Superposition for prescribed Maxwell fields and sources

Assume first that all pairs lie in the smooth branch. For jointly $C^1$ maps $F_1,\dots ,F_n$ from the open set $\mathrm{\Omega }\times I$ to ${\mathbb{R}}^q$ and scalars $a_1,\dots ,a_n$, the combination $F:={\sum }_j{a}_j{F}_j$ is jointly $C^1$ and ${\mathrm{\partial }}_{\mu }F={\sum }_j{a}_j{\mathrm{\partial }}_{\mu }{F}_j$ for every coordinate index $\mu$: componentwise, at a point $a$ the line map $t\mapsto F^i(a+t{e}_{\mu })={\sum }_j{a}_j{F}_j^i(a+t{e}_{\mu })$ is a finite sum of one-variable functions differentiable at $0$, so its derivative at $0$ exists and equals the combination ${\sum }_j{a}_j{\mathrm{\partial }}_{\mu }{F}_j^i(a)$ of derivatives by [F5] through the line-map definition [F4], and each partial ${\mathrm{\partial }}_{\mu }{F}^i={\sum }_j{a}_j{\mathrm{\partial }}_{\mu }{F}_j^i$ is continuous by the finite-sum clause [F6]; Hence $E$ and $B$ are jointly $C^1$ (and every additional derivative required by the branch combines by the same argument), while $\rho$ and $J$ are continuous by [F6]; thus all four combinations meet the regularity hypotheses of the smooth branch without assuming differentiability of the continuous sources.

Assume instead that all pairs lie in the weak branch. For locally integrable scalar fields $f_1,\dots ,f_n$ on $I\times \mathrm{\Omega }$ the combination $f:={\sum }_j{a}_j{f}_j$ is locally integrable: on each compact $K\subseteq I\times \mathrm{\Omega }$ the restrictions of the $f_j$ are $L^1$ functions on $K$, so their finite linear combination is $L^1$ on $K$ by [F7] and ${\int }_K\mathrm{\mid }f\mathrm{\mid }<\mathrm{\infty }$. By the linearity in $f$ of the regular functional [F8] one has $u_f={\sum }_j{a}_j{u}_{f_j}$, and by the defining formula and the bilinearity of the pairing [F9] the distributional derivatives satisfy ${\mathrm{\partial }}^{\alpha }{u}_f={\sum }_j{a}_j{\mathrm{\partial }}^{\alpha }{u}_{f_j}$, so that $\mathrm{div}$, $\mathrm{curl}$ and ${\mathrm{\partial }}_t$ of the combination are the corresponding combinations of the summands' distributional derivatives; componentwise the same holds for the vector fields $E_j$, $B_j$, $J_j$ and the scalar fields ${\rho }_j$.

The operators $\mathrm{div}$ and $\mathrm{curl}$ are linear on jointly $C^1$ vector fields: by the component formulas [F3] together with step 1.1, $\mathrm{div}({\sum }_j{a}_j{F}_j)={\sum }_i{\mathrm{\partial }}_i({\sum }_j{a}_j{F}_j^i)={\sum }_j{a}_j{\sum }_i{\mathrm{\partial }}_i{F}_j^i={\sum }_j{a}_j\mathrm{div}{F}_j$, and the same finite rearrangement applied to the two-term difference in each curl component gives $\mathrm{curl}({\sum }_j{a}_j{F}_j)={\sum }_j{a}_j\mathrm{curl}{F}_j$; the same argument gives ${\mathrm{\partial }}_t({\sum }_j{a}_j{F}_j)={\sum }_j{a}_j{\mathrm{\partial }}_t{F}_j$.

In the weak branch, for each $\psi \in C_c^{\mathrm{\infty }}(I\times \mathrm{\Omega })$, step 1.2 expands every pairing of the combination into the finite sum of the corresponding pairings of the summands, for instance $⟨\mathrm{div}E-\rho \mathrm{/}{\epsilon }_0,\psi ⟩={\sum }_j{a}_j⟨\mathrm{div}{E}_j-{\rho }_j\mathrm{/}{\epsilon }_0,\psi ⟩$, and likewise for $⟨\mathrm{div}B,\psi ⟩$, $⟨\mathrm{curl}E+{\mathrm{\partial }}_{t}B,\psi ⟩$ and $⟨\mathrm{curl}B-{\mu }_{0}J-c^{-2}{\mathrm{\partial }}_{t}E,\psi ⟩$; since each summand pairing vanishes by [F2], all four pairings of the combination vanish, so the combination satisfies the branch in the distributional sense, which is clause (a) in that branch.

In the smooth branch, applying step 2.1 to the combinations $E={\sum }_j{a}_j{E}_j$, $B={\sum }_j{a}_j{B}_j$, $\rho ={\sum }_j{a}_j{\rho }_j$ and $J={\sum }_j{a}_j{J}_j$, and using [F1] for each summand, gives $\mathrm{div}E={\sum }_j{a}_j{\rho }_j\mathrm{/}{\epsilon }_0=\rho \mathrm{/}{\epsilon }_0$, $\mathrm{div}B={\sum }_j{a}_j\cdot 0=0$, $\mathrm{curl}E=-{\sum }_j{a}_j{\mathrm{\partial }}_t{B}_j=-{\mathrm{\partial }}_{t}B$ and $\mathrm{curl}B={\sum }_j{a}_j({\mu }_0{J}_j+c^{-2}{\mathrm{\partial }}_t{E}_j)={\mu }_{0}J+c^{-2}{\mathrm{\partial }}_{t}E$, while step 1.1 supplies the regularity hypotheses; thus the combination satisfies the branch pointwise, which is clause (a) in that branch.

The two branches are exactly those the hypothesis permits, so clause (a) holds in every case; clause (b) records that only the finite linearity actually computed is asserted, with no nonlinear constitutive law and no infinite superposition, and clause (c) records that the result is a conditional consequence of the adopted postulate, which completes the proof.