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An extension of the Helmholtz theorem is proved, which states that tworetarded vector fields ${\bf F}_1$ and ${\bf F}_2$ satisfying appropriateinitial and boundary conditions are uniquely determined by specifying theirdivergences $\nabla\cdot{\bf F}_{1}$ and $\nabla\cdot{\bf F}_{2}$ and theircoupled curls $-\nabla\times{\bf F}_{1}-\partial {\bf F}_{2}/\partial t$ and$\nabla\times{\bf F}_{2}-(1/c^2)\partial {\bf F}_{1}/\partial t$, where $c$ isthe propagation speed of the fields. When a corollary of this theorem isapplied to Maxwell's equations, the retarded electric and magnetic fields aredirectly obtained. The proof of the theorem relies on a novel demonstration ofthe uniqueness of the solutions of the vector wave equation.

Helmholtz's theorem for two retarded fields and its application to Maxwell's equations