  In the past five years, there has been considerable research on the collider
phenomenology of TeV-scale extra compact dimensions which are universal - i.e.
accessible to all the Standard Model fields. We consider in detail a
fundamental conceptual issue in the higher-dimensional theory: Since the cross
product is only physically meaningful in three spatial dimensions, how do
traditional vector formulas involving the cross product or curl operation
generalize in the case of universal extra dimensions? For example, is angular
momentum physically meaningful, and, if so, how is it related to the position
and momentum vectors? We find that if angular velocity, torque, angular
momentum, and magnetic field, for example, are regarded as anti-symmetric
second-rank tensors instead of pseudovectors, we can construct tensor forms of
the standard cross product and curl equations that do generalize to higher
dimensions. Although Maxwell's equations have been generalized to extra
dimensions in terms of differential forms, we show that our simple technique
casts the generalized Maxwell's equations in a form very similar to the
standard vector calculus. We also provide direct integrals for the fields in a
practical tensor calculus form. We explore the electric and magnetic fields in
detail to show that there are no fundamental conceptual problems hidden in the
theory - e.g. how the Lorentz force can be meaningfully expressed in terms of
the velocity and magnetic field when the traditional cross product and
right-hand rule do not apply. Finally, we explore the effects that
compactification has on the effective form of Maxwell's equations, which could
be useful for direct tests of Coulomb's law.
