i only can integrate a k-form in a n-dimensional manifold, if k=n right?
I think you can integrate a k-form in n-dim if k<=n but its not guaranteed:
from the wikipedia article:
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.
are the integral of a 2-form associate with a vector field the same thing to surface integral of that vector field?
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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