Some HN readers might be interested to know that Maxwell's equations can be written in a single expression (with one auxiliary condition).

The first step is to define the Faraday tensor F, which can be written as a 4x4 antisymmetric matrix. Such a matrix has 16 elements, but the diagonal is all zeroes (by antisymmetry: x = -x => x = 0), and the 6 off-diagonal elements on one side of the diagonal are simply the negatives of their mirrored values. This leaves 16 - 4 - 6 = 6 independent elements—exactly enough to contain the 6 elements of the electric and magnetic field vectors (3 each for E and B), which in fact are exactly the elements of the Faraday tensor.

In terms of F, the Maxwell equations for the divergence of B and the curl of E can be combined in the single equation

    dF = 0

where d is the exterior derivative. As with ordinary potential theory, where a vector field with zero curl can be written as the gradient of a scalar function, a tensor whose exterior derivative is zero can be derived from another tensor, called in this case the 4-potential A (which contains both the ordinary electric potential and the magnetic vector potential):

    F = dA      (1)

(This is the auxiliary condition alluded to above.) The second expression for F, corresponding to the divergence of E and the curl of B, combines the exterior derivative and the Hodge star operator ⭑ [1]:

    d⭑F = J     (2)

where J is the 4-current density (which contains both the charge density and the 3-dimensional current density) [2]. Substituting (1) for F in (2) then yields Maxwell's equations in a single expression:

    d⭑dA = J    Maxwell's equation(s)

This equation is perhaps a more compact and elegant choice for those T-shirts: "God said: d⭑dA = J—and there was light!"

[1]: The ⭑ character may not display in some browsers. In this case, you can use an asterisk instead.

[2]: Eq. (2) is written in "God's units", equivalent to SI with μ₀ = ε₀ = 1 => c = 1.