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.