The Maxwell equations of electromagnetism are usually written with vectors:
nabla.D = rho, dD/dt - nablaxH = -J,
dB/dt + nablaxE = 0, nabla.B = 0.
We rewrite them in terms of the Clifford algebra Cl_3 of the Euclidean space
R^3. Perpendicular unit vectors e1,e2,e3 in R^3 induce a basis for Cl_3:
1, e1,e2,e3, e12,e13,e23, e123
containing a scalar 1, the vectors e1,e2,e3, the bivectors e12,e13,e23 and
a volume element e123. The multiplication rules of Cl_3 are fixed by
e1e1 = e2e2 = e3e3 = 1, e1e2 = -e2e1, e1e3 = -e3e1, e2e3 = -e3e2
and
e12 = e1e2, e13 = e1e3, e23 = e2e3, e123 = e1e2e3.
Two vectors E = E1e1+E2e2+E3e3 and B = B1e1+B2e2+B3e3 have a Clifford product,
EB = E1B1+E2B2+E3B3+(E1B2-E2B1)e12+(E1B3-E3B1)e13+(E2B3-E3B2)e23, which is a
sum of a scalar and a bivector,
EB = E.B + E^B or EB = E.B + (ExB)/e123,
where
E.B = (EB+BE)/2, E^B = (EB-BE)/2 and 1/e123 = -e123.
Note that Be123 = B1e23+B2e31+B3e12, a bivector. Using
nabla^E = (nablaxE)e123, and
nabla_|(Be123) = -nablaxB, nabla^(Be123) = (nabla.B)e123,
we get
grade
nabla.D = rho, 0 scalar
dD/dt + nabla_|(He123) = -J, 1 vector
d(Be123)/dt + nabla^E = 0, 2 bivector
nabla^(Be123) = 0. 3 volume element.
Next, we assume that the medium is a vacuum, so that D = E and H = B.
Summing up the above grades, and using nablaE = nabla.E + nabla^E, we get
d/dt(E+Be123) + nablaE + nabla_|(Be123) + nabla^(Be123) = rho - J.
Use nabla(Be123) = nabla_|(Be123) + nabla^(Be123) to get
(d/dt + nabla)(E+Be123) = rho - J,
and we have condensed all the four Maxwell equations into a single equation,
in a vacuum. Taking the grade involution of both sides, results in
(d/dt - nabla)(-E+Be123) = rho + J.