  In this first of two papers, we develop a steady-state version of classical
electrodynamics on the 3-sphere and in hyperbolic 3-space, including an
explicit formula for the vector-valued Green's operator, an explicit formula of
Biot-Savart type for the magnetic field, and a corresponding Ampere's Law
contained in Maxwell's equations.
  We then use this to obtain explicit integral formulas for the linking number
of two disjoint closed curves in these spaces.
  All the formulas, like their prototypes in Euclidean 3-space, are geometric
rather than just topological, in the sense that their integrands are invariant
under orientation-preserving isometries of the ambient space.
  In the second paper, we obtain integral formulas for twisting, writhing and
helicity, and prove the theorem LINK = TWIST + WRITHE in the 3-sphere and in
hyperbolic 3-space. We then use these results to derive upper bounds for the
helicity of vector fields and lower bounds for the first eigenvalue of the curl
operator on subdomains of these two spaces.
  An announcement of most of these results, and a hint of their proofs, can be
found in the Math ArXiv, math.GT/0406276.
