understanding differential form from do carmo

Question

I am recently read the differential form book of do carmo and found the following

Here I can not understand what is $(dx_i)_p$ here?Is it the derivative map of $x_i$.
And I also can not understand that how did author calculate $$(dx_i)_p(e_j)=\frac{\partial x_i}{\partial x_j}.$$
And how did become it the dual basis of {$(e_i)_p$}.
Can someone explain more clearly please

Answer 1

It's usual to regard what he's written, as the definition (see e.g. Introduction to Smooth Manifolds, John M Lee). I believe do Carmo's trying to connect up with the total differential/derivative as defined in calculus courses (e.g. Thomas & Finney) where suitably interpreted expressions such as $df = f_1dx_1 + \cdots+f_ndx_n$, the $f_i$'s are the partials of $f$, occur and the $dx_i$ which represent the increments in the $x_i$ variables are taken to be "independent" of one another. Writing $(df)_p$ as the $1\times n$ matrix $[f_1(p) \ \dots f_n(p)]$, we see that $(df)_p$ can be regarded as a linear functional acting on $\mathbb{R}^n$ and the $(dx_i)_p = [\frac{\partial{x_i}}{\partial x_1} \ \cdots \ \frac{\partial{x_i}}{\partial x_n}] = [0 \ \cdots 1\ \cdots 0]$ can be taken to be the basis vectors of the dual of $\mathbb{R}^n$. It should now be clear that $(dx_i)_p(e_j) = \frac{\partial{x_i}}{\partial x_j} = \delta_{ij}$ (Kronecker delta).

Answer 2

$x_{i}:M^n\to\mathbb{R},p\mapsto x_{i}$ is a coordinate function, $dx_{i}$ - it's differential. It's action on a tangent vector equals the directional derivative: $df(v)=v(f)$, substitute $f=x_{i}$ and $v=e_{j}=\frac{\partial}{\partial x_{j}}$ to get $\frac{\partial x_{i}}{\partial x_{j}}$ and see that the set $\{dx_{i}\}$ is dual to $\{e_{j}\}$ and therefore can be used to read off the components of any tangent vectors: $dx_{k}(v)=dx_{k}(\sum_{i=1}^{n}v^{i}e_{i})=v^{k}$, allowing the tangent to be rewritten as $v=\sum_{i=1}^{n}dx_{i}(v)e_{i}$.

Any linear functional can be expressed as a linear combination of $\{dx_{i}\}$ making the set a basis of the dual space: $f(v)=f(\sum_{i=1}^{n}dx_{i}(v)e_{i})=\sum_{i=1}^{n}f(e_{i})dx_{i}(v)$.