So, I've been struggling over the following problem from the chapter of integration from Lee's Introduction to Smooth Manifolds.
14-4. Define maps $F_+, F_- : \mathbb{B}^2 \rightarrow \mathbb{S}^2$ by $$F_{\pm} = ( u, v, \pm \sqrt{1 - u^2 - v^2}).$$
If $\omega$ is a smooth 2-form on $\mathbb{S}^2$, show that
$$\int_{\mathbb{S}^2} \omega = \int_{\mathbb{B}^2} F_+^* w - \int_{\mathbb{B}^2}F_-^* w,$$
where the integrals on the left-hand side are defined as the limits as $R \nearrow 1$ of the integrals over $\bar{B}_R (0)$. Be sure to justify the limits.
This problems is supposed to show that the hypothesis of the theorem of integrations of parametrizations can be relaxed somewhat.
But I just can't see how the classical definition of integrals over manifolds (with charts and partitions of unity) can be related to those limits. Actually my best chance so far is that it's related through some partition of unity, that has support on some open (upper or lower) hemisphere and that when you take the limit, eventually you will integrate over "everything that has to be integrated". But the chart definition (which involves partitions of unity) would implie involving 6 charts, and the theorem of parametrization doesn't work.
Any help or tip on how to carry on will be really appreciated. Thanks in advance.