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Math Help - Property of derivative

  1. #1
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    Property of derivative

    Hello,

    i have a question about a immersion or submersion.
    A function is a immersion/submersion at a point p if its derivative at that point p is injective/surjective.

    I have read that if a function is a immersion/submersion at p, then it is a immersion/submersion in a nbh. of p.

    Can you explain me, why there must be a open nbh. s.t. the function is a immersion/submersion for all points in this nbh.?

    Regards
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  2. #2
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    Quote Originally Posted by Sogan View Post
    Hello,

    i have a question about a immersion or submersion.
    A function is a immersion/submersion at a point p if its derivative at that point p is injective/surjective.


    What exactly does it mean for you that a function is injective/subjective ON ONE POINT? For me this

    is senseless, but perhaps you have a definition I don't.

    Tonio



    I have read that if a function is a immersion/submersion at p, then it is a immersion/submersion in a nbh. of p.

    Can you explain me, why there must be a open nbh. s.t. the function is a immersion/submersion for all points in this nbh.?

    Regards
    .
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  3. #3
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    Hello,

    i didn't talk about injectivity at some point.
    I mean injectivity of THE DERIVATIVE at some point.

    Perhaps my statement was not so clear.


    Regards
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  4. #4
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    Quote Originally Posted by Sogan View Post
    i didn't talk about injectivity at some point.
    I mean injectivity of THE DERIVATIVE at some point.
    What is the difference? That is the question you were asked to address.
    I dare say that most of us have never seen local injectivity used.
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  5. #5
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    Ok, i try to explain the difference.

    if i have a function f:E->F between two banchspaces. Then the derivative (at x)
    Df(x) is a linear map between E and F, for all x in E. (think of the jacobian-Matrix)
    If we consider the derivative of f, we have Df: E->Hom(E,F), i.e. a function on E.

    Now it makes no sense to talk about injectivity of f in x.
    But it makes sense to talk about the injectivity of Df at the point x, since Df(x) is a linear map between vector-spaces.

    I hope you understand the situation.

    Regards
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  6. #6
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    Quote Originally Posted by Sogan View Post
    Hello,

    i have a question about a immersion or submersion.
    A function is a immersion/submersion at a point p if its derivative at that point p is injective/surjective.

    I have read that if a function is a immersion/submersion at p, then it is a immersion/submersion in a nbh. of p.

    Can you explain me, why there must be a open nbh. s.t. the function is a immersion/submersion for all points in this nbh.?

    Regards
    In the context of Euclidean spaces (real or complex, and finite-dimensional), it is easy to see that the set of linear transformations of maximal rank (in the case of immersions or submersion) are open in the set of linear transformations. This same argument applies to finite-dimensional differentiable manifolds. I'm not so sure about the general Banach space setting though.
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  7. #7
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    Ok thank you for your help.
    You say, "In the context of Euclidean spaces (real or complex, and finite-dimensional), it is easy to see that the set of linear transformations of maximal rank (in the case of immersions or submersion) are open in the set of linear transformations."

    Can you please explain me, why the sets are open? I don't see it. Thank you!
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  8. #8
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    Quote Originally Posted by Sogan View Post
    Ok thank you for your help.
    You say, "In the context of Euclidean spaces (real or complex, and finite-dimensional), it is easy to see that the set of linear transformations of maximal rank (in the case of immersions or submersion) are open in the set of linear transformations."

    Can you please explain me, why the sets are open? I don't see it. Thank you!
    Think of the linear trnasformations as matrices then they are surjective (it works the same for injective) iff they have a minor with nonzero determinant (after possibly relabeling coordinates) and since the determinant is a continous function we can pick a neighbourhood in which said minor is still invertible.
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