Directional and Partial Derivatives ....Notation .... D&K ...

[Math Amateur]

I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ...

I am focused on Chapter 2: Differentiation ... ...

I need help with an aspect of D&K's notation for directional and partial derivatives ... ...

D&K's definition of directional and partial derivatives reads as follows:

In a previous post I have demonstrated that$D_j f(a) = D_{ e_j} f(a) = D f(a) e_j = \begin{pmatrix} D_j f_1 (a) \\ D_j f_2 (a) \\ D_j f_3 (a) \\ ... \\ ... \\ ... \\ D_j f_p (a) \end{pmatrix}$
I am assuming that in the common 'partials' notation ( Jacobi notation ) that the above can be expressed as follows:
$D_j f(a) = \frac{ \partial f }{ \partial j } = \begin{pmatrix} \frac{ \partial f_1 }{ \partial j } (a) \\ \frac{ \partial f_2 }{ \partial j } (a) \\ \frac{ \partial f_3 }{ \partial j } (a) \\ ... \\ ... \\ ... \\\frac{ \partial f_p }{ \partial j } (a) \end{pmatrix}$Is that correct use of notation/terminology ...?

Peter

 

[andrewkirk]

In my experience it would be more usual to write $\frac{\partial f_i}{\partial x_j}$ rather than $\frac{\partial f_i}{\partial j}$. Similarly we would tend to write $\frac{\partial f}{\partial x_j}$ rather than $\frac{\partial f}{\partial j}$.

Which is a pity, because the way you wrote it is clearer since it does not require implicit assumption of a dummy variable $x_j$. Just, unfortunately, not common practice.

 

[Math Amateur]

In my experience it would be more usual to write $\frac{\partial f_i}{\partial x_j}$ rather than $\frac{\partial f_i}{\partial j}$. Similarly we would tend to write $\frac{\partial f}{\partial x_j}$ rather than $\frac{\partial f}{\partial j}$.

Which is a pity, because the way you wrote it is clearer since it does not require implicit assumption of a dummy variable $x_j$. Just, unfortunately, not common practice.

Thanks for pointing that out Andrew ...

I was originally intending to write:

$D_j f(a) = \frac{ \partial f }{ \partial x_j } = \begin{pmatrix} \frac{ \partial f_1 }{ \partial x_j } (a) \\ \frac{ \partial f_2 }{ \partial x_j } (a) \\ \frac{ \partial f_3 }{ \partial x_j } (a) \\ ... \\ ... \\ ... \\\frac{ \partial f_p }{ \partial x_j } (a) \end{pmatrix}$Phew! Learning about ... or further ... getting a good understanding of ... the differentiation of functions/mappings from $\mathbb{R}^n$ to $\mathbb{R}^p$ ... is harder than I thought it would be ... ... ... ...

Peter

 

[Math Amateur]

Thanks for pointing that out Andrew ...

I was originally intending to write:

$D_j f(a) = \frac{ \partial f }{ \partial x_j } = \begin{pmatrix} \frac{ \partial f_1 }{ \partial x_j } (a) \\ \frac{ \partial f_2 }{ \partial x_j } (a) \\ \frac{ \partial f_3 }{ \partial x_j } (a) \\ ... \\ ... \\ ... \\\frac{ \partial f_p }{ \partial x_j } (a) \end{pmatrix}$Phew! Learning about ... or further ... getting a good understanding of ... the differentiation of functions/mappings from $\mathbb{R}^n$ to $\mathbb{R}^p$ ... is harder than I thought it would be ... ... ... ...

Peter

Andrew, fresh_42 and other readers

Given your comments on notation I just thought I would share with you both (and with other readers) Duistermaat and Kolk "Remark on notation". This remark is after the definition of directional and partial derivatives and reads as follows: