Scalar potential functions in electromagnetism are related only

to electrostatic problems because when curlE=0 (electrostatic case)

then E=grad(psi) where psi - electric potential function. {curl(grad(psi))=0

for any smooth diffentiable function}

On the other hand curlH=j where j - electric current density so

the magnetic field can not be expressed as a gradient of a scalar function.

With one exception when j=0 then scalar magnetic function may exist (permanent

magnets problems).

Thanks a lot for reply. It is fine. I would like to see more physical insights than mathematical ones. Classic references for further reading are welcome.

You are right but physical meaning have electromagnetic

fields, not potential functions.

Note that even general definition of such physical quantity as voltege

is expresses as the contour integral of the electric field from point A to B.

It generally depends on the path of integration (which is well know in the electromagnetic field theory). In some cases when curlE=0 and

the scalar potential exists the voltage can be expressed as

the potential difference between points B and A. In such case the contour integral of the electric field is independent on the integration path.

Thanks a lot.

In particular :DO NOT get fooled by "scalar electromagnetic waves"..they are "pushed" by certain people and are based on simple mathematical errors