A Family of Nontrivial Homology Classes (part 2)
We continue investigating the differential forms we defined last time. Recall that we started with the position vector field 
I’ve asserted that 
![\displaystyle\begin{aligned}{}[\omega(p)]&(v_1,\dots,v_n)\\&=[[\pi^*\hat{\omega}](p)](v_1,\dots,v_n)\\&=[\hat{\omega}(\pi(p))](\pi_{*p}v_1,\dots,\pi_{*p}v_n)\\&=\left[\hat{\omega}\left(\frac{p}{\lvert p\rvert}\right)\right]\left(\frac{1}{\lvert p\rvert}\mathcal{I}_{\frac{p}{\lvert p\rvert}}\mathcal{I}_p^{-1}v_1,\dots,\frac{1}{\lvert p\rvert}\mathcal{I}_{\frac{p}{\lvert p\rvert}}\mathcal{I}_p^{-1}v_n\right)\\&=\frac{1}{\lvert p\rvert^n}\left[\hat{\omega}\left(\frac{p}{\lvert p\rvert}\right)\right]\left(\mathcal{I}_{\frac{p}{\lvert p\rvert}}\mathcal{I}_p^{-1}v_1,\dots,\mathcal{I}_{\frac{p}{\lvert p\rvert}}\mathcal{I}_p^{-1}v_n\right)\\&=\frac{1}{\lvert p\rvert^n}\left[[du^0\wedge\dots\wedge du^n]\left(\frac{p}{\lvert p\rvert}\right)\right]\left(P\left(\frac{p}{\lvert p\rvert}\right),\mathcal{I}_{\frac{p}{\lvert p\rvert}}\mathcal{I}_p^{-1}v_1,\dots,\mathcal{I}_{\frac{p}{\lvert p\rvert}}\mathcal{I}_p^{-1}v_n\right)\\&=\frac{1}{\lvert p\rvert^n}\left[[du^0\wedge\dots\wedge du^n]\left(\frac{p}{\lvert p\rvert}\right)\right]\left(\mathcal{I}_{\frac{p}{\lvert p\rvert}}\mathcal{I}_p^{-1}P(p),\mathcal{I}_{\frac{p}{\lvert p\rvert}}\mathcal{I}_p^{-1}v_1,\dots,\mathcal{I}_{\frac{p}{\lvert p\rvert}}\mathcal{I}_p^{-1}v_n\right)\\&=\frac{1}{\lvert p\rvert^n}[du^0\wedge\dots\wedge du^n]\left(\mathcal{I}_p^{-1}P(p),\mathcal{I}_p^{-1}v_1,\dots,\mathcal{I}_p^{-1}v_n\right)\\&=\frac{1}{\lvert p\rvert^n}\left[[du^0\wedge\dots\wedge du^n](p)\right](P(p),v_1,\dots,v_n)\\&=\left[\left[\frac{1}{\lvert p\rvert^n}\hat{\omega}\right](p)\right](v_1,\dots,v_n)\end{aligned}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cbegin%7Baligned%7D%7B%7D%5B%5Comega%28p%29%5D%26%28v_1%2C%5Cdots%2Cv_n%29%5C%5C%26%3D%5B%5B%5Cpi%5E%2A%5Chat%7B%5Comega%7D%5D%28p%29%5D%28v_1%2C%5Cdots%2Cv_n%29%5C%5C%26%3D%5B%5Chat%7B%5Comega%7D%28%5Cpi%28p%29%29%5D%28%5Cpi_%7B%2Ap%7Dv_1%2C%5Cdots%2C%5Cpi_%7B%2Ap%7Dv_n%29%5C%5C%26%3D%5Cleft%5B%5Chat%7B%5Comega%7D%5Cleft%28%5Cfrac%7Bp%7D%7B%5Clvert+p%5Crvert%7D%5Cright%29%5Cright%5D%5Cleft%28%5Cfrac%7B1%7D%7B%5Clvert+p%5Crvert%7D%5Cmathcal%7BI%7D_%7B%5Cfrac%7Bp%7D%7B%5Clvert+p%5Crvert