= Velocity-shell identity for dynamical friction
{title2=$\int f(v)\frac{\mathbf V-\mathbf v}{|\mathbf V-\mathbf v|^3}\,d^3v=\frac{4\pi\mathbf V}{V^3}\int_0^Vf(v)v^2\,dv$}
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.
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