  The concept "Classical Electromagnetism" in the title of the paper here
refers to a theory built on three foundations: relativity principles, the
original Maxwell's equations, and the mathematics of exterior calculus. In this
theory of electromagnetism the general laws are expressed as geometric
(coordinate-free) relations between quantities on a four-dimensional spacetime
manifold. Based on these laws, and exterior calculus on metric spaces, we
derive fields and relations that describe electromagnetism from the inside
perspective of Lorentz observers, inside the Minkowski spacetime (the ordinary
flat spacetime). It is shown how fields and relations defined on the spacetime
are sliced into three-dimensional fields and relations on time slices, and that
Lorentz observers perceive these as time-dependent fields and relations on a
single three-dimensional Euclidean space. In exterior calculus (on metric
spaces) each field is associated to four field variants from which one can
choose how a field is to appear. Using this feature, we derive alternative
equation systems describing electromagnetism from the inside perspective. In
one system, A, all field quantities are form fields. In another system, C, the
fields are represented as vector and scalar fields. But in contrast to the
traditional Maxwell's equations with div and curl operators, there is here no
need for pseudo-vectors and pseudo-scalars. Some theorems of a pseudo-free
vector analysis, adapted to the C-system, are derived. The paper concludes with
a section where we analyse properties of the electromagnetic field and its laws
with respect to mirror transformations.
