Integration on manifolds

lavinia

Gold Member
Thanks, I'd really appreciate that!
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.

Consider the two 1-forms defined in the plane minus the origin

$ω =( xdx + ydy)/ (x^2 + y^2)$

and

$τ = (-ydx + xdy)/(x^2 + y^2)$

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?

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