I'm trying to learn the Divergence/Stoke's theorem and I can't wrap my head around the meaning of a differential form in this context. What does it mean that a differential form represents a vector field? What does it mean to derive a differential form? Why do we sometimes take an integral wherein of the inner product of a form and its derivative? What does it mean to integrate over a form? Lastly, sometimes in the context of a form I see dx ^ dy . What does the ^ mean?
Can anyone give me some intuition about this? I've gone over my class notes and it all goes straight to formalities and I'm having trouble grasping these concepts. Any help would be welcomed!
Advice: First, you need understand some basic concepts of differential forms. I recommend two excellent readings of about differential forms, in my point-view.
1: Differential Forms, by Henri Cartan. This book is ideal for understand differential forms in various contexts, for example, Cartan develops the theory of forms in space of finite and infinite dimension.
2: Differential Forms, by Manfredo do Carmo. This book is ideal to learn the concepts of differential forms to apply in Differential Geometry.
So a quick reading of Manfredo's book will be great for you and your doubts, but I recommend, principally for geometry study, a reading in Cartan's book.