Multipole expansion in magnetostatics

We derive the multipole expansions of the magnetostatic field and vector potential of an arbitrary steady current density. A simplifying parametrization of the (l + l)th-order tensor of lth-order moments of the current density in terms of an lth-order tensor b_il...il allows us to derive all orders in the multipole expansions using only Cartesian coordinates of tensors. We do not use a magnetic scalar potential or spherical harmonics. The field B^(l)(r) of the lth-order magnetostatic multipole depends on only the 2l + 1 independent components of the symmetric traceless part b^s0_il...il of b_il...il in exactly the same way as the field E^(l)(r) of the lth-order electrostatic multipole depends on the lth-order symmetric traceless tensor ρ^s0_il...il of multipole moments of the charge density. The vector potential that depends on only the symmetric traceless tensors b^s0_il...il differs from the vector potential in the Coulomb gauge. Our derivation shows that the fact that only the symmetric traceless part of b_il...il contributes to the magnetostatic field is a consequence of charge conservation and gauge invariance.