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that's pretty much the proof of Stolkes Theorem given in Spivak

but I'm having a lot of difficulty understanding the details

specifically...when the piecewise function is defined for j>1 the integral is 0

and for j=1 the integral is nontrivial...why is it defined like that?

Also, Im having difficulty understanding what the inclusion map does (spivak defines it

as I(j,alpha) which is a continuous function or a chain of some sort) but the pull back

I*(j,alpha)fdx^1....dx^n is taken and integrated over in that piecewise function

could someone shed some light on that?

Thanks

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# Spivak (Calculus on Manifolds) proof of stolkes theorem

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