Maxwell and Geometry
can be represented geometrically. Define two simple cubic lattices - unit cells shown on the left. Along the edges of one arrange the E fields, along the edges of the other, the H fields.
Next arrange the two lattices so that they interpenetrate: one lattice
centres the other. Note that each face of the H lattice is pierced by a
line of force of the E lattice. Applying Stokes theorem around the
edges of the face gives the first Maxwell equation. Similarly the
second equation can be obtained by considering faces of the E
lattice. Even if the lattice is distorted, we can still apply Stokes
and get some equations.
These can be transformed back to the original Maxwell's equations on
the undistorted lattice provided that we
modify and and the fields.