Boundary of a chain, Stokes' theorem.

[Kim.S.H]

Hi, I'm studying multivariable analysis using Spivak's book "calculus on manifolds"
When I see this book, one strange problem arouse.
Thank you for seeing this.
Here is the problem.

c₀ , c₁ : [0,1] → ℝ² - {0}
c : [0,1]² → ℝ² - {0}
given by
c₀(s) = (cos2πs,sin2πs) : a circle of radius 1
c₁(s) = (3cos4πs,3sin4πs) : a circle of radius 3 that winds twice.
c(s,t) = (1-t)*c₀(s) + t*c₁(s).

Then c(s,0) = c₀(s) , and c(s,1) = c₁(s).

Now the boundary of a 2-chain c is c₀-c₁.
Is this right?

If so, there is a problem.

If w = (-ydx + xdy)/(x²+y²) is a 1-form on ℝ² - {0} , then it is closed.(i.e. dw = 0)

Then by the Stokes' theorem we get

-2π = ∫_{c0-c1w =} ∫_∂cw = ∫_cdw = 0

Why this happened?

 

[Kim.S.H]

oh,,, c wasn't well-defined on R^2 - 0

 

[mathwonk]

you have proved that if a cylinder maps into the complement of the origin, then both boundary circles wind the same number of times about the origin. this is quite interesting and can be used to prove the fundamental theorem of algebra. I.e. this can be used to prove that any polynomial of degree n winds a large circle the same number of times about the origin as does its lead term z^n, in particular it winds a non zero number of times if the polynomial is not constant.

It follows that the polynomial acting on the disc forming the interior of that circle must hit the origin, since if not, the winding number on the large circle bounding the disc, would equal the winding number on a very small circle near the center of the disc, namely zero.The general statement that a continuous map from a disc to the plane must hit the origin, if it wraps the boundary of the disc a non zero number of times about the origin, is a generalization of the intermediate value theorem, which states that a continuous function from a closed interval tot he line, must hit the origin if it sends the boundary points of the interval to opposite sides of the origin.

There are also higher dimensional generalizations, for which one needs to define wrapping number of an n-1 sphere about the origin in n-space. Usually winding numbers are defined by integrating the "angle form" dtheta around the curve, so in general one wants to integrate a version of a "solid angle form" over a surface. see problem 5-31 of spivak.

by integrating the solid angle form over a sphere in 3 space, it follows that the identity map of the 2-sphere cannot be extended to a 3 dimensional "cylinder" in such a way that the map on the other 2 sphere bounding the cylinder is the antipodal map. I.e. the antipodal map wraps the sphere around the origin a different number of times from the identity map, (-1 times as opposed to 1 time). it follows that there is no everywhere non zero continuous vector field on the sphere, since such a vector field would allow one to define the non existent map above.

Technically these arguments using integrals of differential forms require hypotheses of smoothness rather than just continuity, but they can probably be tweaked a bit to get the stronger results. (E.g. since dtheta is a "closed" form, it can actually be integrated over continuous, not just smooth curves. The point is that closed forms have integrals that do not change under small deformation, so you can make a close approximation of a continuous curve by a smooth one.)