= Solution
In a contractible region, the <Maxwell equations> $\nabla\cdot\mathbf B=0$ and the <Poincaré lemma> imply the existence of a <magnetic vector potential> $\mathbf A$ such that $\mathbf B=\nabla\times\mathbf A$. <Faraday's law of induction> then gives
$$
\nabla\times\left(\mathbf E+\partial_t\mathbf A\right)=0.
$$
The expression in parentheses is a <gradient> there, so a scalar <electric potential> $\phi$ can be chosen with
$$
\boxed{\mathbf B=\nabla\times\mathbf A,\qquad \mathbf E=-\nabla\phi-\partial_t\mathbf A.}
$$
These statements are local, or global when the domain has the required topology and the <electromagnetic fields> are regular. A nonzero flux through a topologically nontrivial closed surface can obstruct a single globally defined <vector potential>, even though local expressions remain valid. The <electric potential> and <magnetic vector potential> are not unique: $\mathbf A\mapsto\mathbf A+\nabla\chi$ and $\phi\mapsto\phi-\partial_t\chi$ leave both <electromagnetic fields> unchanged.
For the <Lorenz gauge condition> $\nabla\cdot\mathbf A=-c^{-2}\partial_t\phi$, <Gauss's law> becomes
$$
-\nabla^2\phi-\partial_t(\nabla\cdot\mathbf A)=\frac{\rho}{\epsilon_0},\qquad
\left(\nabla^2-\frac1{c^2}\partial_t^2\right)\phi=-\frac{\rho}{\epsilon_0}.
$$
The <Ampere-Maxwell law> gives
$$
\nabla(\nabla\cdot\mathbf A)-\nabla^2\mathbf A
=\mu_0\mathbf J-\frac1{c^2}\nabla\partial_t\phi-\frac1{c^2}\partial_t^2\mathbf A.
$$
Cancel the two <gradient> terms using the <Lorenz gauge condition>. Hence
$$
\boxed{\left(\nabla^2-\frac1{c^2}\partial_t^2\right)\phi=-\frac{\rho}{\epsilon_0},\qquad
\left(\nabla^2-\frac1{c^2}\partial_t^2\right)\mathbf A=-\mu_0\mathbf J.}
$$
Both potential equations have the same <d'Alembert operator>, with their respective <charge density> and <current density> sources. In a source-free region these are homogeneous <wave equations> with propagation speed $c$.
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