I would like to have a intuitive view of the definition of differential in the page 127 of Carmo's differential geometry book:
Why does we define differential in this way? In other words, why is this particular vector $d_p(w)$ so important? This choice seems arbitrary to me.
EDIT:
Definition of differential map:
We say $F$ is differentiable at $p\in U$ if its component functions are differentiable at $p$; that is, by writing
$$F(x_1,\ldots,x_n)=(f_1(x_1,\ldots,x_n),\ldots,f_m(x_1\ldots,x_n))$$
the functions $f_i,i=1,\ldots,m$, have continuous partial derivatives of all orders at $p$. $F$ is differentiable in $U$ if it is differentiabe at all points in $U$.