In the more general formulation of the field integration equations, both electric and magnetic currents are included. In that case, the total electric and magnetic fields are given by the following equations:
:<math>\mathbf{E = E^{i}} + \mathbf{\iiint_V \overline{\overline{G}}_{EJ}(r|r') \cdot J(r') } + \mathbf{\iiint_V \overline{\overline{G}}_{EM}(r|r') \cdot M(r') }</math>Â Â :<math>\mathbf{H = H^{i}} + \mathbf{\iiint_V \overline{\overline{G}}_{HJ}(r|r') \cdot J(r') } + \mathbf{\iiint_V \overline{\overline{G}}_{HM}(r|r') \cdot M(r') }</math><!--[[File:image001_tn.gif]]-->
The above coupled equations involve four types of dyadic Green's functions that represent the electric and magnetic field radiated by an electric or a magnetic current. The incident or impressed electric and magnetic fields Ei and Hi exist independently of the given structures and are related to each other depending on the type of excitation source.
Enforcing the boundary conditions on the integral definitions of the '''E''' and '''H''' fields results in a system of integral equations as follows:
:<math>\mathcal{L}_E(E) = \mathcal{L}_E \left(\mathbf{E = E^{i}} + \mathbf{\iiint_V \overline{\overline{G}}_{EJ}(r|r') \cdot J(r') } + \mathbf{\iiint_V \overline{\overline{G}}_{EM}(r|r') \cdot M(r') } \right)= 0</math>Â Â :<math>\mathcal{L}_H(H) = \mathcal{L}_H \left(\mathbf{E = E^{i}} + \mathbf{\iiint_V \overline{\overline{G}}_{HJ}(r|r') \cdot J(r') } + \mathbf{\iiint_V \overline{\overline{G}}_{HM}(r|r') \cdot M(r') } \right)= 0</math><!--[[File:image016_tn.gif]]-->
where [[File:05_3d-integrals_tn.gif]] <math>\mathcal{L}_E(E)</math> is the boundary value operator for the electric field and [[File:05_3d-integrals_tn.gif]] <math>\mathcal{L}_H(H)</math> is the boundary value operator for the magnetic field. For example, they may require that the tangential components the '''E'''field vanish on perfect electric conductors. Or they may require that the tangential components the '''E''' and '''H''' fields be continuous across an aperture in a perfect ground plane. Given the fact that the dyadic Greenâs functions and the incident or impressed fields are all known, one can solve the above system of integral equations to find the unknown currents '''J''' and '''M'''. Therefore, through these relationships you can easily cast the above integral equations in terms of unknown '''E''' and '''H''' fields.
=== Galerkin Testing ===