Results 1 to 3 of 3

Math Help - Differentiable Functions on Manifolds

  1. #1
    Newbie
    Joined
    Apr 2010
    Posts
    12

    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!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,693
    Thanks
    1466
    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}
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Apr 2010
    Posts
    12
    Well it makes sense now. What about the second problem? How do you use the Lie bracket?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Differentiable Functions
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: January 9th 2011, 07:13 PM
  2. Replies: 9
    Last Post: December 17th 2010, 08:13 AM
  3. Differentiable Functions
    Posted in the Calculus Forum
    Replies: 1
    Last Post: April 26th 2010, 01:28 AM
  4. Mobius n-manifolds within Complex n manifolds
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: January 20th 2010, 07:15 AM
  5. Sequence of differentiable functions, non-differentiable limit
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: April 3rd 2009, 05:13 AM

Search Tags


/mathhelpforum @mathhelpforum