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Important connection between computational and mathematical electromagnetics is presented. The newly developed well-conditioned electromagnetic frequency domain surface integral equation formulations, the current and charge integral equations, are shown to be related to Picard's extended Maxwell system, an extended partial differential equation system that has the correct static behavior. Electromagnetic surface integral representations are derived in this paper for traditional surface integral equation formulations and for the Picard system using the fundamental solution approach, i.e., from the definition of Dirac's delta function. The surface integral representations are constructed with proper solid angle coefficients starting from the scalar Helmholtz equation. The traditional surface integral equation formulations are shown to be derived from Maxwell's curl equations and are thus lacking the contribution of the divergence equations at zero frequency. It is shown that the new current and charge formulations can be derived from the surface integral representation of the Picard system.