This chapter develops the mathematical structures needed to treat electromagnetism as a field theory on physical space and time. It begins by formalizing the geometry of position, distance, area, and volume in three-dimensional Euclidean space and emphasizes the distinction between coordinates and physical vectors. The electromagnetic independent variables r and t, and the fields ρ, J, E, and B, are introduced as scalar and vector fields with well-defined transformation properties. Coordinate transformations, rotation matrices, and the distinction between polar and axial vectors are used to connect covariance with the form of physical laws. The chapter then builds the differential and integral vector operators grad, div, curl, and the Laplacian, derives Gauss’s and Stokes’ theorems, and rewrites Maxwell’s equations in integral form with clear geometric meaning. Finally, it introduces the wave operator and wave equations for E and B, and develops Fourier and Poisson–Helmholtz decompositions, including Jefimenko-type source representations, to show how fields are determined by their sources.

This appendix develops the mathematical machinery of screening and Green’s functions needed for electrostatics and magnetostatics. Starting from the screened Coulomb potential, it computes its Fourier transform, takes the unscreened limit, and thereby derives the standard free-space Green’s function and its representation as a Fourier integral. The appendix then examines convergence issues and the definition of the delta function, showing how to regularize Fourier representations with exponential factors. Translational invariance is used to motivate partial Fourier transforms when only some coordinates are symmetric, reducing Poisson’s equation to an ordinary differential equation in the remaining variable. These methods are applied to construct the Green’s function for half-space bounded by a grounded conducting plane, from which one obtains the induced surface charge, the potential, and the force on a point charge near the conductor.

This chapter builds the integral solution of Maxwell’s equations using Green’s functions and makes the causal structure of electromagnetism explicit. Starting from the wave equations for E and B, it reduces the problem to a generic scalar wave equation and introduces the Green’s function, G.

G as the response to a spacetime impulse. Solving the three-dimensional wave equation yields the retarded and advanced Green’s functions and leads to retarded integral solutions for arbitrary source distributions. Applying these to charge and current produces the Jefimenko equations, which cleanly separate quasistatic near fields from radiation fields and show how fields at a point depend on sources on its past light cone. A spacetime picture in terms of light cones clarifies retardation and the domain of dependence. The chapter then specializes to moving point charges, deriving compact expressions for their electric and magnetic fields and the associated Jacobian factors. An alternative Fourier–space derivation of the Green’s function connects these classical results to methods used in quantum field theory.