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Jul 10, 2019 at 16:24 comment added Bazin Look, the Lie derivative of a tensor is a very intuitive and simple geometric operation which relies only on the definition of a local flow for a smooth enough vector field. Then you get for free the exterior differentiation. It is a bit unfair to Elie Cartan, but it is also very easy to check Cartan's formula in coordinates for the $X=\partial/\partial x_j$ with the usual formula for the exterior differentiation, so that you get a full proof.
Jul 10, 2019 at 12:45 comment added Lo Scrondo I do prefer your demonstration...unfortunately, I've to follow the one present in the book :-(
Jul 9, 2019 at 16:08 history answered Bazin CC BY-SA 4.0