Understanding Difference b/w Derivative & Differential in D&K Definition 9.1.3

[Math Amateur]

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

 

[Ackbach]

Re: Multivariable Analysis ...the derivative and the differential ... D&K Definition 9.1.3 ... ...

Derivatives and differentials are definitely not the same thing. As Berkeley put it in The Analyst, regarding differentials, "May we not call them the ghosts of departed quantities?" In the single-variable case, if you have $y=f(x)$, where $f$ is a differentiable function, then the differential $dx$ is simply a real independent variable, and the differential $dy$ is defined as $dy=f'(x)\,dx$ - a dependent variable. While we often think about differentials as infinitesimals, there's nothing about this definition that requires it. Moreover, this definition has the virtue of corresponding somewhat with the concept of sensitivity of dependent variables to changes in the independent variable. I use differentials in two contexts: integration, where it's simply a notation to A. remind me that I'm integrating w.r.t. a particular variable, and B. assuming we are not using that horrid physicist notation of $\displaystyle\int dx\,f(x)$, it envelopes the integrand, making it abundantly clear exactly what the integrand is.

It's no different in the multivariable case. Instead of $dx$, you have $\mathbf{h}$ - which is just a variable in $\mathbf{R}^n$ - and instead of $dy$, you have $df_a(\mathbf{h})=f'(\mathbf{a})\cdot\mathbf{h}$.

A derivative is a function obtained from another function by a particular limiting process. A differential is a variable, not a function. You definitely can't equate a derivative with a differential. In fact, one rule about differentials is that you can't have a differential on one side of an equation without one on the other side - kind of an unwritten rule, but I find it helps students, particularly when they're doing $u$-substitutions in integration.

Does that help?

 

[Math Amateur]

Re: Multivariable Analysis ...the derivative and the differential ... D&K Definition 9.1.3 ... ...

Derivatives and differentials are definitely not the same thing. As Berkeley put it in The Analyst, regarding differentials, "May we not call them the ghosts of departed quantities?" In the single-variable case, if you have $y=f(x)$, where $f$ is a differentiable function, then the differential $dx$ is simply a real independent variable, and the differential $dy$ is defined as $dy=f'(x)\,dx$ - a dependent variable. While we often think about differentials as infinitesimals, there's nothing about this definition that requires it. Moreover, this definition has the virtue of corresponding somewhat with the concept of sensitivity of dependent variables to changes in the independent variable. I use differentials in two contexts: integration, where it's simply a notation to A. remind me that I'm integrating w.r.t. a particular variable, and B. assuming we are not using that horrid physicist notation of $\displaystyle\int dx\,f(x)$, it envelopes the integrand, making it abundantly clear exactly what the integrand is.

It's no different in the multivariable case. Instead of $dx$, you have $\mathbf{h}$ - which is just a variable in $\mathbf{R}^n$ - and instead of $dy$, you have $df_a(\mathbf{h})=f'(\mathbf{a})\cdot\mathbf{h}$.

A derivative is a function obtained from another function by a particular limiting process. A differential is a variable, not a function. You definitely can't equate a derivative with a differential. In fact, one rule about differentials is that you can't have a differential on one side of an equation without one on the other side - kind of an unwritten rule, but I find it helps students, particularly when they're doing $u$-substitutions in integration.

Does that help?

Thanks for the help Ackbach ...

Still reflecting on what you have written ...

Peter