Higher Order Operators and Second Order Derivatives

There are five ways to combine the del operator in order to generate second derivatives.

1.

The divergence of the gradient: $\nabla \cdot (\nabla f)$

2.

The curl of the gradient: $\nabla \times (\nabla f)$

3.

The gradient of the divergence: $\nabla(\nabla \cdot \vec{F})$

4.

The divergence of the curl: $\nabla \cdot (\nabla \times \vec{F})$

5.

The curl of the curl: $\nabla \times (\nabla \times \vec{F})$

I will only discuss the first of these here. It is left as an exercise to show (correctly) that second of these always results in the zero vector and that the fourth of these is always zero (as a scalar.)

The divergence of the gradient appears so often that it has been given a special name: the Laplacian. It is written as $\nabla^2$ or $\Delta$ and, in Cartesian components has the form

$$\Delta f = \frac{\partial^2f}{\partial x^2} +\frac{\partial^2f}{\partial y^2} +\frac{\partial^2f}{\partial z^2}.$$ (7)

It operates on scalar functions and produces a scalar result. It is also possible to take the Laplacian of a vector field, $\vec{F} = F_1 \hat{i} + F_2 \hat{j} + F_3 \hat{k}$:

$$\Delta \vec{F} = (\nabla^2F_1)\hat{i} + (\nabla^2F_2)\hat{j} + (\nabla^2F_3)\hat{k}.$$ (8)