Textbooks frequently use the Helmholtz theorem to derive expressions for electrostatic and
magnetostatic fields but they do not usually apply this theorem to derive expressions for
time-dependent electric and magnetic fields, even when there is no formal objection to doing so
because the proof of the theorem does not involve time derivatives but only spatial derivatives.
Here we address the question as to whether the Helmholtz theorem is useful in deriving expressions
for the fields of Maxwell’s equations. We show that when this theorem is applied to Maxwell’s
equations we obtain instantaneous expressions of the electric and magnetic fields, which are
formally correct but of little practical usefulness. We then discuss two generalizations of the
theorem which are shown to be useful in deriving the retarded fields.