Let $ p: M \to B $ be a smooth vector bundle over B.
I'm wondering how to express a differential form on M in local coordinates. I'm aware of the local expression of a differential form on a arbitrary smooth manifold, but I'm not sure if in the case of vector bundle over B there will be something special in the local expression of a differential form by using the base space B for example .
Thank you!
Since I can't comment, I will write my comment as an answer.
Vector bundle also has a smooth manifold structure where the local trivialisation defines a smooth structure. Now try to write 1-form in terms of local coordinates.
For example, Suppose $(U,(x_1, x_2, \cdots, x_n))$ be a smooth chart containing $p \in M$. Then $\{dx_i, 1 \leq i \leq n\}$ denotes the dual basis of cotangent spaces ${T^*}_p M$. Since 1- form is a covector field ( i.e. it assigns a covector at each point $p \in M$), so we can express our 1-form $\alpha = \Sigma_{i=1}^{i=n} a_i(p) dx_i|_p $ for each $p \in U$.