the four-gradient
The ordinary gradient bundles the three spatial derivatives into a vector that points uphill. Relativity needs a version that also includes the time derivative and transforms correctly under boosts. The four-gradient is that operator — the spacetime derivative that turns scalar fields into four-vectors and lets you write conservation laws and wave equations covariantly.
The four-gradient is partial_mu = partial/partial x^mu = ((1/c) partial/partial t, partial/partial x, partial/partial y, partial/partial z). A subtlety: differentiating with respect to a contravariant coordinate x^mu produces a COVARIANT (lower-index) object — that is why partial_mu naturally carries a lower index, and the contravariant version partial^mu = eta^munu partial_nu = ((1/c) partial/partial t, -partial/partial x, ...) has the opposite spatial sign. Acting on a scalar field phi it gives a genuine four-vector partial_mu phi; contracted with a four-vector field it gives an invariant divergence partial_mu A^mu = (1/c) partial A^0/partial t + div(A_space).
The four-gradient makes physics manifestly covariant. The continuity equation becomes simply partial_mu J^mu = 0 (charge conservation); the Lorenz gauge condition is partial_mu A^mu = 0; and contracting the four-gradient with itself gives the d'Alembertian wave operator. Caveat: the index placement is the usual trap — partial_mu (the derivative with respect to x^mu) is lower-index/covariant even though it 'looks like' it should match the upper index of x^mu; keeping the sign of the spatial part straight (it flips between partial_mu and partial^mu) is essential.
For a plane wave phi = exp(-i k_mu x^mu) with k^mu = (omega/c, k_vector), the four-gradient brings down the wave four-vector: partial_mu phi = -i k_mu phi. The dispersion relation of a massless field, k_mu k^mu = 0, is then just the light-cone condition on k^mu.
The four-gradient converts a plane-wave field into its wave four-vector.
Watch the sign convention: partial_mu has a PLUS spatial derivative and partial^mu has a MINUS (in the +,-,-,- signature). The time component keeps its sign because eta^00 = +1.