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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:

$\displaystyle \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:

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

Next we have:

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

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

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

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

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

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

$\displaystyle \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!

Robert G. Brown 2017-07-11