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

[Math Amateur]

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, $$\bigtriangledown f$$ , the differential/derivative for real-valued functions of several real variables, $$f$$ ... ...

Help will be appreciated ...

Peter

 

[S.G. Janssens]

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".

Further to the above ... is the gradient, $$\bigtriangledown F$$ , 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.

 

[Math Amateur]

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

 

[Math Amateur]

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!

Appreciate your assistance...

Peter