I am trying to prove that for a manifold $M$ with or without boundary of dimension $n$ ,the k-th exterior power bundle $\Lambda^k(M)$ which is defined as disjoint union of the vector spaces $\Lambda^k(\tau(M)_p)$ as $p$ varies over $M$ , is a manifold of dimension $n+\binom{n}{k}$.Here is my attempt----
A coordinate chart $\phi:U\rightarrow R^n$ in $M$ induces isomorphisms $d\phi_p:\tau(M)_p\rightarrow R^n$ and $\lambda_p=(d\phi_p)^*:\Lambda^k(R^n)\rightarrow \Lambda^k(\tau(M)_p)$. Then a coordinate chart for $\Lambda^k(M)$ is given by $\psi:\Lambda^k(U)\rightarrow \phi(U)×\Lambda^k(R^n)$,where $\Lambda^k(U)=\pi^{-1}(U)$ and $\psi(p,\omega)=(\phi(p),\lambda_p^{-1}(\omega))$. Now how do I prove transition maps are diffeomorphism.
I define a subset of $\Lambda^k(M)$ to be open iff it is of the form $\phi^{-1}(U)$ for some coordinte chart $\phi:U\rightarrow R^n$ of $M$. Am I right.
Is there any other alternative way to prove exterior power bundle is a manifold of given dimension.