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**Question: Vector field and (n-1)-form representation of current density**

Electric current density can be represented by both a vector field and by a 2-form. Integrating them on a given surface must lead the same result. My question is, what is the relation between this vector field and the 2-form. More generally, in an n-dimensional space, between a vector field and an (n-1)-form, which results the same value when integrated on a (n-1) dimensional surface. Is the latter the Hodge-dual of the vector field regarded as an 1-form ? If yes, how can one see this?

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