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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:

View attachment 7856

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

\(\displaystyle = \begin{pmatrix} D_1 f_1(a) v_1 + D_2 f_1(a) v_2 + \ ... \ ... \ + D_n f_1(a) v_n \\ D_1 f_2(a) v_1 + D_2 f_2(a) v_2 + \ ... \ ... \ + D_n f_2(a) v_n \\ ... \ ... \ ... \ ... \ ... \\ ... \ ... \ ... \ ... \ ... \\ ... \ ... \ ... \ ... \ ... \\ D_1 f_p(a) v_1 + D_2 f_p(a) v_2 + \ ... \ ... \ + D_n f_p(a) v_n \end{pmatrix} \)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:View attachment 7857

View attachment 7858