Total divergence

It is a notion related to divergence but in a jet bundle context. Given a jet space $J^n({\mathbb{R}}^p,{\mathbb{R}}^q)$ and $p$ smooth functions defined on it $P=(P_1,\dots ,P_p)$ the total divergence of $P$ is

with $D_{x_j}$ the total derivative operators.

I think that the $p$-tuple $P$ should be thought as a total vector field.