Exploring the Relationship between k-forms and l-forms on m Manifold

[blendecho]

So I was wondering about this... if $$\omega$$ is a $$k$$-form and $$\eta$$ is a $$l$$-form, and $$m$$ is a $$k+l+1$$ manifold in $$\mathbb{R}^n$$, what's the relationship between $$\int_M \omega\wedge d\eta$$ and $$\int_M d\omega\wedge \eta$$
given the usual niceness of things being defined where they should be, etc. etc. The manifold has no boundary, so am I correct in writing $$\int_{\partial M}\omega\wedge\eta=0$$?

 

[explain]

So I was wondering about this... if $$\omega$$ is a $$k$$-form and $$\eta$$ is a $$l$$-form, and $$m$$ is a $$k+l+1$$ manifold in $$\mathbb{R}^n$$, what's the relationship between $$\int_M \omega\wedge d\eta$$ and $$\int_M d\omega\wedge \eta$$
given the usual niceness of things being defined where they should be, etc. etc. The manifold has no boundary, so am I correct in writing $$\int_{\partial M}\omega\wedge\eta=0$$?

I think it's basically integration by parts. You start with the identity
$$\textrm{d}(\omega\wedge \eta) = \textrm{d}\omega\wedge \eta+(-1)^{k}\omega\wedge\textrm{d}\eta$$. Then you integrate both sides over M, taking into account that $$\int_M \textrm{d}(...)=0$$ since $$\partial M=0$$.

 

[tacman]

The relationship between k-forms and l-forms on an m-manifold can be better understood through the concept of exterior derivative and wedge product. The exterior derivative of a k-form \omega is a (k+1)-form denoted by d\omega, while the wedge product of a k-form \omega and an l-form \eta is a (k+l)-form denoted by \omega\wedge\eta.

In terms of integration, the integral of a k-form \omega over an m-manifold M can be written as \int_M \omega, while the integral of an l-form \eta over the same manifold can be written as \int_M \eta. Now, if we consider the integral of the wedge product \omega\wedge\eta over M, it can be written as \int_M \omega\wedge\eta.

Using the properties of exterior derivative and wedge product, we can rewrite this integral as \int_M \omega\wedge d\eta + \int_M d\omega\wedge\eta. This shows that the integral of a wedge product of a k-form and an l-form is equal to the sum of the integral of the exterior derivative of the k-form wedged with the l-form and the integral of the exterior derivative of the l-form wedged with the k-form.

Now, coming to the relationship between \int_M \omega\wedge d\eta and \int_M d\omega\wedge\eta, we can see that they are not equal in general. The integral of \omega\wedge d\eta involves integrating a (k+l+1)-form over an m-manifold, while the integral of d\omega\wedge\eta involves integrating a (k+l)-form over the same manifold. These two forms are not necessarily equal, unless there are certain conditions imposed on the manifold and the forms.

As for your question about the boundary of the manifold, since the manifold has no boundary, then \partial M = \emptyset and therefore \int_{\partial M}\omega\wedge\eta=0. This follows from the fact that the boundary of a manifold is a lower dimensional manifold and therefore cannot contain forms of higher degree.

In conclusion, the relationship between k-forms and l-forms on an m-manifold can be understood through the properties of exterior derivative and wedge product. The integral of a wedge product of forms is equal to