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I Electric Field and manifolds

  1. Mar 22, 2016 #1
    This is from Jackson, "Electrodynamics"
    a field is a fuction mapping


    phi: M -> T, x -> phi(x) from a base manifold M into a target manifold T.

    field X: R3 * R1 -> R3
    x(r,t) ->X(x)


    I think this is eucledian R4 to R3 so I wonder why Jackson explained this with the concept of manifolds?
    Is it necessary?
     
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  3. Mar 23, 2016 #2

    Orodruin

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    Impossible to say unless you specify what X is.
     
  4. Mar 23, 2016 #3
  5. Mar 24, 2016 #4
    It is necessary if you want to consider spaces which have topologies other than that of R^n. For example a field on an n-dimensional sphere S^n.
    If you restrict yourself to defining a field as a function from R^m to R^n, then you could not define a field on S^n as a global function because S^n is not homeomorphic to R^m for any n, m >= 0. In the example you mention here, the domain space is just R^4 with the usual topology and diferentiable structure, but this space is of course a manifold so it fits in the definition the book gives.
     
  6. Mar 26, 2016 #5
    Consider also that in practical applications, the boundaries of a particular region might not be made up of chunks of planes isometric to R2. Often the boundary of a cavity or waveguide, or the surfaces of antennas and reflectors are best described as two dimensional manifolds rather than as subsets of R3 or R3,1.
    Also, thinking in terms of manifolds makes some nice simplifications and generalizations easier later on, like using the generalized Stokes theorem for all integration over bounded regions, curves, or surfaces.
     
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