Mathematics
The Intuition Behind Curl and Divergence of Vector Fields
Quick fact
A vector field can have nonzero curl even if it carries no net circulation around a large loop, and it can have zero divergence even if it has a constant flow. These local measures capture behavior at a single point, not global motion.
Why this is interesting
Imagine you're floating in a river. Sometimes the water swirls you in a circle, sometimes it stretches you out. Curl and divergence are the mathematical tools that describe those two very different motions.