# How to calculate the Poincare dual of a ray on R2-{0}?

1. Oct 4, 2011

### kakarotyjn

S={(x,0)|x>0} on R^2-{0},I need to calculate the closed Poincare dual of S.

Assume $$\omega=f(x,y)dx+g(x,y)dy$$ on $$R^2$$-{0} have compact support.Then we need to find a form $$\eta$$in $$H^1 (R^2 - {0} )$$ satisfying $$\int\limits_S {i^* \omega = \int\limits_M {\omega \wedge \eta } }$$,

The book let me prove $$d\theta /2\pi$$($$\theta$$is the angle function) is the poincare dual.

But in my calculation,$$\int\limits_S {i^* \omega = \int_0^{ + \infty } {f(x,0)dx} }$$ and $$1/2\pi \int\limits_M {\omega \wedge d\theta } = 1/2 \pi \int_0^{ + \infty } {dr} \int_0^{2\pi } {(f(r\cos \theta ,r\sin \theta )\cos \theta + g(r\cos \theta ,r\sin \theta )\sin \theta )d\theta }$$ ,how to prove they are equal?

Last edited: Oct 4, 2011