Velocity space ($\mathbf{v}\in {\mathbb{R}}^3$)

Velocity space is the space of velocity vectors at a fixed spatial position. Integrating a galactic distribution function over velocity space produces mass density; its normalized moments determine mean velocity and velocity dispersion.

For a gyrophase-independent phase-space distribution function, $d^{3}v=2\pi v_{\perp }d{v}_{\perp }d{v}_{\parallel }$. With first adiabatic invariant $\mu =v_{\perp }^2/(2B)$, this gives the measure in the title. When the magnetic field changes, both the measure and any explicit factor $\mu B$ in a moment must be perturbed.