Relation between codifferential and boundary operator

1. Jul 18, 2008

navigator

As we know,the codifferential $$\delta$$ is the adjoint of the exterior derivative,and the boundary operator $$\partial$$ is also the adjoint of exterior derivative according to stokes' theorem, then what is the relation between codifferential and boundary operator?

2. Jul 19, 2008

HallsofIvy

Staff Emeritus
?? The only relation is exactly what you have given: Stoke's theorem.

3. Jul 20, 2008

navigator

Stokes' theorem states that $$<D,d \omega >=<\partial D,\omega>$$ ($$\partial$$ is the boundary operator), exterior derivative d and codifferential $$\delta$$ hold the relation $$(\theta ,d \omega)=(\delta \theta ,\omega)$$,then could we form a formula between $$\partial$$ and $$\delta$$ directly?

4. Jul 20, 2008

HallsofIvy

Staff Emeritus
No, they are completely different things. In fact, it really does not make sense to talk about "d" without the $\omega$ or $\delta$ with out the $\theta$.

5. Aug 24, 2008

navigator

So B=Dual(A) and C=Dual(A) do not imply B=C, right? I once thought that the dual of one object must be unique, it is not true?