Intro to differential forms (part 19)

A quick but beautiful consequence of Stokes' theorem, and then the start of its proof in the general case.

[davidmetzler]

Most (though not all) textbooks in introductory differential geometry cover this material. There are also a few texts on multivariable/advanced calculus that emphasize differential forms...do a search on Amazon. Have fun!

[davidmetzler]

It goes back to earlier videos, but briefly, you first use the product rule and d^2 = 0 to show that it's dP wedge dx_2 wedge dx_3 ... wedge dx_k. Then you know that for a function, dP = sum_i (partal P/partial x_i) dx_i. But all of those except the term with i=1 die since you get a factor such as dx_2 wedge dx_2, which is zero.

[alxjones]

Thanks. Yeah, when I commented I hadn't watched the rest of the videos, but after watching through the first 18, this one made a lot more sense. This is a really interesting course. What would be a good way to follow up these videos, such as a text book or college course? I'm a math major and I want to study more calculus-esque maths like this.

[alxjones]

Can you explain more clearly how to calculate da? I don't quite understand why we have dp/dx1*x1^...^xk

[meep]

David! you son of a bitch! *shakes hand* <-- Predator

I finally understand it this