- #1

cianfa72

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- About the definition of differential operator of a scalar function as one-form or covector field

Hi, I'd like to ask for clarification about the definition of

As far as I know, the differential of a scalar function ##f## can be understood as:

In both cases ##d()## operator is actually the

Does it make sense ? Thank you.

*differential*of a smooth scalar function ##f: M \rightarrow \mathbb R## between smooth manifolds ##M## and ##\mathbb R##.As far as I know, the differential of a scalar function ##f## can be understood as:

- a linear map ##df()## between tangent spaces defined at each point of domain and target manifolds (##T_{p}M## and ##T_{q}\mathbb R##)
- a one-form or covector field ##df## defined on the domain manifold ##M##

In both cases ##d()## operator is actually the

*exterior derivative operator*and both definitions should be actually equivalent.Does it make sense ? Thank you.