Differential forms as sections

[Kreizhn]

Hey,

A quick question. In the definition of a differential form, we normally require that they be sections of the k-th exterior power of the cotangent bundle. However, on page 14 of Jurdjevic's book on http://books.google.ca/books?id=PpZ...6AEwAA#v=onepage&q=differential form&f=false", he defines them simply as sections of the cotangent projection.

Is there a mistake in his notes or does this represent an alternative way of examining differential forms?

 

[zhentil]

His definition is for 1-forms (covectors).

 

[Kreizhn]

Ah, so do you mean to say that a when the "dimension" of the form is omitted, it should be assumed to be a 1-form?

 

[zhentil]

That's not usual, no. I think the author was being sloppy.

 

[blue_raver22]

Hello,

Thank you for your question. In the definition of a differential form, it is common to require that they be sections of the k-th exterior power of the cotangent bundle. This is because differential forms are geometric objects that describe the behavior of vector fields on a manifold. By requiring them to be sections of the exterior power, we ensure that they are compatible with the geometric structure of the manifold.

However, Jurdjevic's definition of differential forms as sections of the cotangent projection is also valid. It may represent a different way of examining differential forms, possibly in the context of geometric control theory. It is important to note that there are various ways of defining and studying differential forms, and each approach may have its own advantages and applications.

I would suggest further researching and understanding both definitions to gain a deeper understanding of differential forms and their applications. I hope this helps answer your question.