Let , and , be two functions.

If are two manifolds of dimension respectively and a differentiable function, while an open set. Let be a point at and and be the respective tangent spaces. We define , where if then is differentiable at an area of and for each we have

Calculate a matrix of and . Then find the vector

Also if is a differentiable bijection with also differentiable and are two vector fields onto M, then , are also vector fields onto N. Show that where is the Lie bracket.

If someone could explain me the definition a bit, as well as to solve this (simple, I think) problem I would be grateful!