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Curl in 5D using levi-civita tensor

  1. Jun 22, 2014 #1
    i really lost with this. i see two possibilities:

    (1) something like,

    [itex]\epsilon_{abc}\partial_{a}A_{b}e_{c}[/itex] with a,b,c between 1 and 5

    or

    (2)like that

    [itex]\epsilon_{abcde}\partial_{a}A_{b}[/itex]

    one of the options nears correct?

    thank's a lot
     
  2. jcsd
  3. Jun 22, 2014 #2
    Unfortunately there is no universally accepted way (at least not that I'm aware of) to generalize the curl to higher dimensions so if someone uses this terminology you just have to check what definitions they are using.

    Probably the easiest framework to try and generalize vector calculus to higher dimensions is to use differential forms and the exterior derivative. To reproduce the standard 3-dimensional results, just use the obvious identifications of 1-forms and 2-forms with vector fields and then by simply doing the computations, it is immediate that
    [itex] d:\Omega^0(\mathbb{R}^3) \to \Omega^1(\mathbb{R}^3) [/itex]
    is the gradient,
    [itex] d:\Omega^1(\mathbb{R}^3) \to \Omega^2(\mathbb{R}^3) [/itex]
    is the curl, and
    [itex] d:\Omega^2(\mathbb{R}^3) \to \Omega^3(\mathbb{R}^3) [/itex]
    is the divergence.

    The upshot is that this immediately generalizes to arbitrary dimensions since the exterior derivative is defined on any manifold and so you can think of these as being the analogs of the required operators. In this case, the "curl" is just the exterior derivative from 1-forms to 2-forms which is given in coordinates by
    [itex] d\left( A_udx^u\right)=\frac{\partial A_u}{\partial x^\nu} dx^\nu \wedge dx^u [/itex]

    However, if you are looking for an operator which takes two objects on a manifold and returns an object of the same type (ie. curl takes two vector fields and returns another vector field) then the exterior derivative alone won't work since it takes p-forms to (p+1)-forms. If you take the above formula for the exterior derivative on [itex] \mathbb{R}^3[/itex], and apply the hodge star operator you get a 1-form back and the components of the one form are precisely the components of the curl (if you are being very precise here, I am also using the isomorphism of one-forms and vector fields to make this statement). The Levi-Civita symbol appears precisely here since locally one can express the hodge star operator using it. However in higher dimensions, the Hodge star of a 2-form is an (n-2)-form not a 1-form so this construction does not yield an operator which takes in two 1-forms and returns another 1-form.

    There are lots of things you can do to try and define your operator using the exterior derivative, hodge star and the musical isomorphisms (in fact I have seen some generalizations of the curl that even use a covariant derivative) however there is no standard way to define what you are looking for. So to answer your question you first need to tell us exactly what properties of the curl you want to preserve in the generalization so that we can get a well-defined object to work with.
     
  4. Jun 22, 2014 #3
    thanks buddy, I'll use your text as a starting point, I have to study much more, much more
     
  5. Jun 22, 2014 #4

    Matterwave

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    As an aside, generally you want repeated indices to be one upstairs and one downstairs to imply the summation (Einstein summation convention).
     
  6. Jun 23, 2014 #5
    No problem. If you want a book that goes over everything in my post in much more detail you could try the first section of "Gauge Fields, Knots and Gravity" by Baez and Muniain. It develops the basics of manifolds (although it leaves out a few proofs you can find in all the standard differential geometry books) with a view towards doing electrodynamics/gauge theory.
     
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