satisfies the continuity equation $$\begin{aligned}
\partial_\nu J^\nu=\frac{1}{\mu_0}\partial_\mu\partial^\mu \partial_\nu A^\nu=0,\end{aligned}$$ because of (24). Equation (25) is a second-order equation that causally connects the four-potential $A^\nu$ with the four-current $J^\nu$. Are there two first-order equations equivalent to the equation (25)? The answer is in the affirmative. We combine (24) and (25) to get the equation $$\begin{aligned}
\partial_\mu\big\{\partial^\mu A^\nu-\partial^\nu A^\mu\big\}=\mu_0J^\nu.\end{aligned}$$ We strongly suspect that the antisymmetric tensor $\partial^\mu A^\nu-\partial^\nu A^\mu$ could be physically significant. We find convenient to label this antisymmetric tensor as $$\begin{aligned}
F^{\mu\nu}=\partial^\mu A^\nu-\partial^\nu A^\mu,\end{aligned}$$ in terms of which (27) takes the elegant form $$\begin{aligned}
\partial_\mu F^{\mu\nu} =\mu_0J^\nu.\end{aligned}$$ This provides us another expression for the four-current $J^\nu=\partial_\mu F^{\mu\nu}/\mu_0$ that satisfies the continuity equation $$\begin{aligned}
\partial_\nu J^\nu=\frac{1}{\mu_0}\partial_\mu\partial_\nu  F^{\mu\nu}=0,\end{aligned}$$ because $\partial_\mu\partial_\nu  F^{\mu\nu}\equiv 0$ since the operator $\partial_\mu\partial_\nu$ is symmetric in the indices $\mu$ and $\nu$ and the tensor $F^{\mu\nu}$ is antisymmetric in these indices. On the other hand, any antisymmetric tensor field $F^{\mu\nu}$ in the four-space has an associated a dual tensor defined by $^*\!{F}^{\mu\nu}\!=\!(1/2)\varepsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}$, where $\varepsilon^{\mu\nu\alpha\beta}$ is the four-dimensional Levi-Civita symbol with $\varepsilon^{0123}\!=\!1$. A generalised Helmholtz theorem [@22; @24] states that an antisymmetric tensor field is completely determined by specifying its divergence and the divergence of its dual. We can show that 