%7D%7D%5Cmathcal%7BI%7D_p%5E%7B-1%7Dv_1%2C%5Cdots%2C%5Cfrac%7B1%7D%7B%5Clvert+p%5Crvert%7D%5Cmathcal%7BI%7D_%7B%5Cfrac%7Bp%7D%7B%5Clvert+p%5Crvert%7D%7D%5Cmathcal%7BI%7D_p%5E%7B-1%7Dv_n%5Cright%29%5C%5C%26%3D%5Cfrac%7B1%7D%7B%5Clvert+p%5Crvert%5En%7D%5Cleft%5B%5Chat%7B%5Comega%7D%5Cleft%28%5Cfrac%7Bp%7D%7B%5Clvert+p%5Crvert%7D%5Cright%29%5Cright%5D%5Cleft%28%5Cmathcal%7BI%7D_%7B%5Cfrac%7Bp%7D%7B%5Clvert+p%5Crvert%7D%7D%5Cmathcal%7BI%7D_p%5E%7B-1%7Dv_1%2C%5Cdots%2C%5Cmathcal%7BI%7D_%7B%5Cfrac%7Bp%7D%7B%5Clvert+p%5Crvert%7D%7D%5Cmathcal%7BI%7D_p%5E%7B-1%7Dv_n%5Cright%29%5C%5C%26%3D%5Cfrac%7B1%7D%7B%5Clvert+p%5Crvert%5En%7D%5Cleft%5B%5Bdu%5E0%5Cwedge%5Cdots%5Cwedge+du%5En%5D%5Cleft%28%5Cfrac%7Bp%7D%7B%5Clvert+p%5Crvert%7D%5Cright%29%5Cright%5D%5Cleft%28P%5Cleft%28%5Cfrac%7Bp%7D%7B%5Clvert+p%5Crvert%7D%5Cright%29%2C%5Cmathcal%7BI%7D_%7B%5Cfrac%7Bp%7D%7B%5Clvert+p%5Crvert%7D%7D%5Cmathcal%7BI%7D_p%5E%7B-1%7Dv_1%2C%5Cdots%2C%5Cmathcal%7BI%7D_%7B%5Cfrac%7Bp%7D%7B%5Clvert+p%5Crvert%7D%7D%5Cmathcal%7BI%7D_p%5E%7B-1%7Dv_n%5Cright%29%5C%5C%26%3D%5Cfrac%7B1%7D%7B%5Clvert+p%5Crvert%5En%7D%5Cleft%5B%5Bdu%5E0%5Cwedge%5Cdots%5Cwedge+du%5En%5D%5Cleft%28%5Cfrac%7Bp%7D%7B%5Clvert+p%5Crvert%7D%5Cright%29%5Cright%5D%5Cleft%28%5Cmathcal%7BI%7D_%7B%5Cfrac%7Bp%7D%7B%5Clvert+p%5Crvert%7D%7D%5Cmathcal%7BI%7D_p%5E%7B-1%7DP%28p%29%2C%5Cmathcal%7BI%7D_%7B%5Cfrac%7Bp%7D%7B%5Clvert+p%5Crvert%7D%7D%5Cmathcal%7BI%7D_p%5E%7B-1%7Dv_1%2C%5Cdots%2C%5Cmathcal%7BI%7D_%7B%5Cfrac%7Bp%7D%7B%5Clvert+p%5Crvert%7D%7D%5Cmathcal%7BI%7D_p%5E%7B-1%7Dv_n%5Cright%29%5C%5C%26%3D%5Cfrac%7B1%7D%7B%5Clvert+p%5Crvert%5En%7D%5Bdu%5E0%5Cwedge%5Cdots%5Cwedge+du%5En%5D%5Cleft%28%5Cmathcal%7BI%7D_p%5E%7B-1%7DP%28p%29%2C%5Cmathcal%7BI%7D_p%5E%7B-1%7Dv_1%2C%5Cdots%2C%5Cmathcal%7BI%7D_p%5E%7B-1%7Dv_n%5Cright%29%5C%5C%26%3D%5Cfrac%7B1%7D%7B%5Clvert+p%5Crvert%5En%7D%5Cleft%5B%5Bdu%5E0%5Cwedge%5Cdots%5Cwedge+du%5En%5D%28p%29%5Cright%5D%28P%28p%29%2Cv_1%2C%5Cdots%2Cv_n%29%5C%5C%26%3D%5Cleft%5B%5Cleft%5B%5Cfrac%7B1%7D%7B%5Clvert+p%5Crvert%5En%7D%5Chat%7B%5Comega%7D%5Cright%5D%28p%29%5Cright%5D%28v_1%2C%5Cdots%2Cv_n%29%5Cend%7Baligned%7D&bg=e6e6e6&fg=333333&s=0&c=20201002)
as asserted. Along the way we’ve used two things that might not be immediately apparent. First: the derivative 
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