It is a very common practice in the analysis procedure to introduce auxiliary functions, known as vector potentials, which will aid in the solution of the problems. The most common vector potential functions are the A (magnetic vector potential) and F (electric vector potential). Another pair is the Hertz potentials Пe and П h. Although the electric and magnetic field intensities (E and H) represent physically measurable quantities, among most engineers the potentials are strictly mathematical tools. The introduction of the potentials often simplifies the solution even though it may require determination of additional functions. While it is possible to determine the E and H fields directly from the source-current densities J and M, as shown in Figure, it is usually much simpler to find the auxiliary potential functions first and then determine the E and H.

This two-step procedure is also shown in Figure. The one-step procedure, through path 1, relates the E and H fields to J and M by integral relations. The two-step procedure, through path 2, relates the A and F (or Пe and Пh) potentials to J and M by integral relations. The E and H are then determined simply by differentiating A and F (or Пe and Пh). Although the two-step procedure requires both integration and differentiation, where path 1 requires only integration, the integrands in the two-step procedure are much simpler. The most difficult operation in the two-step procedure is the integration to determine A and F (or Пe and П h). Once the vector potentials are known, then E and H can always be determined because any well-behaved function, no matter how complex, can always be differentiated. The integration required to determine the potential functions is restricted over the bounds of the sources J and M. This will result in the A and F (or Пe and П h) to be functions of the observation point coordinates; the differentiation to determine E and H must be done in terms of the observation point coordinates. The integration the one-step procedure also requires that its limits be determined by the bounds of the sources. The vector Hertz potential Пe is analogous to A and Пh is analogous to F. The functional relation between them is a proportionality constant which is a function of the frequency and the constitutive parameters of the medium.