Electromagnetic Fields and Waves: Notes, Solved Problems and Interactive Lab

2.2.5 Examples on gradient, divergence and curl

The following examples exercise formulas (2.2.2), (2.2.6) and (2.2.8). The last two lead to two identities that hold for every sufficiently smooth field and that we shall use many times. All of them follow the same short procedure.

Recipe · Computing a gradient, divergence or curl in Cartesian coordinates

Check the type of the input: the gradient needs a scalar $f$ , whereas divergence and curl need a vector $A$ . Read off the components $A_{x}, A_{y}, A_{z}$ .
Take the partial derivatives you need, holding the other two coordinates constant (use the chain rule for composite functions).
Assemble: the gradient uses the three derivatives with their unit vectors, the divergence is the sum $\partial A_{x} / \partial x + \partial A_{y} / \partial y + \partial A_{z} / \partial z$ , and the curl consists of the three brackets of (2.2.8).
Check the type of the output (vector, scalar, vector) and, if possible, a value at a simple point.

Example 2.2.1Gradient of a scalar function Example 2.2.2Divergence of a vector field Example 2.2.3Curl of a vector field Example 2.2.4Curl of a gradient Example 2.2.5Divergence of a curl

Two null identities

\nabla \times (\nabla f) = 0, \nabla \cdot (\nabla \times A) = 0. (2.2.9)

The converses are also true in a region without "holes" (for example all of space): a field with zero curl everywhere can be written as the gradient of a scalar, and a field with zero divergence everywhere can be written as the curl of another vector field. (Precisely: the first statement needs a region in which every closed loop can be shrunk to a point without leaving it, the second a region that encloses no cavity.) The first statement is the reason why the electrostatic field can be described by a scalar potential (Section 3.3).

2.2.5 Examples on gradient, divergence and curl

In this part

Examples · 5