I'm trying to work with the first jet prolongation of a $k$-distribution on a manifold $M$ of dimension $n$. My intuition is to consider the Grassmann bundle $X=Gr_k(TM)\to M$ and look at the first jet space $X^{(1)}$. But I'm stuck at how to write down a point in the jet space. From the definition, a typical point has to be of the form $j^1_\mathcal{D}(p)$, for some germ of local section $\mathcal{D}$ of $X$ at the point $p\in M$. Thus $\mathcal{D}$ is a locally defined $k$-distribution around $p$. But how do I write $j^1_\mathcal{D}(p)$ in local coordinates?
Another approach could be using the $k$-frame bundle and the take jet prolongation. But again, I cannot think about the local coordinates.
Any help is appreciated.