My question arises from "An Introduction To Manifolds" 2nd edition P313
quote begin
Let $\omega_\alpha$ be a k-form on $M\times R$ and $(U_\alpha, x^1, ..., x^n)$ is a local chart on $M$, then $\omega_\alpha$ can be written as:
$\omega_\alpha = \sum_I a_Idx^I + \sum_J b_Jdt\wedge dx^J$
quote end
The decomposition is directly used without proof. How to understand the decomposition? I think $dx^I$ is a k-1 form, but $dt\wedge dx^J$ is a k-form. How can a k-form $\omega_\alpha$ be decomposed as sum of k-forms and k-1 forms? And how to determine the coefficients $a_I$ and $b_J$?