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Thread: Question about local parameterization of a manifold

  1. #1
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    Question about local parameterization of a manifold

    Just to lay the backdrop, suppose that I have an n-dimensional manifold $M^n$ and choose a particular coordinate chart on the manifold $(U,\phi)$, $\phi: U \rightarrow \mathbb{R^n}$.

    I understand that $\phi$ is the local system of coordinates. My question concerns $\phi^{-1}$. In many books or lecture notes, $\phi^{-1}$ is called a local parameterization of $M$, and it is left at that. I am having trouble envisioning how the inverse map of a coordinate map is a parameterization.

    Would someone please walk me through a simple example to clear up my confusion as to $\phi$ and $\phi^{-1}$? The thing that makes the confusion worse is that I somehow think that it should be obvious and that I already know this from coordinate transformations I have done in calculus and in physics.

    Thanks.
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    Re: Question about local parameterization of a manifold

    Hey weirdanalysis.

    Are you familiar with an inverse operator in linear algebra? If you are then it will make a lot more sense of what this inverse mapping will mean when extended to multiple dimensions.
    Thanks from weirdanalysis
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    Re: Question about local parameterization of a manifold

    Quote Originally Posted by chiro View Post
    Hey weirdanalysis.

    Are you familiar with an inverse operator in linear algebra? If you are then it will make a lot more sense of what this inverse mapping will mean when extended to multiple dimensions.
    Until just this moment I thought I was but now I'm not so sure because I can't see what you're driving at. I mean, I know what an inverse map is and I'm comfortable even with it being multidimensional (I think). But here again, I feel like I should know this ... like I'm at the threshold of understanding but I just can't quite get there.
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    Re: Question about local parameterization of a manifold

    What it means for a linear operator is that you have a matrix (call it A) and if you are mapping a vector X to B then the inverse operator will be A_inverse.

    If it is an n-dimensional system you need an n-dimensional operator and the operator need not be linear (i.e. not be represented solely as a matrix - but often a function of matrices).

    That's the basic idea.
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    Re: Question about local parameterization of a manifold

    I am having trouble envisioning how the inverse map of a coordinate map is a parameterization.

    Practice on the unit circle \mathbb{S}^1\subseteq \mathbb{R}^2.
    Construct a local parametrization (say x=cosu,y=sinu, 0\leq u\leq \pi/2) and inverse it, to obtain (x/\sqrt(x^2+y^2),y/\sqrt(x^2+y^2)).

    Do note that many writers choose to define local charts using \phi^{-1} instead of \phi.
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