There are five second derivatives. Two are important, and a third could conceivably be important but will often vanish for the same reason. The first rule defines and operator that is arguably the most important second derivative in physics:

The operator is called the Laplacian and it

Next we have:

(not precisely trivial to prove but important). Also:

which has no simpler form but which is often zero for in electrodynamics. Next:

(not precisely trivial to prove but important). Finally:

which is *very* important - a key step in the derivation of the 3d
wave equation from Maxwell's equations in differential form!

Robert G. Brown 2017-07-11