Here is the first example - presented as problems. All are designed to make you think about integrating forms - what that means - and how it is done.Thanks, I'd really appreciate that!
Consider the two 1-forms defined in the plane minus the origin
##ω =( xdx + ydy)/ (x^2 + y^2)##
##τ = (-ydx + xdy)/(x^2 + y^2)##
1) Integrate each form over the unit circle using Cartesian coordinates
2) Integrate each form over the unit circle using polar coordinates
3) Consider the 1- form dθ is defined on the unit circle and let ψ be the mapping of the plane minus the origin onto the unit circle that divides each vector by its Euclidean length.
What is the pull back,ψ*(dθ), of dθ to the plane minus the origin?
What does this mapping tell you about the integral of ##ψ^*(dθ)## around a closed curve that does not enclose the origin?