Action stationarity implies the inhomogeneous Maxwell equations

The data of [F1] meet the hypotheses of [F2] at $\kappa :={\mu }_0$ and $J:=j$: $U$ is a bounded open subset of ${\mathbb{R}}^4$ whose closure is a compact subset of the chart image, $N$ is an open neighbourhood of $\bar{U}$, ${\mu }_0>0$ by [F7], and $A,j$ are smooth on $N$; the integrand of [F1] is term-by-term the integrand of [F2] at these parameters, so $S[A]=c^{-1}{S}_{\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{h}}[A]$ and both integrals are finite real numbers by the finiteness clause of [F2].

Fix $h\in C_c^{\mathrm{\infty }}(U;{\mathbb{R}}^4)$. Clause (a) of [F2] presents $s\mapsto S_{\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{h}}[A+sh]=a_0+a_{1}s+a_2{s}^2$ with finite real coefficients $a_0=S_{\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{h}}[A]$, $a_1={\int }_U(-{\mu }_0^{-1}{F}^{\mu \nu }{\mathrm{\partial }}_{\mu }{h}_{\nu }+j^{\nu }{h}_{\nu })d\lambda$ and $a_2={\int }_U{\mathrm{\ell }}_{2}d\lambda$, hence step 1.1 gives $S[A+sh]=c^{-1}{a}_0+c^{-1}{a}_{1}s+c^{-1}{a}_2{s}^2$; in particular the first variation of $S$ at $A$ in the direction $h$ is $c^{-1}{a}_1$, the first variation of $S_{\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{h}}$ multiplied by $c^{-1}>0$.

Because $A$ is smooth on the open $N\supseteq U$, every second partial derivative ${\mathrm{\partial }}_{\lambda }{\mathrm{\partial }}_{\mu }{A}_{\nu }$ exists and is continuous on $U$, so [F8] applies: expanding $F_{\mu \nu }={\mathrm{\partial }}_{\mu }{A}_{\nu }-{\mathrm{\partial }}_{\nu }{A}_{\mu }$ from [F1] gives ${\mathrm{\partial }}_{\lambda }{F}_{\mu \nu }+{\mathrm{\partial }}_{\mu }{F}_{\nu \lambda }+{\mathrm{\partial }}_{\nu }{F}_{\lambda \mu }=({\mathrm{\partial }}_{\lambda }{\mathrm{\partial }}_{\mu }{A}_{\nu }-{\mathrm{\partial }}_{\lambda }{\mathrm{\partial }}_{\nu }{A}_{\mu })+({\mathrm{\partial }}_{\mu }{\mathrm{\partial }}_{\nu }{A}_{\lambda }-{\mathrm{\partial }}_{\mu }{\mathrm{\partial }}_{\lambda }{A}_{\nu })+({\mathrm{\partial }}_{\nu }{\mathrm{\partial }}_{\lambda }{A}_{\mu }-{\mathrm{\partial }}_{\nu }{\mathrm{\partial }}_{\mu }{A}_{\lambda })=0$ on $U$, the six terms cancelling in three commuting pairs. This is clause (b).

Since $c^{-1}>0$, one has $c^{-1}{a}_1=0$ exactly when $a_1=0$; as $h$ was an arbitrary element of $C_c^{\mathrm{\infty }}(U;{\mathbb{R}}^4)$ and the two variation classes coincide by [F1] and [F2], the field $A$ is stationary for $S$ in the sense of clause (b) of [F1] if and only if $A$ is stationary for $S_{\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{h}}$ in the sense of [F2].

Applying clause (c) of [F2] with $\kappa ={\mu }_0$, $J=j$ and the field $A$ of step 1.1, the field $A$ is stationary for $S_{\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{h}}$ if and only if ${\mathrm{\partial }}_{\mu }{F}^{\mu \nu }=-{\mu }_0{j}^{\nu }$ on $U$ for every $\nu \in \{0,1,2,3\}$, the derivatives being classical partial derivatives of the smooth $A$ by [F2]; combined with step 3.1 this proves (i)$\iff$(ii), and it uses ${\mathrm{A}\mathrm{C}}_{\omega }$ of [F9] exactly through the cited lemma.

If $A$ is stationary, then (ii) holds by the equivalence of step 4.1; that consequent statement is clause (a).

Put $\phi :=-c{A}_0$, $\mathbf{A}:=(A_1,A_2,A_3)$, $E:=-\mathrm{\nabla }\phi -{\mathrm{\partial }}_t\mathbf{A}$ and $B:=\mathrm{curl}\mathbf{A}$. Since $A_{\mu }$ is the lowering of the smooth $A^{\mu }$ and satisfies $F_{\mu \nu }={\mathrm{\partial }}_{\mu }{A}_{\nu }-{\mathrm{\partial }}_{\nu }{A}_{\mu }$ by [F1], clause (c) of the potential dictionary [F5] identifies these $E,B$ with the fields whose component dictionary [F4] reproduces the same $F_{\mu \nu }$ and $F^{\mu \nu }$; the sources $\rho ,J$ satisfy $j=(c\rho ,J)$ by [F6], and $E,B$ are smooth and $\rho ,J$ continuous on $U$. The source convention and the constant identity ${\epsilon }_0=({\mu }_0{c}^2{)}^{-1}$ of [F7] place these data in the class of [F3]; the tensor system (TM1) holds by step 5.1 and the cyclic identity (TM2) by step 2.2, since the equivalence of clause (a) of [F3] is a pointwise component statement it restricts from the laboratory region to the open subset $U$, so that clause yields the four smooth-branch equations $\mathrm{div}E=\rho \mathrm{/}{\epsilon }_0$, $\mathrm{div}B=0$, $\mathrm{curl}E=-{\mathrm{\partial }}_{t}B$ and $\mathrm{curl}B={\mu }_{0}J+c^{-2}{\mathrm{\partial }}_{t}E$ on $U$, equivalently the eight scalar equations identified there. This is clause (c).

The argument used exactly the smooth class, the fixed component dictionary, the identification $S=c^{-1}{S}_{\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{h}}$ and the positive factor $c^{-1}$; it asserted no existence of a stationary field, imposed no boundary or decay condition, and used no weak, distributional, singular-source or non-potential-represented branch. The equivalence is conditional on the adopted stationarity principle [F10] and makes no empirical claim; Countable Choice enters only through [F2], as declared in the statement.