Infinitesimal Stoke's theorem

Let $\omega$ be a 1-form and $X,Y\in {\mathrm{\Omega }}^1(M)$:

Proof: see [Boothby 1986 page 224]

Idea:

Another explanation: @needham2021visual page 392.

More useful formulas here.

The finite case is in Stokes' theorem.