Multivariable Analysis ....the derivative & the differential

[Math Amateur]

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

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

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

Definition 9.1.3 and the relevant accompanying text read as follows:

In the above text from Junghenn we read the following:

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

$df_a(h) = f'(a) \cdot h, \ \ \ \ \ \ (h \in \mathbb{R}^n )$ ... ... ... "My question is as follows:Is the derivative essentially equivalent to the differential ... can we write $df_a = f'(a)$ ... 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=============================================================================

In another post it was pointed out to me that the terms total derivative and differential are sometimes 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 $f'(a)$ and the differential is $df_a(h) = f'(a) \cdot h$ ... but then Junghenn states that the differential is $df_a$ ... and hence not $df_a(h)$ ...

Maybe I am making too much of the difference between $df_a$ and $df_a(h)$ ... ...

But my apologies to mathwonk and others if I have misunderstood their posts ...

Peter

 

[andrewkirk]

Is the derivative essentially equivalent to the differential ... can we write $df_a = f'(a)$ ... if if we can't ... then why not?

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

This particular author has chosen to call the derivative the vector in $\mathbb R^n$ that is labelled $f'(\mathbf a)$, whereas he calls the differential the linear transformation $df_a\in\mathscr L(\mathbb R^n,\mathbb R)$. So one is a vector and the other is a linear map.

The relationship between the two is that the components of the Jacobian matrix of the linear map $df_a$ are the same as those of the vector $f'(\mathbf a)$.

We can't write $df_a=f'(\mathbf a)$ because they are different types of objects.

Bear in mind that the use of these terms is not universal and will vary between authors, so swapping from one book to another can cause confusion. There will very likely be inconsistencies.

 

[Math Amateur]

This particular author has chosen to call the derivative the vector in $\mathbb R^n$ that is labelled $f'(\mathbf a)$, whereas he calls the differential the linear transformation $df_a\in\mathscr L(\mathbb R^n,\mathbb R)$. So one is a vector and the other is a linear map.

The relationship between the two is that the components of the Jacobian matrix of the linear map $df_a$ are the same as those of the vector $f'(\mathbf a)$.

We can't write $df_a=f'(\mathbf a)$ because they are different types of objects.

Bear in mind that the use of these terms is not universal and will vary between authors, so swapping from one book to another can cause confusion. There will very likely be inconsistencies.

Thanks Andrew ...

Reflecting on what you have written...

Thanks again...

Peter

 

[fresh_42]

We can't write $df_a=f'(\mathbf a)$ because they are different types of objects.

One additional remark. As for any vector $v$ there is a unique linear mapping $w \rightarrow \langle w,v \rangle = \sum w_iv_i$ and vice versa. This duality establishes a one-to-one correspondence (matrix - linear function) between the two, which is especially in physics important to distinguish - not so much in mathematics (IMO) as usually the context makes clear which one is meant.

The confusion normally arises when the differential, resp. derivative is called a linear map. Take for example $f(x)=x^3\,.$ We are used to write $f\,'(x)=3x^2$ and no linear map is in sight, so where is it? The difference lies in $a$. We better should have written $f\,'(a)=3a^2$ because the derivative takes place at a certain point, a location where the differential is evaluated, where $f\,'$ represents a linear approximation to $f$. Here we have $f\,'(a)=\left. \dfrac{d}{dx}\right|_{x=a} x^3 =3x^2|_{x=a}=3a^2$ as the derivative and $df_a = D_af \, : \, (\,w \longmapsto 3a^2\cdot w\,)$ which is a linear transformation (in $w$). Although our vector here has only one component, and the Jacobi matrix is a $(1\times 1)-$matrix, everything said has still to be right, because we simply have $n=1\,.$

This means

This particular author has chosen to call the derivative the vector in $\mathbb R^n$ that is labelled $f'(\mathbf a)$,...

$(3a^2)_{1\leq i \leq 1}$ is the derivative and vector $f\,'(a)$ here ...

... whereas he calls the differential the linear transformation $df_a\in\mathscr L(\mathbb R^n,\mathbb R)$. So one is a vector and the other is a linear map.

... and $df_a = \text{ times }3a^2$ is the differential and linear transformation.
The only difference here is a tiny multiplication dot.