Differentiable Functions on Manifolds

**Ricko** · May 6th 2010, 09:12 AM

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

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!

**HallsofIvy** · May 7th 2010, 02:35 AM

$F_*(p)$ 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 $f_*(x_1, x_2)= \begin{bmatrix}2x_1 & -2 \\ 12x_1^2x_2^2 & 8x_1^3x_2\end{bmatrix}$

**Ricko** · May 10th 2010, 05:22 PM

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

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