Velocity-shell identity for dynamical friction ($\int f(v)\frac{\mathbf{V}-\mathbf{v}}{|\mathbf{V}-\mathbf{v}{|}^3}{d}^{3}v=\frac{4\pi \mathbf{V}}{V^3}{\int }_0^{V}f(v)v^{2}dv$)

For an isotropic, integrable background phase-space distribution function, rotational symmetry makes the integral radial in velocity space. Its divergence with respect to $\mathbf{V}$ is $4\pi f(V)$ by the Dirac delta identity for the divergence of the inverse-square vector field. Integrating over a velocity-space sphere and using the divergence theorem proves the displayed identity. Exterior velocity shells have zero contribution, just as in the spherical shell theorem. This explains why the leading-log Chandrasekhar dynamical friction integral contains only particles slower than the test body.