Multivariable Analysis .... Directional and Partial Derivatives .... D&K Propostion 2.3.3 ....

In summary: I hope this helps and provides a formal and rigorous demonstration of Assertion (i) following from Formula (2.11). Let me know if you have any further questions. In summary, in the above conversation, Peter is seeking help with understanding and proving a specific proposition and its proof in a book he is reading about multidimensional real analysis. He provides a link to the proof and asks if someone can demonstrate formally and rigorously how the proof follows from a certain formula. Another user responds by breaking down the notation and providing a step-by-step proof for Assertion (i) in the proposition.
  • #1
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I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ...

I am focused on Chapter 2: Differentiation ... ...

I need help with an aspect of the proof of Proposition 2.3.2 ... ...

Duistermaat and Kolk's Proposition 2.3.2 and its proof read as follows:
View attachment 7847
https://www.physicsforums.com/attachments/7848
In the above proof by D&K we read the following:

" ... ... Assertion (i) follows from Formula (2.11). ... ..."Can someone please demonstrate (formally and rigorously) that this is the case ... that is that assertion (i) follows from Formula (2.11). ... ...Help will be appreciated ...

Peter==========================================================================================***NOTE***

It may help readers of the above post to have access to the start of Section "2.3: Directional and Partial Derivatives" ... in order to understand the context and notation of the post ... so I am providing the same ... as follows:https://www.physicsforums.com/attachments/7849Hope that the above helps readers of the post understand the context and notation of the post ...

Peter
 
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  • #2
Hi, Peter.

Peter said:
" ... ... Assertion (i) follows from Formula (2.11). ... ..."

Can someone please demonstrate (formally and rigorously) that this is the case ... that is that assertion (i) follows from Formula (2.11). ... ...

Try dividing the first line of (2.11) through by $t$. Now take the limit as $t\rightarrow 0$, use the definition of directional derivative and the second part of (2.11) to obtain the desired result.
 
  • #3


Hello Peter,

I am not familiar with the book you are reading, but I can try to provide some assistance with your question.

Firstly, let's define the notation used in the proof. In Formula (2.11), we have the expression $\partial_i f(x)$, which represents the $i$th partial derivative of $f$ at the point $x$. This can also be written as $\frac{\partial f}{\partial x_i}(x)$.

Now, let's look at Assertion (i) in the proof. It states that for any fixed $i$, the function $\partial_i f$ is continuous. In other words, for any point $x_0$, the limit $\lim_{x \to x_0} \partial_i f(x)$ exists. This can be proved using Formula (2.11) as follows:

By definition, the limit $\lim_{x \to x_0} \partial_i f(x)$ exists if and only if for any given $\epsilon > 0$, there exists a $\delta > 0$ such that whenever $|x - x_0| < \delta$, we have $|\partial_i f(x) - \partial_i f(x_0)| < \epsilon$.

Now, using Formula (2.11), we have $\partial_i f(x) - \partial_i f(x_0) = \frac{\partial f}{\partial x_i}(x) - \frac{\partial f}{\partial x_i}(x_0)$. By the mean value theorem, there exists a point $c$ between $x$ and $x_0$ such that $\frac{\partial f}{\partial x_i}(x) - \frac{\partial f}{\partial x_i}(x_0) = \frac{\partial^2 f}{\partial x_i^2}(c)(x-x_0)$. Since $f$ is continuously differentiable, $\frac{\partial^2 f}{\partial x_i^2}$ is continuous. Hence, for any given $\epsilon > 0$, there exists a $\delta > 0$ such that whenever $|x-x_0| < \delta$, we have $|\frac{\partial^2 f}{\partial x_i^2}(c)| < \epsilon$. Therefore, $|\partial_i f(x) - \partial_i f(x_0)| < \epsilon$,
 

Related to Multivariable Analysis .... Directional and Partial Derivatives .... D&K Propostion 2.3.3 ....

What is multivariable analysis?

Multivariable analysis is a statistical technique used to analyze relationships between multiple variables. It involves examining how changes in one variable affect changes in another while controlling for other variables. It is commonly used in fields such as economics, psychology, and biology.

What are directional and partial derivatives?

Directional derivatives refer to the rate of change of a function in a specific direction, while partial derivatives refer to the rate of change of a function with respect to one of its variables, holding all other variables constant. Both are important concepts in multivariable analysis as they help us understand how a function changes with respect to its variables.

What is D&K Proposition 2.3.3?

D&K Proposition 2.3.3 is a mathematical theorem that states that if a function has continuous partial derivatives, then its directional derivative exists in all directions and is equal to the dot product of the gradient vector and the unit vector of the direction.

How is D&K Proposition 2.3.3 related to multivariable analysis?

D&K Proposition 2.3.3 is an important tool in multivariable analysis as it allows us to calculate directional derivatives, which are essential for understanding the behavior of a function in different directions. It also highlights the importance of having continuous partial derivatives in order to ensure the existence of directional derivatives.

What are some real-world applications of multivariable analysis?

Multivariable analysis has a wide range of applications in various fields. For example, it can be used to analyze the relationship between income, education, and job satisfaction in economics, or to understand how different factors affect the growth of a tumor in biology. It is also used in engineering, finance, and social sciences to analyze complex systems and make predictions based on multiple variables.

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