A scalar cannot extrapolate into a vector, nor the converse, so the scalar and vector parts of the 4-potential extrapolate separately. To first order, the extrapolation formula for a scalar is
Ψ(r,t) = (∇ Ψ0) dot dr + dΨ0/dt dt.
The coordinate system is orthogonal, so the space and time parts extrapolate separately, but space-time cross terms appear anyway in the quadratic extrapolations. A vector extrapolates as three scalars
A(r,t) = (dr dot ∇)A0 + dA0/dt dt.
The order of the extrapolation needed depends on the rank of the tensor. For example, the third rank contravariant tensor requires quadratic extrapolations. The following computer program will perform the extrapolations up to the fifth order. The solutions represent the most general time-varying 4-potential.