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In cylindrical coordinates, the Newtonian gravitational stress tensor gives radial angular-momentum flux $r{T}_{r\varphi }=r({\partial }_r\Phi )(r^{-1}{\partial }_{\varphi }\Phi )/(4\pi G)$. Nonaxisymmetric structure can correlate the field components and transport angular momentum. The isotropic negative-pressure part has zero off-diagonal component, so it gives no direct radial transport. This separates a shear flux from a contribution to normal force balance.

For an isolated self-gravitating fluid obeying ideal magnetohydrodynamics, let $I_{ij}=\int \rho x_i{x}_{j}dV$ and $K_{ij}=\frac{1}{2}\int \rho u_i{u}_{j}dV$. Use the force-stress convention $\rho D_t{u}_i={\partial }_j{T}_{ij}$, with pressure, magnetic and negative Newtonian gravitational stress tensor contributions. Two time derivatives of $I$ give $4K_{ij}$ plus $\int (x_i{\partial }_k{T}_{jk}+x_j{\partial }_k{T}_{ik})dV$. Integration by parts and symmetry yield the theorem when surface terms vanish. This fluid version complements the collisionless formulation of the tensor virial theorem.

The magnetic field satisfies $\nabla \cdot \mathbf{B}=0$. Expanding the divergence of its Maxwell stress tensor givesFor the Newtonian gravitational field $\mathbf{g}=-\nabla \Phi$, Poisson equation for Newtonian gravity gives ${\partial }_j{g}_j=-4\pi G\rho$, and the gradient representation gives ${\partial }_j{g}_i={\partial }_i{g}_j$. ConsequentlyThe negative of the Newtonian gravitational stress tensor, in this force-stress convention, therefore supplies $\rho g_i$. Adding the pressure stress supplies $-{\partial }_{i}p$, soEach summand is symmetric. Absence of external gravitational sources is needed to represent the full gravitational force by this self-gravitating stress.

First eliminate the density-dependent definition of the auxiliary $\mathbf{g}$. The Newtonian gravitational stress tensor can be written without derivatives of its normalization:Here ${\delta }_{ij}$ is the Kronecker delta, and repeated indices are summed. Differentiating givesThe first and third terms cancel because mixed partial derivatives commute; the final step uses the Poisson equation for Newtonian gravity. Therefore the gravitational force density is a stress divergence: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 isIn the momentum conservation equation, $r{T}_{r\varphi }$ 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.