Charged-fluid current and total stress conservation

Given: The smooth fields and equations of Strict charged barotropic tensor system, including $g(u,u)=-1$, antisymmetric $F$, $j^a=cNZ{u}^a$, $u^a{\mathrm{\nabla }}_{a}Z=0$, and ${\mathrm{\nabla }}^a{T}_{m,ab}=F_{bc}{j}^c$.

Strict charged barotropic tensor system: $T_{m,ab}=(ϵ+p)u_a{u}_b+p{g}_{ab}$, $j^a=cNZ{u}^a$, $N=N_{\ast }\exp ({\int }_{{ϵ}_{\ast }}^{ϵ}(z+p(z){)}^{-1}dz)$, $g(u,u)=-1$, $u^a{\mathrm{\nabla }}_{a}Z=0$, and ${\mathrm{\nabla }}^a{T}_{m,ab}=F_{bc}{j}^c$.

The first fundamental theorem: if $f$ is integrable on $[a,b]$ and continuous at $c$, then $F^{\mathrm{\prime }}(c)=f(c)$; in particular a continuous $f$ has $F$ as a primitive: If an integrand is continuous on the interval between a base point and a variable endpoint, the derivative of its integral function at that endpoint is the integrand value.

Maxwell stress algebra and covariant conservation: The current draft CP01 statement, for smooth $F,j$ satisfying $dF=0$ and ${\mathrm{\nabla }}_a{F}^{ab}=-{\mu }_0{j}^b$, is that $T_{\mathrm{E}\mathrm{M},ab}={\mu }_0^{-1}(F_{ac}{F}_b{}^c-\frac{1}{4}{g}_{ab}{F}_{cd}{F}^{cd})$ satisfies ${\mathrm{\nabla }}^a{T}_{\mathrm{E}\mathrm{M},ab}=-F_{bc}{j}^c$. Its actual draft proof differentiates the quadratic tensor by the product rule, uses the cyclic identity from $dF=0$ to cancel the scalar-invariant derivative, and substitutes the stated Maxwell source sign. The supplier carrier is present but remains draft with precheck pending, so the total-stress conclusion below is used conditionally and its acceptance is not inferred here.