Tangent Bundle vs. Vector Field

Discussion in 'Physics & Math' started by mannyfold, Apr 23, 2006.

  1. mannyfold Registered Member

    Messages:
    7
    Does anyone know the exact difference between a tangent bundle and a vector field? Mathworld says:
    A vector field is a tangent bundle section of its tangent bundle.

    The problem is that using the ordinary definitions, it appears that a vector field is equivalent to a tangent bundle, not just a section.

    Can someone clarify this?
     
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  3. DaleSpam TANSTAAFL Registered Senior Member

    Messages:
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    Hi Mannyfold, welcome to SciForums,

    A vector field in physics is just a vector-valued function over space. I don't know what a tangent bundle is. Are you trying to describe some surface to which a vector field is tangent at all points?

    -Dale
     
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  5. Physics Monkey Snow Monkey and Physicist Registered Senior Member

    Messages:
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    Hi mannyfold,

    Let me try to clarify things for you. The tangent space to manifold M at a point p, T<SUB>p</SUB>M, is the place where all the tangent vectors at p live. The tangent bundle is then the set of all ordered pairs (p,v) where p is a point in M and v is a point in T<SUB>p</SUB>M. In other words, each element of the tangent bundle is a pair consisting of a point p and a tangent vector from T<SUB>p</SUB>M. A vector field is a function f:M -> TM which takes a point p in M and gives you an ordered pair (p,v(p)) in TM. Different vector fields correspond to different ways to choose a vector in the tangent space at p. So while a vector field is a function whose range is contained in TM, it is not the tangent bundle itself.

    For a general fiber bundle F over M, a section of that bundle is a map s:M -> F which satisfies the addition requirement that s(p) "lies over p," or in technical terms, &pi;(s(p)) = p, where &pi; is the projection associated with the fiber bundle. You can also say that a section maps a point p to an element of the fiber &pi;<SUP>-1</SUP>(p). How ever you phrase it, this is exactly what a vector field does.

    Hope this helps.
     
    Last edited: Apr 23, 2006
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