I've recently been attempting to self-study Spivak's Calculus on Manifolds and have hit a snag in the section discussing integration on chains. My confusion centers around the nuance between $f^{*}$ and $f_{*}$. Spivak defines their relationship as $f^{*}\omega\left( p \right)\left( v_{1}, \dots, v_{n} \right) = \omega \left( f \left( p \right) \right) \left( f_{*}\left( v_{1} \right), \dots, f_{*}\left( v_{n} \right) \right)$. Here, $f_{*}(p)(v) =\left(Df(p)(v)\right)_{f(p)}$.
(Problem 4-30) If $\omega$ is a 1-form on $\mathbb{R}^2 \setminus \left\{ 0 \right\}$ s.t. $d\omega = 0$, prove $\omega = \lambda d\theta + dg$ for a $\lambda \in \mathbb{R} $ and $g: \mathbb{R}^2 \to \mathbb{R}^2 \setminus \left\{ 0 \right\}$.
Spivak provides a hint to equate $c_{R,1}^*\left( w \right) = \lambda_{R}dx + d \left( g_{R} \right)$ and show all $\lambda_{R} = \lambda$ for all possible $\lambda_{R}$.
I was considering an application of Stoke's Theorem $\int_{\partial c} \omega = \int_{c} d\omega$ as follows. I define $c'=\{(x, y) \in \mathbb{R}^2:|(x, y)| < R\}$
$$\int_{c_{R,1}} \omega = \int_{[0, 1]} c^*_{R,1} \left( \omega \right) = \int_{[0, 1]} \lambda_R dx + d \left( g_R \right) = \int_{c'}dw = 0$$
Do you all have suggestions for materials that could elaborate on k-forms, exterior calculus, dual spaces, $f^*$, and volume elements? I have purchased a copy of Munkres's Analysis on Manifolds, and are there any web resources defining the terms present in each text.
Best Regards.