I was watching this Frederic Schuller's lecture on Grassmann algebra and de Rham cohomology. I don't understand. At 43:40 it looks like a vector field is eating a smooth function.
The $X_i$ should be a vector field, that is a section of a smooth tangent bundle. So it should map a point on the manifold to one of the point's tangent vectors. But omega is a $0$-$n$ tensor field, that is, a function that takes n vector fields and outputs a smooth function (manifold $\to \mathbb R$). So how can you give the $X_i$ a smooth function when it should eat a point of the manifold? What am I missing?