Intro to differential forms (part 5)

Continuation of Part 4, finishing up the derivation of the integral of f dx wedge dy.

[russellbarry]

Thanks David I have both books. If you knew the level I was at you would not believe me. Your lectures are unique. I have never seen the forms approach done so well. You need to write a book. I learned cartans structured equations to get the Ricci tensor. I never got the the nuts and bolts of the subject. I know a lot of people out there that love these lectures. This is never really covered in college physics courses. Suskind at Stanford avoids doing relativity this way. The physics books cover this then kind of glaze over it. Someone needs to write a book that starts at the beginning. Trust me there are a lot of people looking for good books on this stuff.  If it were written like the DeMystified books, it would be a big seller. You have the grounds for a book with these lectures. Bring up differential forms to a PhD student in physics and the will say, what is that. I also like proofs, like the one for d(df) = 0. This is usually just stated, without showing how it is done. Sorry so long, but this is one subject that seems to be a secret, and when presented they make it magical. Zee in his Quantum Field Theory in a nutshell book brings it up and it is like he pulled a rabbit out of a hat. A good book on this is a long time coming. You now have PhD students watching these lectures. And the response, I did not know it at that level. Thanks again.

[davidmetzler]

...as long as we don't flip orientations (which I allude to repeatedly in these videos as being crucial to keep track of) then the standard change of variables formula for ordinary multiple integrals says that you need to put in a factor of the determinant of the matrix of derivatives of the change of variables (the Jacobian determinant). Well, that's _exactly_ what the differential form pullback mechanism will automatically do in changing the du \wedge dv. Hence it's all consistent.

[davidmetzler]

The short answer is that it's a definition, not a theorem---it's where we make contact with the usual notion of integration. But there is more to say. The whole point of differential forms (in terms of integral calculus) is that they can be integrated independently of coordinates. So it had better be true that when we change variables, the integral of f du \wedge dv (thought of as a 2-form) behaves exactly like the integral of f du dv (thought of in the ordinary way). See the next comment...

[russellbarry]

David I do not know if you check comments very often. I have looked for Textbooks covering this that makes it understandable. Do you have a Textbook or can you tell me the one you teach out of. These lectures are the best I have ever seen. Thanks. And thank you for these lectures.

[davidmetzler]

OK, I see better now where you're coming from. Feel free to ask more once you're ready.

[davidmetzler]

Hmm...Response 1: I'm not sure why you are taking the symbol "du" and creating "{du, 0}". If "du" has any independent meaning, it's already a 1-form, so your tilde operation doesn't seem meaningful. Also, you seem to be confusing my style of differential forms with a different (though equivalent) style where one sees a lot of antisymmetric matrices.

[davidmetzler]

Response 2: More important than worrying about various manipulations is to point out that in my presentation, the fact that you can take out the wedge product is not a calculation, it's a *definition*. So the kind of manipulations you're doing are fundamentally unnecessary. One does have to do some work to make sure that this definition is consistent with other known facts about integration, but it really is a definition.