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When I see this book, one strange problem arouse.

Thank you for seeing this.

Here is the problem.

c

_{0}, c

_{1}: [0,1] → ℝ

^{2}- {0}

c : [0,1]

^{2}→ ℝ

^{2}- {0}

given by

c

_{0}(s) = (cos2πs,sin2πs) : a circle of radius 1

c

_{1}(s) = (3cos4πs,3sin4πs) : a circle of radius 3 that winds twice.

c(s,t) = (1-t)*c

_{0}(s) + t*c

_{1}(s).

Then c(s,0) = c

_{0}(s) , and c(s,1) = c

_{1}(s).

Now the boundary of a 2-chain c is c

_{0}-c

_{1}.

Is this right?

If so, there is a problem.

If w = (-ydx + xdy)/(x

^{2}+y

^{2}) is a 1-form on ℝ

^{2}- {0} , then it is closed.(i.e. dw = 0)

Then by the Stokes' theorem we get

-2π = ∫

_{c0-c1w = }∫

_{∂c}w = ∫

_{c}dw = 0

Why this happened?