- #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 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:
In both cases ##d()## operator is actually the exterior derivative operator and both definitions should be actually equivalent.
Does it make sense ? Thank you.
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.