I am reading Guillemin and Pollack's Differential Topology Page 163:
The coordinate functions $x_1, \dots, x_k$ on $\mathbb{R}^k$ yield $1$-form $dx_1, \dots, dx_k$ on $\mathbb{R}^k$. Check $dx_1, \dots, dx_k$ have the specific action $dx_i(z)(a_1, \dots, a_k) = a_i$.
So I am confused - $dx_i$ is the speed of change on the $x_i$ direction. And then I am not sure how it work with $z \in \mathbb{R}^k$, and $a_i$s?
Thanks
It is a bit difficult to know how to answer your question without knowing how your text defines the one form $dx_i$. Some might just define it using the equation you've written. Here is some background that might help see why these formulas even make sense.
A differential $k$-form, $\omega$ on a smooth, real manifold $M$ assigns to each point $p\in M$ a multilinear, alternating map:
$$\omega_p:\underbrace{T_pM\times\ldots\times T_pM}_{k \text{ times}}\rightarrow\mathbb{R}$$
where $T_pM$ is the tangent space of $M$ at $p$. Notice that with this definition, $1$-forms are just linear functionals. Now, if $M=\mathbb{R}^n$, $k=1$, and $p=z$, for each $i=1,\ldots,n$, we can define a differential $1$-form, called $dx_i$, as the following linear functional:
$$(dx_i)_z:\mathbb{R}^n\rightarrow\mathbb{R}\\(a_1,\ldots,a_n)\mapsto a_i$$
This is nothing more than the $i$th coordinate function. Here we have used the fact that $T_z\mathbb{R}^n=\mathbb{R}^n$ for all $z$.