Relation between codifferential and boundary operator

[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?

 

[HallsofIvy]

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

 

[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?

 

[HallsofIvy]

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

 

[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?