17. Divergence, Curl and Potentials

d. Differential Identities

2. Second Order Differential Operators and Identities

These second order differential operators and identities are extremely important in vector analysis and much of higher mathematics.

We start with a scalar or vector field and compute all possible second derivatives using gradient, divergence or curl. Three or four of these are more interesting than the others.

If we start with a scalar field, \(f\), the gradient produces a vector field, \(\vec\nabla f\), and we can compute either its divergence, \(\vec\nabla\cdot\vec\nabla f\), (which is interesting) or its curl, \(\vec\nabla\times\vec\nabla f\), (which is also interesting).

If we start with a vector field, \(\vec F\), we can compute either the divergence or curl. Since the divergence produces a scalar field, \(\vec\nabla\cdot\vec F\), we can only compute its gradient, \(\vec\nabla(\vec\nabla\cdot\vec F)\), (which has no simplier form and so is uninteresting). Since the curl produces a vector field, \(\vec\nabla\times\vec F\), we can compute either its divergence \(\vec\nabla\cdot\vec\nabla\times\vec F\), (which is interesting), or the curl \(\vec\nabla\times\vec\nabla\times\vec F\), (which is a little less interesting).

We discuss the four interesting cases on separate pages.

Interesting Second Order Differential Operators

1. \(\underline{\text{div}\,\text{grad}f =\vec\nabla\cdot\vec\nabla f\equiv\nabla^2 f}\)
2. \(\underline{\text{curl}\,\text{grad}f =\vec\nabla\times\vec\nabla f}\)
3. \(\underline{\text{div}\,\text{curl}\,\vec F =\vec\nabla\cdot\vec\nabla\times\vec F}\)
4. \(\underline{\text{curl}\,\text{curl}\,\vec F =\vec\nabla\times\vec\nabla\times\vec F}\)

Summary

\[\begin{aligned} \text{(1)} \qquad &\vec\nabla\cdot\vec\nabla f=\text{Lap}f \\ \text{(2)} \qquad &\vec\nabla\times\vec\nabla f=\vec 0 \\ \text{(3)} \qquad &\vec\nabla\cdot\vec\nabla\times\vec F=0 \\ \text{(4)} \qquad &\vec\nabla\times\vec\nabla\times\vec F =\vec\nabla(\vec\nabla\cdot\vec F)-\nabla^2\vec F \end{aligned}\]

Here, the Laplacian operator \(\text{Lap}f\equiv\nabla^2 f\) means to compute the sum of the direct second partial derivatives of \(f\) and the operation \(\nabla^2\vec F\) means to apply the Laplacian, \(\nabla^2\), to each component of \(\vec F\).

Formulas (2) and (3) form the foundation of solving the differential equations of electromagnetism, fluid dynamics, geosciences and elementary particle theory. They are also the primary way to determine whether a vector field has a scalar or vector potential as discussed in the rest of this chapter.