# The Differentail and the Derivative in Multivariable Analysis .... ....

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In summary, Kantorovitz's definition of "differential" is the same as the "derivative" in D&K's text, with the uniqueness aspect needing a short proof. The term "differential" and the notation $df$ may be confusing for those coming from a physics background, as it differs from the physicist's definition. The gradient, represented as $\bigtriangledown f$, is the differential/derivative for real-valued functions of several real variables. However, it is often more than just the differential, as it also includes a coordinate representation. Further clarification can be found in the specific text.
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I am reading the book "Several Real Variables" by Shmuel Kantorovitz ... ...

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

I need help with an aspect of Kantorovitz's definition of "differential" ...

Kantorovitz's Kantorovitz's definition of "differential" reads as follows:
View attachment 7821
https://www.physicsforums.com/attachments/7822Is the "differential" as defined by Kantorovitz for real-valued functions of several real variables the same as 'the derivative" ...

If so ... is it the same situation for vector-valued functions of several real variables ...

Further to the above ... is the gradient, $$\displaystyle \bigtriangledown f$$ , the differential/derivative for real-valued functions of several real variables, $$\displaystyle f$$ ... ...

Help will be appreciated ...

Peter

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Peter said:
Is the "differential" as defined by Kantorovitz for real-valued functions of several real variables the same as 'the derivative" ...

If so ... is it the same situation for vector-valued functions of several real variables ...

Yes, the definition of "differential" it is exactly the same as the "new" definition of "derivative" in D&K, for example. Note that the uniqueness part of the definition in Kantorovitz actually would need a short proof, like in the lemma we discussed https://mathhelpboards.com/analysis-50/differentiability-mappings-r-n-r-p-d-amp-k-lemma-2-2-3-a-23535.html.

Also, when I learned this material first, I was very confused by the term "differential" and the notation $df$, as I came from a physics background. There, a "differential" usually refers to an infinitesimal quantity, that is not even rigorously defined in standard analysis. So, the "differential" we are discussing here is not the physicist's "differential".

Peter said:
Further to the above ... is the gradient, $$\displaystyle \bigtriangledown F$$ , the differential/derivative for real-valued functions of several real variables

Yes, the gradient is the differential (= derivative) of a real scalar-valued function of several real variables. One stipulation: Often, the gradient is a bit more than that: It is the coordinate representation (w.r.t. to some basis, such as the standard basis) of the differential of such a function. What is meant specifically will probably be apparent from the particular text.

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Krylov said:
Yes, the definition of "differential" it is exactly the same as the "new" definition of "derivative" in D&K, for example. Note that the uniqueness part of the definition in Kantorovitz actually would need a short proof, like in the lemma we discussed https://mathhelpboards.com/analysis-50/differentiability-mappings-r-n-r-p-d-amp-k-lemma-2-2-3-a-23535.html.

Also, when I learned this material first, I was very confused by the term "differential" and the notation $df$, as I came from a physics background. There, a "differential" usually refers to an infinitesimal quantity, that is not even rigorously defined in standard analysis. So, the "differential" we are discussing here is not the physicist's "differential".
Thanks Krylov ... most clear and helpful ...

Thanks again ...

Peter

Krylov said:
Yes, the definition of "differential" it is exactly the same as the "new" definition of "derivative" in D&K, for example. Note that the uniqueness part of the definition in Kantorovitz actually would need a short proof, like in the lemma we discussed https://mathhelpboards.com/analysis-50/differentiability-mappings-r-n-r-p-d-amp-k-lemma-2-2-3-a-23535.html.

Also, when I learned this material first, I was very confused by the term "differential" and the notation $df$, as I came from a physics background. There, a "differential" usually refers to an infinitesimal quantity, that is not even rigorously defined in standard analysis. So, the "differential" we are discussing here is not the physicist's "differential".
Yes, the gradient is the differential (= derivative) of a real scalar-valued function of several real variables. One stipulation: Often, the gradient is a bit more than that: It is the coordinate representation (w.r.t. to some basis, such as the standard basis) of the differential of such a function. What is meant specifically will probably be apparent from the particular text.
Thanks for the help regarding the gradient!

Peter

## 1. What is the difference between a differential and a derivative in multivariable analysis?

A differential is a small change in a variable, while a derivative is the rate of change of a function with respect to one of its variables. In multivariable analysis, differentials are used to approximate changes in multiple variables, while derivatives are used to calculate the rate of change of a function with respect to each of its variables.

## 2. How are differentials and derivatives used in multivariable analysis?

In multivariable analysis, differentials and derivatives are used to study how a function changes in response to changes in multiple variables. They are also used to calculate the slope of a function at a given point, which can be used to determine the direction of steepest ascent or descent.

## 3. Can differentials and derivatives be applied to functions with more than two variables?

Yes, differentials and derivatives can be applied to functions with any number of variables. In fact, multivariable analysis is specifically focused on studying functions with multiple variables.

## 4. What is the process for finding the derivative of a multi-variable function?

The process for finding the derivative of a multi-variable function involves taking the partial derivatives of the function with respect to each of its variables. This can be done using the rules of differentiation, such as the chain rule and product rule.

## 5. How are differentials and derivatives used in real-world applications?

Differentials and derivatives are used in a wide range of real-world applications, including physics, engineering, economics, and statistics. They can be used to analyze and optimize complex systems, such as predicting the trajectory of a projectile or optimizing a company's production process.

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