propositions preceding the above proposition ...I am most interested in how/why we know that \(\displaystyle \text{df} (h) = \text{df}_1 (h), \ ... \ ... \ ... \ \text{df}_m (h) )\)... and also that ...\(\displaystyle f' (p) = \begin{bmatrix} f'_1 (p) \\ f'_2 (p) \\ . \\ . \\ . \\ f'_n (p) \end{bmatrix} \)... ... ... The definitions and propositions pertaining to the differential preceding the above proposition read as follows:

View attachment 9429
View attachment 9430
Hope that someone can help ...

Help will be much appreciated ...

Peter
 

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  • #2
Hi Peter,

Here is a hint to hopefully help move things along.

Assuming $\bf{f}$ is differentiable at $\bf{p}$, use the inequality $0\leq \max_{1\leq j\leq m}|a_{j}|\leq |\bf{a}|$ and the definition of differentiability to establish that the $f_{j}$ are also differentiable at $\bf{p}.$

When assuming the $f_{j}$ are differentiable at $\bf{p}$, use the other inequality provided by the author to conclude that $\bf{f}$ is as well.

Let me know if anything is unclear.
 
  • #3
GJA said:
Hi Peter,

Here is a hint to hopefully help move things along.

Assuming $\bf{f}$ is differentiable at $\bf{p}$, use the inequality $0\leq \max_{1\leq j\leq m}|a_{j}|\leq |\bf{a}|$ and the definition of differentiability to establish that the $f_{j}$ are also differentiable at $\bf{p}.$

When assuming the $f_{j}$ are differentiable at $\bf{p}$, use the other inequality provided by the author to conclude that $\bf{f}$ is as well.

Let me know if anything is unclear.
Thanks GJA ... appreciate your help ...

Think I have gotten the idea ... so ... will try to proceed ...
Proof of ...

... \(\displaystyle f\) is differentiable at \(\displaystyle p \Longleftrightarrow\) each \(\displaystyle f_j\) is differentiable at \(\displaystyle p\) ...Assume f is differentiable at \(\displaystyle p\) ...... now ... ... \(\displaystyle f\) is differentiable at \(\displaystyle p\)\(\displaystyle \Longrightarrow \lim_{ h \to 0 } \frac{1}{ |h| } ( f(p + h) - f(p) - Lh ) = 0\)


\(\displaystyle \Longrightarrow \lim_{ h \to 0 } \frac{ | f(p + h) - f(p) - Lh | }{ |h| } = 0\) \(\displaystyle \Longrightarrow\) for every \(\displaystyle \epsilon \gt 0 \ \exists \ \delta \gt 0\) such that\(\displaystyle |h| \gt 0 \Longrightarrow \frac{ | f(p + h) - f(p) - Lh | }{ |h| } \lt \epsilon\)But ...\(\displaystyle Lh = \text{df} (h) = ( \text{df}_1 (h), \ldots , \text{df}_m (h) )\)

... and hence ...

\(\displaystyle | f(p + h) - f(p) - Lh | = | f(p + h) - f(p) - \text{df} (h) |\)... so ... we also have ... for \(\displaystyle |h| \lt \delta\) ... since \(\displaystyle |a_j | \leq |a|\) ... ...\(\displaystyle \frac{ | f_j(p + h) - f_j(p) - \text{df}_j (h) | }{ |h| } \leq \frac{ | f(p + h) - f(p) - Lh | }{ |h| } \leq \epsilon\) ... therefore ... each \(\displaystyle f_j\) is differentiable at \(\displaystyle p\) ...

Now prove ...

... each \(\displaystyle f_j\) is differentiable at \(\displaystyle p\) ... \Longrightarrow ... f is differentiable at \(\displaystyle p\) ...Assume ... each \(\displaystyle f_j\) is differentiable at \(\displaystyle p\) ...

Then ... we have ...\(\displaystyle \frac{ | f_j(p + h) - f_j(p) - \text{df}_j (h) | }{ |h| } \lt \epsilon/m\) for \(\displaystyle |h| \lt \delta\) for \(\displaystyle j = 1, \ldots, m\)Therefore ...\(\displaystyle \frac{ | f_1(p + h) - f_1(p) - \text{df}_1 (h) | }{ |h| } + \ldots + \frac{ | f_m(p + h) - f_m(p) - \text{df}_m (h) | }{ |h| } \lt \epsilon\) But since ... \(\displaystyle |a|\leq \sum_{ j = 1 }^m |a_j|\) ... we have \(\displaystyle \frac{ | f(p + h) - f(p) - \text{df} (h) | }{ |h| } \leq \frac{ | f_1(p + h) - f_1(p) - \text{df}_1 (h) | }{ |h| } + \ldots + \frac{ | f_m(p + h) - f_m(p) - \text{df}_m (h) | }{ |h| } \lt \epsilon\) ... so ... \(\displaystyle f\) is differentiable at \(\displaystyle p\) ...
Can someone please confirm that the above is correct and/or point out the errors and shortcomings ...Peter
 
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Related to Differentials/Total Derivatives in R^n .... Browder, Proposition 8.12 ....

1. What is a differential in R^n?

A differential in R^n refers to the linear approximation of a function at a specific point in n-dimensional space. It is represented as a vector of partial derivatives of the function with respect to each variable.

2. How is a total derivative different from a partial derivative?

A total derivative is the sum of all partial derivatives of a function with respect to each variable, while a partial derivative only considers the change in the function with respect to one variable while holding others constant.

3. What is the significance of Proposition 8.12 in Browder's work?

Proposition 8.12 in Browder's work states that if a function is differentiable at a point, then it must also be continuous at that point. This is a fundamental concept in the study of differentials and total derivatives in R^n.

4. How are differentials and total derivatives used in real-world applications?

Differentials and total derivatives are used in various fields of science and engineering, such as physics, economics, and computer science. They are used to approximate and optimize functions, model complex systems, and analyze data.

5. Can differentials and total derivatives be extended to higher dimensions?

Yes, differentials and total derivatives can be extended to higher dimensions, such as R^3, R^4, and beyond. This is known as multivariable calculus and is an essential tool in many advanced mathematical and scientific concepts.

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