Chapter 2 · Section 2.2
2.2 The Nabla Operator: Gradient, Divergence, Curl and Laplacian
The nabla operator and the four operators built from it, gradient, divergence, curl and Laplacian, each derived from its physical definition and read as a picture: the uphill arrow, the net outflow of a small box, the paddle wheel and the comparison with the neighbours.
- what the nabla operator is, and which four operators are built from it,
- how to compute a gradient, and how to read it on a contour map (steepest ascent, normal to the level surfaces),
- what the divergence measures (net outflow per unit volume: sources and sinks) and where its formula comes from,
- what the curl measures (circulation per unit area: the paddle wheel) and where its formula comes from, and
- what the Laplacian measures (a comparison with the average of the neighbours), and the two null identities and .
Imagine you are hiking with a contour map of a hill, or watching a river from a bridge. Three natural questions come to mind. On the hill: which way is straight uphill, and how steep is it? In the river: is water being added or removed somewhere (a spring feeding it, a drain emptying it)? And: does the water swirl, so that a floating leaf would start to spin? Each of these questions is about how a field changes from point to point, and each has a precise mathematical answer: the gradient, the divergence and the curl. All three are built from one object, the nabla operator. Every field law of this course is written with them, so in this section we take our time to understand not only their formulas but, above all, their pictures.