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?