Cartesian Tensor Scalar Product and Spherical Harmonic Expansions in Boltzmann's Equation

The electron or ion distribution function, expanded in a sum of spherical harmonics, is shown to be equivalent to a tensor scalar product expansion. As such, it can be readily substituted into the Boltzmann equation to give transport equations integrated over angle and also the necessary equations for determination of the expansion coefficients. This has been done for terms up to order three, the order of the pressure transport tensor.