How Does One Derive Kantorovitz's Proposition on Pages 61-62?

[Math Amateur]

Existence of Partial Derivatives and Continuity ... Kantorovitz's Proposition pages 61-62 ...

I am reading the book "Several Real Variables" by Shmuel Kantorovitz ... ...

I am currently focused on Chapter 2: Derivation ... ...

I need help with another element of the proof of Kantorovitz's Proposition on pages 61-62 ...

Kantorovitz's Proposition on pages 61-62 reads as follows:View attachment 7808
https://www.physicsforums.com/attachments/7809In the above proof we read the following:

" ... ... Formula 2.4 is trivially true in case \(\displaystyle h_j = 0\), and by (2.2) - (2.4)

\(\displaystyle f(x + h) - f(x) = \sum_{ j = 1}^k [ F_j ( h_j ) - F_j (0) ]\)

\(\displaystyle = \sum_j h_j \frac{ \partial f }{ \partial x_j } ( x + h^{ j - 1 } + \theta_j h_j e^j )\) ... ... ... ... ... "I have tried to derive \(\displaystyle f(x + h) - f(x) = \sum_{ j = 1}^k [ F_j ( h_j ) - F_j (0) ]\) but did not succeed ...

... can someone please show how \(\displaystyle f(x + h) - f(x)\) equals \(\displaystyle \sum_{ j = 1}^k [ F_j ( h_j ) - F_j (0) ]\) ...Also can someone show how the above equals \(\displaystyle \sum_j h_j \frac{ \partial f }{ \partial x_j } ( x + h^{ j - 1 } + \theta_j h_j e^j )\) ... ...

Help will be much appreciated ... ...

Peter

 

[jvicens]

Dear Peter,

Thank you for reaching out for help with understanding Kantorovitz's Proposition on pages 61-62 of "Several Real Variables". I am happy to assist you with this element of the proof.

First, let's review the notation used in the proof. We have the function f defined on a subset of the k-dimensional real vector space, denoted by \Omega \subset \mathbb{R}^k. The variables x and h are k-dimensional vectors, with x being the point at which we are evaluating the function f and h being a small displacement vector. The notation e^j denotes the j-th standard basis vector in \mathbb{R}^k, and \theta_j is a real number between 0 and 1.

Now, let's break down the proof step by step:

1. We start with the definition of the directional derivative, which is given by:

D_h f(x) = \lim_{t \to 0} \frac{f(x + th) - f(x)}{t}

2. Using the definition of the directional derivative, we can write:

f(x + h) - f(x) = t D_h f(x)

3. Next, we use the fact that the function f is differentiable at x to write:

f(x + h) - f(x) = t \sum_{j=1}^k \frac{\partial f}{\partial x_j}(x) h_j + o(t)

4. Now, we substitute t = h_j into the above equation and rearrange terms to obtain:

f(x + h) - f(x) = h_j \frac{\partial f}{\partial x_j}(x) + o(h_j)

5. Using the definition of the error term o(h_j), we can write:

f(x + h) - f(x) = h_j \frac{\partial f}{\partial x_j}(x) + (h_j) \epsilon_j(h_j)

where \epsilon_j(h_j) is a function that tends to 0 as h_j tends to 0.

6. Finally, we use the fact that the function F_j(h_j) is defined as:

F_j(h_j) = \frac{f(x + h) - f(x)}{h_j}

to write:

f(x + h) - f(x) = h_j F_j(h_j)

7. Substituting