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I am reading Hugo D. Junghenn's book: "A Course in Real Analysis" ...

I am currently focused on Chapter 9: "Differentiation on \(\displaystyle \mathbb{R}^n\)" ... ...

I need some help with another aspect of Definition 9.1.3 ...

Definition 9.1.3 and the relevant accompanying text read as follows:

https://www.physicsforums.com/attachments/7874

View attachment 7875

In the above text from Junghenn we read the following:

" ... ... The vector \(\displaystyle f'(a)\) is called the derivative of \(\displaystyle f\) at \(\displaystyle a\). The differential of \(\displaystyle f\) at \(\displaystyle a\) is the linear transformation \(\displaystyle df_a \in \mathscr{L} ( \mathbb{R}^n, \mathbb{R} )\) defined by

\(\displaystyle df_a(h) = f'(a) \cdot h, \ \ \ \ \ \ (h \in \mathbb{R}^n )\) ... ... ... "

My question is as follows:Is the derivative essentially equivalent to the differential ... can we write \(\displaystyle df_a = f'(a)\) ... if if we can't ... then why not?

... ... indeed, what is the exact difference between the derivative and the differential ...

(I know I have asked a general question like this before ... but this is now in the specific context of Junghenn ...)

Hope someone can help to clarify the above ...

Peter

=========================================================================================***NOTE***

I am aware that the term total derivative and differential are terms used for the same concept ... but this author seems to employ both the term derivative (and Junghenn seems to be defining a total derivative for a scalar function) and differential ...It may also be that the derivative is \(\displaystyle f'(a)\) and the differential is \(\displaystyle df_a(h) = f'(a) \cdot h\) ... but then Junghenn states that the differential is \(\displaystyle df_a\) ... and hence not \(\displaystyle df_a(h)\) ...Maybe I am making too much of the difference between \(\displaystyle df_a\) and \(\displaystyle df_a(h)\) ... ...Peter

I am currently focused on Chapter 9: "Differentiation on \(\displaystyle \mathbb{R}^n\)" ... ...

I need some help with another aspect of Definition 9.1.3 ...

Definition 9.1.3 and the relevant accompanying text read as follows:

https://www.physicsforums.com/attachments/7874

View attachment 7875

In the above text from Junghenn we read the following:

" ... ... The vector \(\displaystyle f'(a)\) is called the derivative of \(\displaystyle f\) at \(\displaystyle a\). The differential of \(\displaystyle f\) at \(\displaystyle a\) is the linear transformation \(\displaystyle df_a \in \mathscr{L} ( \mathbb{R}^n, \mathbb{R} )\) defined by

\(\displaystyle df_a(h) = f'(a) \cdot h, \ \ \ \ \ \ (h \in \mathbb{R}^n )\) ... ... ... "

My question is as follows:Is the derivative essentially equivalent to the differential ... can we write \(\displaystyle df_a = f'(a)\) ... if if we can't ... then why not?

... ... indeed, what is the exact difference between the derivative and the differential ...

(I know I have asked a general question like this before ... but this is now in the specific context of Junghenn ...)

Hope someone can help to clarify the above ...

Peter

=========================================================================================***NOTE***

I am aware that the term total derivative and differential are terms used for the same concept ... but this author seems to employ both the term derivative (and Junghenn seems to be defining a total derivative for a scalar function) and differential ...It may also be that the derivative is \(\displaystyle f'(a)\) and the differential is \(\displaystyle df_a(h) = f'(a) \cdot h\) ... but then Junghenn states that the differential is \(\displaystyle df_a\) ... and hence not \(\displaystyle df_a(h)\) ...Maybe I am making too much of the difference between \(\displaystyle df_a\) and \(\displaystyle df_a(h)\) ... ...Peter

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