Line Integral vs Path

A line integral adds up how much a vector field pushes you along as you walk a path: at every step you take the part of the field pointing your way and sum it up. The deep question is whether the total depends on the route or only on where you start and finish. For a special class of fields, the conservative ones that are the gradient of some potential, it depends only on the endpoints: every path from A to B gives the same answer, and a round trip gives exactly zero. For everything else the path matters, and walking out one way and back another leaves you with a net gain or loss, the circulation, which by Stokes' theorem equals the curl swept up inside the loop. The scene shows the field with two endpoints and several routes you can drag between them; the diagnostic accumulates the integral along each route, so you watch the curves either land on the same value or split apart.