If $M^n$ is a smooth manifold, the tangent bundle $TM$ is defined to be $$TM = \bigcup_{p\in M} \{p\}\times T_pM$$ where $T_pM$ is the tangent space at $p$. In order to talk about a vector field (a map $v: M\to TM$ with $\pi(v(p)) =p$) being smooth, one needs to define a smooth structure on $TM$. As I learned it, the smooth structure on $TM$ is given as follows: if $(p,v)\in TM$ and $(U, \varphi)$ is a chart for $M$ at $p$, then we assert that $(\pi^{-1}(U), d\varphi_p)$ is a chart for $TM$.

My question is this: how do we generalize this process to other important vector bundles over a smooth manifold? For example, if $V$ is a vector space, let $\mathcal{A}^k(V)$ denote the vector space of alternating $k$-tensors over $V$, and let $$\mathcal{A}^kM = \bigcup_{p\in M} \{p\}\times\mathcal{A}^k(T_pM)$$ How do we define the smooth structure on $\mathcal{A}^k M$, so that we may have a notion of smooth sections of this bundle (i.e. differential $k$-forms)?


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