Second mass moment tensor ($I_{ij}=\int \rho (\mathbf{x})x_i{x}_j{d}^{3}x$)

The second mass moment tensor records the quadratic spatial distribution of mass. Its trace-free part is the mass quadrupole moment $Q_{ij}=I_{ij}-{\delta }_{ij}{I}_{kk}/3$. The mechanical moment-of-inertia tensor instead equals $J_{ij}={\delta }_{ij}{I}_{kk}-I_{ij}$. A rigid rotation $L$ transforms the second mass moment tensor as $I(t)=L(t)I(0)L(t{)}^T$.

The scalar second mass moment is the trace of the second mass moment tensor. The moment of inertia about a specified axis instead integrates squared distance from that axis. For an isotropic distribution each axial moment of inertia is $2I/3$; the sum over three orthogonal axes is $2I$.