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Integration over manifolds is commonly defined with object called chains. What about if I want to integrate the exterior derivative of a $k-form$ over the n-sphere and use Stokes theorem:

\begin{eqnarray} \int_{\sigma} d\omega &=& \int_{\partial \sigma} \omega \end{eqnarray}

I found in several books that this integral is zero, they argue it's because the sphere is compact so its boundary is zero.

My question is how can I relate this with chains??, I menan, how is the chain for a n-sphere and why its boundary is zero...

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