  Textbooks frequently use the Helmholtz theorem to derive expressions for the
electrostatic and magnetostatic fields but they do not usually apply this
theorem to derive expressions for the 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 to
derive 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 to derive the retarded fields.
