  This paper studies the relativistic angular momentum for the generalized
electromagnetic field, described by $r$-vectors in $(k,n)$ space-time
dimensions, with exterior-algebraic methods. First, the angular-momentum tensor
is derived from the invariance of the Lagrangian to space-time rotations
(Lorentz transformations), avoiding the explicit need of the canonical tensor
in Noether's theorem. The derivation proves the conservation law of angular
momentum for generic values of $r$, $k$, and $n$. Second, an integral
expression for the flux of the tensor across a $(k+n-1)$-dimensional surface of
constant $\ell$-th space-time coordinate is provided in terms of the normal
modes of the field; this analysis is a natural generalization of the standard
analysis of electromagnetism, i. e. a three-dimensional space integral at
constant time. Third, a brief discussion on the orbital angular momentum and
the spin of the generalized electromagnetic field, including their expression
in complex-valued circular polarizations, is provided for generic values of
$r$, $k$, and $n$.
