= Solution
First eliminate the density-dependent definition of the auxiliary $\mathbf g$. The <Newtonian gravitational stress tensor> can be written without derivatives of its normalization:
$$
T_{ij}=\frac1{4\pi G}\left(\partial_i\Phi\,\partial_j\Phi-\frac12\delta_{ij}\partial_k\Phi\,\partial_k\Phi\right).
$$
Here $\delta_{ij}$ is the <Kronecker delta>, and repeated indices are summed. Differentiating gives
$$
\partial_jT_{ij}=\frac1{4\pi G}\left(\partial_j\partial_i\Phi\,\partial_j\Phi+\partial_i\Phi\,\nabla^2\Phi-\partial_k\Phi\,\partial_i\partial_k\Phi\right)
=\frac{\partial_i\Phi}{4\pi G}\nabla^2\Phi=\rho\partial_i\Phi.
$$
The first and third terms cancel because mixed <partial derivatives> commute; the final step uses the <Poisson equation for Newtonian gravity>. Therefore \b[the gravitational force density is a stress divergence]:
$$
\boxed{-\rho\nabla\Phi=-\nabla\cdot\mathbf T.}
$$
The identity holds for spatially varying <mass density>: substituting the definition of $\mathbf g$ before differentiating prevents erroneous extra density-gradient terms.
For the <gravitational stress contribution to angular-momentum transport>, the anisotropic term $\rho g_i g_j$ supplies <angular momentum transport>. In cylindrical components its radial–azimuthal entry is
$$
T_{r\phi}=\frac{\partial_r\Phi\,(r^{-1}\partial_\phi\Phi)}{4\pi G}=\rho g_rg_\phi.
$$
In the momentum conservation equation, $rT_{r\phi}$ is the radial stress contribution to the <angular-momentum flux>, whose sign depends on the correlated radial and azimuthal field components. It vanishes for a perfectly axisymmetric potential but can be nonzero for spiral disturbances.
The other term is isotropic and acts as an effective negative <pressure>, $P_g=-\rho g^2/2=-|\nabla\Phi|^2/(8\pi G)$. Its force contribution is $-\nabla P_g=+\nabla(\rho g^2/2)$, modifying normal compression and force balance. It has no off-diagonal shear component and therefore no direct radial <angular momentum transport>. In an axisymmetric averaged disk it supplies no azimuthal torque. The complete symmetric <Newtonian gravitational stress tensor> also expresses <conservation of angular momentum> without an internal couple.
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