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I read here that an electric current is not a vector and is not a scalar

  • an electric current do not obey vector addition
  • an electric current is not invariant under rotation, in other words, do not respect this proof

Why use an electric current or "extract" an electric current from fossil sites (as an isobaric process etc), for example, if electric current is neither vector and not even scalar ?

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  • $\begingroup$ Current density is a 2-form. Which is usually perceived as a vector in 3D under the right correspondence between 2-forms and vectors. $\endgroup$
    – Bort
    Mar 20 '20 at 9:34
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The electric current is a scalar (in the ordinary sense).

an electric current is not invariant under rotation,

It is invariant under rotation. If you have a circuit that draws 2 mA from a battery and you rotate the circuit by 90 degrees then you are still drawing 2 mA from the battery.

One important thing to remember is that circuit theory does not involve space at all. There is no concept of rotation or translation or distance or geometry. There is a topology, but nothing else. In that sense the designation between scalar and vector is rather meaningless in circuit theory.

It is possible to consider circuits in the context of Maxwell’s equations instead of circuit theory. In that case rotations and translations do exist, but in that theory current is no longer a primary quantity. In Maxwell’s equations the important quantity is current density which is a vector for the usual purposes.

Maxwell’s equations can also be formulated quite elegantly in terms of tensors or differential forms and so forth. In those cases current density is no longer a separate distinct entity, but is part of something else. For example, in the tensor formulation current density and charge density are merged to form the four-current-density vector and then current density is just the spatial component of that vector.

So the point is that context is important. I have given a brief overview of several contexts, but if you want me to delve deeper into a specific context then just leave a comment.

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As said in one of the answers to the linked, current density is a vector. I think you will find that current is sometimes stated in the text, when the formulae make clear that current density is actually used.

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