Divergence and Curl
Two derivative operations on a field: divergence measures net outflow at a point, curl measures local rotation. Both are the language of the theorems that follow.
What a learner can do afterwards
- Compute the divergence and curl of a given field
- Describe the physical reading of each at a point
- Show that a conservative field has zero curl
1 · Read
Divergence is the net outflow of a field at a point. You add the three outward rates, one per direction, and the total says how much flow the point creates or swallows. A positive value marks a source, and a negative value marks a sink. No integrals are needed: you read it straight off the partial derivatives.
Curl measures the local rotation instead. It assembles the cross derivatives into a vector that points along the spin axis, with a size equal to twice the local rotation rate. Train your eye first: arrows spreading apart signal positive divergence, arrows bunching together signal negative divergence, and arrows looping around signal curl. A field can carry one without the other.
Try the radial field (x, y, z). Each component differentiates to 1, so the divergence is 1 plus 1 plus 1, which is 3 everywhere: a pure source with no spin. Now try the spin field (minus y, x, 0). Its curl keeps only a k component: dQ/dx minus dP/dy is 1 minus minus 1, which is 2, so the curl is (0, 0, 2). That is pure rotation with no outflow.
Use the pair as vocabulary for everything that follows. Zero curl flags a field that could be a gradient, and nonzero divergence flags a source sitting inside. Uniform rotation has curl but no divergence, while radial outflow has divergence but no curl, so always test both before judging a field.
Divergence totals the outflow at a point and curl vectors the spin, both read straight from partial derivatives.
2 · Watch
Take it off screen
Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.