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Diffeomorphisms beginner

  1. Apr 20, 2013 #1
    I'm a complete rookie here, and i'd like some help.
    For starters , can a diffeomorphic mapping be represented via a matrix , like say a transformation?
    If so, how would it be parameterised?
     
  2. jcsd
  3. Apr 20, 2013 #2

    dx

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    No you cannot represent it by a matrix, since it is not a linear transformation.
     
  4. Apr 20, 2013 #3

    dx

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    A diffeomorphism is a linear transformation of all the tangent spaces, so you can give the infinite collection of matrices

    ∂Xi/∂Yj

    at each point of the manifold
     
  5. Apr 20, 2013 #4

    WannabeNewton

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    Hi garrus. dx already answered the crux of your question but let's specialize to the case of ##\mathbb{R}^{n}##. Let ##U\subseteq \mathbb{R}^{n}## be open, ##p\in U##, and ##F:U\rightarrow \mathbb{R}^{m}## a map differentiable at ##p##. Recall that ##F## is differentiable at ##p## if there exists a linear map ##DF(p)## such that ##\lim _{v\rightarrow 0}\frac{|F(p + v) - F(p) - DF(p)v|}{|v|} = 0##. As you may remember, we call ##DF(p)## the total derivative of ##F## at ##p##. Now, as dx noted we may not be able to represent ##F## itself as a matrix if it isn't itself linear on ##\mathbb{R}^{n}## (in which case it agrees with its total derivative) but ##DF(p)## is linear and one can show that the standard matrix representation of ##DF(p)## is given by ##[DF(p)]_{S} = (\frac{\partial F^{j}}{\partial x^{i}}(p))##. This is none other than the Jacobian matrix. You can think of the linear map ##DF(p)## as being the best linear approximation of ##F## in a neighborhood of ##p##. As dx noted above, you can then develop such a formalism on arbitrary smooth manifolds.
     
  6. Apr 20, 2013 #5
    Thanks for your responses, but i think i'm way out of my league :/
    I want to apply a diffeomorphism in image analysis and i'm looking for a way to build a function to map pixel positions.

    edit: disregard that.
     
    Last edited: Apr 20, 2013
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