I Directional 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 derivatives ... ...

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

I am assuming that under D&K's definitions and notation one can write:$D_v f(a) = \begin{pmatrix} D_v f_1 (a) \\ D_v f_2 (a) \\ D_v f_3 (a) \\ ... \\ ... \\ ... \\ D_v f_p (a) \end{pmatrix}$$= Df(a)v$$= \begin{pmatrix} D_1 f_1(a) & D_2 f_1(a) & ... & ... & D_n f_1(a) \\ D_1 f_2(a) & D_2 f_2(a) & ... & ... & D_n f_2(a) \\ ... & ... & ... & ... &... \\ ... & ... & ... & ... &... \\ ... & ... & ... & ... &... \\ D_1 f_p(a) & D_2 f_p(a) & ... & ... & D_n f_p(a) \end{pmatrix} \begin{pmatrix} v_1 \\ v_2 \\ ... \\ ... \\ ... \\ v_n \end{pmatrix}$

$= \begin{pmatrix} D_1 f_1(a) v_1 + D_2 f_1(a) v_2 + \ ... \ ... \ + D_n f_1(a) v_n \\ D_1 f_2(a) v_1 + D_2 f_2(a) v_2 + \ ... \ ... \ + D_n f_2(a) v_n \\ ... \ ... \ ... \ ... \ ... \\ ... \ ... \ ... \ ... \ ... \\ ... \ ... \ ... \ ... \ ... \\ D_1 f_p(a) v_1 + D_2 f_p(a) v_2 + \ ... \ ... \ + D_n f_p(a) v_n \end{pmatrix}$Is the above a correct use of notation according to D&K's schema of notation ...

Peter
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Proposition 2.3.2 may well be relevant to the above post ... so I am providing the same ... as follows:

 

[andrewkirk]

What you have written makes sense. You have broken things up more than the text did, into component functions $D_jf_i(a)$ for $1\leq i\leq p$. The text has not stated a choice of notion for those component functions. Other possibilities for writing them would be $D_jf(a)_i$ and $D_jf(a)^i$. The latter uses a superscript rather than subscript to align with how these things tend to be written in tensor calculus.

 

[Math Amateur]

What you have written makes sense. You have broken things up more than the text did, into component functions $D_jf_i(a)$ for $1\leq i\leq p$. The text has not stated a choice of notion for those component functions. Other possibilities for writing them would be $D_jf(a)_i$ and $D_jf(a)^i$. The latter uses a superscript rather than subscript to align with how these things tend to be written in tensor calculus.

Thanks Andrew ...

Appreciate your help ...

Peter