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Condition to integrate a k-form

  1. Sep 18, 2014 #1
    i only can integrate a k-form in a n-dimensional manifold, if k=n right?
     
  2. jcsd
  3. Sep 18, 2014 #2

    jedishrfu

    Staff: Mentor

    I think you can integrate a k-form in n-dim if k<=n but its not guaranteed:

    from the wikipedia article:

    http://en.wikipedia.org/wiki/Differential_form
     
  4. Sep 19, 2014 #3
    You can integrate k-form on k dimensional manifold. You can have k-form on n dimensional manifold, where n>k. This can be integrated on k-dimensional submanifold of original manifold. For example you can integrate 2-form on a surface in three dimensional space.
     
  5. Sep 19, 2014 #4
    thanks dudes
     
  6. Sep 20, 2014 #5
    are the integral of a 2-form associate with a vector field the same thing to surface integral of that vector field?
     
  7. Sep 22, 2014 #6

    lavinia

    User Avatar
    Science Advisor
    Gold Member

    The integral of a k form is defined on a smooth k chain. A smooth k chain is a formal algebraic sum of smooth oriented k simplices.
    An oriented k manifold can be expressed as a smooth k chain so integration of k forms is defined. Not so for an unorientable k manifold. It can not be expressed as a smooth k chain.

    Sometimes an k form can be integrated over lower dimensional manifolds. The result is a lower dimensional differential form. For example integration along the fibers of a fiber bundle reduces the dimension of the form by the dimension of the fiber.
     
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