Second Derivatives

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:

$$\deldot \grad f = \lapl f$$

The $\lapl$ operator is called the Laplacian and it enormously important in both electrodynamics and quantum mechanics. It is the 3d equivalent of $\dddx{}$ , given explicitly by:

$$\lapl = \ppartialdiv{}{x} + \ppartialdiv{}{y} + \ppartialdiv{}{z}$$

Next we have:

$$\curl (\grad f) = 0$$

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

$$\grad(\deldot \vA)$$

which has no simpler form but which is often zero for $\vA = \vE,\vB$ in electrodynamics. Next:

$$\deldot (\curl \vA) = 0$$

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

$$\curl (\curl \vA) = \grad(\deldot \vA) - \lapl \vA$$

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