If , are vector fields on a manifold and , differentiable functions, show that

Also if , , are vector fields on , calculate the components of , , .

Finally, let be a map of the -dimensional manifold and , be two dimensional fields onto . If are the basic vector fields of and , where , show that .

( denotes the Lie bracket on vector fields) I am not familiar with the Lie bracket at all, so I would be happy if someone helped!