MHB 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:View attachment 7855In 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:
$$D_j f(a) = \frac{ \partial f }{ \partial x_j } = \begin{pmatrix} \frac{ \partial f_1 }{ \partial x_j } (a) \\ \frac{ \partial f_2 }{ \partial x_j } (a) \\ \frac{ \partial f_3 }{ \partial x_j } (a) \\ ... \\ ... \\ ... \\\frac{ \partial f_p }{ \partial x_j } (a) \end{pmatrix}$$Is that correct use of notation/terminology ...?

Peter

 

[DrWahoo]

Yes. Check out Wolfgang Walter's book on DE's. Or Roseenwasser's books on autonomous control systems using sensitivity analysis. Notation is similar in both.