I'm drowning in confusion trying to understand the simple example of a differential 1-form (see image below) given in Arnold's Mathematical Methods of Modern Mechanics (p174). He says “if $x=1$ and the coordinate of the tangent vector $\mathbf{\xi}$ is equal to 1, then $df=2$, and if the coordinate of $\mathbf{\xi}$ is equal to 10, then $df=20$”. So he appears to be equating “the coordinate of the tangent vector $\mathbf{\xi}$” with $dx$ in the differential 1-form $df=2xdx$.
I thought things like $dx^{j}$ are basis 1-forms which act on basis vectors according to$$dx^{j}\left(\frac{\partial}{\partial x^{i}}\right)=\delta_{i}^{j}.$$
But Arnold seems to be treating $dx$ as a vector component. Bachman appears to be doing the same thing (in A Geometric Approach to Differential Forms, p26) when he says:
“In general, we may refer to the coordinates of an arbitrary vector in $T_{p}P$ as $\left\langle dx,dy\right\rangle$, just as we may refer to the coordinates of an arbitrary point in $P$ as $\left\langle x,y\right\rangle$.”
What am I missing? Am I confusing components with coordinates?
