
19.2. First-Order Decompositions. Examples 1249
where L
1
and L
2
are linear differential operators in time t. For operators of this form,
the following two identities have been used in the derivation of the definitive results stated
below:
∆(xg) ≡ x∆g + 2∇g, curl[∆(xg)] ≡ curl(x∆g),
where g is an arbitrary scalar function.
For all solutions of system (19.1.2.5)–(19.1.2.6) based on formula (19.2.1.19) and for
four distinct representations of the vector function Ψ
n
, the unknown functions ψ
(1)
and
ψ
(2)
satisfy two independent (identical) equations (19.2.1.1), and the functions ϕ and p are
described by Eqs. (19.2.1.3)–(19.2.1.4).
Note that the above-described