# Condition to integrate a k-form

1. Sep 18, 2014

### davi2686

i only can integrate a k-form in a n-dimensional manifold, if k=n right?

2. Sep 18, 2014

### 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

3. Sep 19, 2014

### Blazejr

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.

4. Sep 19, 2014

### davi2686

thanks dudes

5. Sep 20, 2014

### davi2686

are the integral of a 2-form associate with a vector field the same thing to surface integral of that vector field?

6. Sep 22, 2014

### lavinia

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.