Definition of partial derivatives from Rudin's PMA 
It's the definition of partial derivative from Rudin's PMA. 
Why he consider $(25)$ for real functions $f_i$? What about if $f_i$ in $(25)$ replaced by vector-valued function $\mathbf{f}=(f_1,f_2,\dots,f_m)$? I guess that it could be more general.
Any answer is greatly appreciated.
 A: You can do linear algebra in terms of $n$-tuples $(x_1,x_2,\ldots, x_n)$ and matrices $\bigl[a_{ik}\bigr]$, or in terms of "abstract" vectors ${\bf x}$ and linear maps $A$.
Similarly with derivatives in a multivariate setting: You can consider component functions $f_i$ $(1\leq i\leq m)$ and their partial derivatives $f_{i.k}$, as defined in $(25)$, or you can talk about the derivative as a linear map between tangent spaces, and satisfying
$${\bf f}({\bf p}+{\bf X})-{\bf f}({\bf p})=d{\bf f}({\bf p}).{\bf X}+o\bigl(|{\bf X}|\bigr)\qquad({\bf X}\to{\bf 0})\ .$$
What you are proposing is a chimerical mixed version, which certainly  makes mathematical sense, but is only seldom used. At any rate, you would have
$${\bf f}_{.k}({\bf p})=d{\bf f}({\bf p}).{\bf e}_k\ .$$
(The dots have the following significance: In $d{\bf f}({\bf p}).{\bf X}$ I write a dot to enhance that one should first compute the derivative $d{\bf f}({\bf p})$ and then apply this linear map to the vector ${\bf X}$. Some people write $d{\bf f}_{\bf p}\,{\bf X}$ instead. In $f_{i.k}$ the $i$ denotes the $i^{\rm th}$ component of ${\bf f}$, and $k$ denotes the number of the variable with respect to which one differentiates. This then allows to write ${\bf f}_{.k}$ with the exact meaning the OP had in mind in his question.)
