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Thread: What is a surface in Maths and/or Physics?

  1. #1 What is a surface in Maths and/or Physics? 
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    I have learned in another forum that a ID lines can be a surface on a 2D plane and a 2D plane can be a surface on a 3D Volume.

    Can this be generalized mathematically as we introduce more (higher) dimensions?

    Is it the case (as it seems to me to be true in the examples I have given) that when we set one of the dimensional variables to a constant that what we have is a surface ?(I think they may be referred to as hypersurfaces in higher dimensions)

    By setting one of the dimensional variables to a constant (decreasing its value range to smaller and smaller quantities in the same way as is done in Calculus *,do we effectively transform the n-Dimensional model to a n-1 Dimensional model and create a (hyper)surface that "straddles" the 2 models?

    *ie as a limit
    Last edited by geordief; 11-16-2017 at 01:29 PM.
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  2. #2  
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    Quote Originally Posted by geordief View Post
    I have learned in another forum that a ID lines can be a surface on a 2D plane and a 2D plane can be a surface on a 3D Volume.

    Can this be generalized mathematically as we introduce more (higher) dimensions?

    Is it the case (as it seems to me to be true in the examples I have given) that when we set one of the dimensional variables to a constant that what we have is a surface ?(I think they may be referred to as hypersurfaces in higher dimensions)

    By setting one of the dimensional variables to a constant (decreasing its value range to smaller and smaller quantities in the same way as is done in Calculus *,do we effectively transform the n-Dimensional model to a n-1 Dimensional model and create a (hyper)surface that "straddles" the 2 models?

    *ie as a limit
    Here is the rigorous mathematical treatment of the concept called "embedding".
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  3. #3  
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    I am still not clear if a hypersurface is 2 dimensional or [n-1] dimensional for a n-dimensional space.

    In particular in 4-D space-time does one work with 2-D surfaces (for example in calculating curvature by means of parallel transporting of a vector) or is use also made of 3-D surfaces (presumably ,at least in my mind constructed by setting one of the dimensional variables to a constant).



    If the 3-D surface does exist and is useful can it be built up successively from the 6(?) combinations of 2-D at any point (event)?
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  4. #4  
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    Quote Originally Posted by geordief View Post
    I am still not clear if a hypersurface is 2 dimensional
    Only in 3 dimensions

    or [n-1] dimensional for a n-dimensional space.
    This is the answer.
    The math is quite clear on the subject. Have you read the web page I linked for you?
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  5. #5  
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    No but I will now.
    And I'll get back then......
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  6. #6  
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    Quote Originally Posted by AndrewC View Post
    Only in 3 dimensions



    This is the answer.
    The math is quite clear on the subject. Have you read the web page I linked for you?
    OK I have read the main body of that link and think I understand that a hypersurface is embedded in the manifold with 1 more dimensions than it has.

    Am I right though to think that there are nevertheless also 2D surfaces in higher dimensional manifolds (esp 4D space-time manifolds) even if is incorrect to refer to them as "hypersurfaces".

    Are they just referred to as surfaces or 2D surfaces?
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  7. #7  
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    Quote Originally Posted by geordief View Post
    OK I have read the main body of that link and think I understand that a hypersurface is embedded in the manifold with 1 more dimensions than it has.

    Am I right though to think that there are nevertheless also 2D surfaces in higher dimensional manifolds (esp 4D space-time manifolds) even if is incorrect to refer to them as "hypersurfaces".

    Are they just referred to as surfaces or 2D surfaces?
    yes
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