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I need to integrate the following form over the 2-sphere: $$w=\frac{xdy\wedge dz-ydx\wedge dz+zdx\wedge dy}{(x^2+y^2+z^2)^\frac{3}{2}}$$ now a direct calculation shows that $dw=0$, thus from stokes' theorem we can integrate $dw$ over the unit ball instead and get $0$. however when i try the direct calculation via pullback i reach a problem. I define $$\phi:H\rightarrow \mathbb{D}^2, (x,y,z)\mapsto (x,y)$$ $$ \psi:N\rightarrow \mathbb{D}^2, (x,y,z)\mapsto (x,y)$$ the projections, where $\mathbb{D}^2=\{(x,y)\in\mathbb{R}^2 |x^2+y^2<1\}$ is the unit disk, $H=\{(x,y,z)\in S^2|z>0\}$ and $N=\{(x,y,z)\in S^2|z<0\}$. by considerations of orientation we get $$\int_{S^2}w=\int_\mathbb{D^2}(\phi^{-1})^*w-\int_\mathbb{D^2}(\psi^{-1})^*w$$ since $\phi$ preserves orientation while $\psi$ reverses orientation.

however, pulling $w$ back via $\psi^{-1}$ admits another minus sign arising from linearity in $z,dz$. so the two terms in the lastequation sum up instead of calcel each other out.

where am i going wrong?

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  • $\begingroup$ Is your two form sufficiently regular at the origin to use Stokes' theorem? $\endgroup$ – mickep Dec 15 '17 at 12:36

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