Differentiable Functions on Manifolds

Let http://data.artofproblemsolving.com/...bf94102d1b.gif, http://data.artofproblemsolving.com/...37c18dc40f.gif and http://data.artofproblemsolving.com/...40e93223f0.gif, http://data.artofproblemsolving.com/...4e0c5c8d8f.gif be two functions.

If http://data.artofproblemsolving.com/...d765d308af.gif are two manifolds of dimension http://data.artofproblemsolving.com/...fa0e482290.gif respectively and http://data.artofproblemsolving.com/...a172522f9b.gif a differentiable function, while http://data.artofproblemsolving.com/...63d9131c1b.gif an open set. Let http://data.artofproblemsolving.com/...5310b19759.gif be a point at http://data.artofproblemsolving.com/...63d9131c1b.gif and http://data.artofproblemsolving.com/...beb80e7b54.gif and http://data.artofproblemsolving.com/...92297da132.gif be the respective tangent spaces. We define http://data.artofproblemsolving.com/...7c7495d774.gif , where if http://data.artofproblemsolving.com/...8b7b24c219.gif then http://data.artofproblemsolving.com/...1e14f66356.gif is differentiable at an area of http://data.artofproblemsolving.com/...5310b19759.gif and for each http://data.artofproblemsolving.com/...b2be9c6b96.gif we have http://data.artofproblemsolving.com/...bea61c90ba.gif

Calculate a matrix of http://data.artofproblemsolving.com/...285a37cbd5.gif and http://data.artofproblemsolving.com/...cc4bc358e4.gif. Then find the vector http://data.artofproblemsolving.com/...c76ed9ff75.gif

Also if http://data.artofproblemsolving.com/...e2492d5434.gif is a differentiable bijection with http://data.artofproblemsolving.com/...d73aef9192.gif also differentiable and http://data.artofproblemsolving.com/...3cf5b507ea.gif are two vector fields onto M, then http://data.artofproblemsolving.com/...a2d163706c.gif, http://data.artofproblemsolving.com/...709656709f.gif are also vector fields onto N. Show that http://data.artofproblemsolving.com/...c419730e78.gif where http://data.artofproblemsolving.com/...45206835df.gif 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!

$F_*(p)$ is the "linearization" of F= <f(x,y), g(x,y)> at point p. It's matrix is $\begin{bmatrix}\frac{\partial f}{\partial x}(p) & \frac{\partial f}{\partial y}(p)\\ \frac{\partial g}{\partial x} & \frac{\partial g}{\partial y}\end{bmatrix}$ .

Here, $f(x_1,x_2)= (x_1^2- 2x_2, 4x_1^3x_2^2)$ so $f_*(x_1, x_2)= \begin{bmatrix}2x_1 & -2 \\ 12x_1^2x_2^2 & 8x_1^3x_2\end{bmatrix}$

Well it makes sense now. What about the second problem? How do you use the Lie bracket?