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

In summary: F_j(h_j)which is the desired result.I hope this summary has helped you understand the proof of Kantorovitz's Proposition on pages 61-62. Remember, when dealing with partial derivatives and continuity, it is important to carefully follow the definitions and notations to avoid any confusion. Good luck with your studies! In summary, the proof of Kantorovitz's Proposition on pages 61-62 shows that for a differentiable function f defined on a subset of the k-dimensional real vector space, the difference between f evaluated at two points can be expressed as a sum
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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
 
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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
 

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

1. What is the significance of partial derivatives in mathematics?

Partial derivatives are important in mathematics because they allow us to study the rate of change of multivariable functions in a specific direction. This is useful in various fields such as physics, economics, and engineering, where multiple variables are involved in a single system.

2. How do partial derivatives relate to continuity?

In order for a function to be continuous at a point, all its partial derivatives must exist at that point. This means that the function must have a well-defined rate of change in all directions at that point. Without this, the function would not be considered continuous.

3. Can a function have partial derivatives but still not be continuous?

Yes, it is possible for a function to have partial derivatives at a point but still not be continuous at that point. This can happen if the partial derivatives exist but are not continuous themselves. In this case, the function would not satisfy the requirements for continuity.

4. What is Kantorovitz's Proposition in relation to partial derivatives and continuity?

Kantorovitz's Proposition states that if a function has continuous partial derivatives on an open set, then it is continuous on that set. This means that if a function has well-defined rates of change in all directions on an open set, it will also be continuous on that set.

5. How can the existence of partial derivatives be used to determine continuity?

If a function has continuous partial derivatives at a point, then it is automatically continuous at that point. This is a useful tool in determining continuity, as it allows us to focus on the existence of partial derivatives rather than checking for continuity in every direction separately.

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