Thread: Differentiable Functions on Manifolds

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!

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

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

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