A Line Integral Sums Along a Path
Picture walking across a windy field. When the wind shoves your back, each step comes for free; when it blows in your face, you have to push. Step by step, add up how much the local wind helped your forward motion, and you get the total help you got along the path. That is a line integral, ∫ F·dr: the work a field does, accumulated along a route. The strange part is that even with the same start and finish, the total can change depending on which path you took. Yet in certain special fields the path doesn't matter at all. This lesson shows where that difference comes from.
Inside a swirling field, keep the start A and end B fixed, then drag the gold point in the middle to bend the route. The work along the path is recomputed at once. Each little segment of the route is tinted green where the local arrow helps your motion and orange where it fights it. Bend the path to run with the swirl and the whole route glows green as W climbs positive; bend it against the swirl and the whole route turns orange as W drops negative. Even though you go from the same A to the same B, W rises and falls. That is the first sign that start and finish alone do not pin down the answer.
Now put two routes from A to B side by side: an arc that loops high and one that loops low. Hit the toggle to switch the active route, and it lights up bright while the other fades. The work along each route and the gap between them show up as numbers. Because the field is a swirl, the high route and the low route give different values even though they join the same two points. That gap is exactly the circulation once around the closed loop made by joining the two routes. In a rotating field, you see with your own eyes that the path changes the answer.
Look point by point at how a line integral actually piles up. This time the field is a steady wind blowing north, and the path is an arc that rises and then comes back down. The slider t picks one point on the curve. There the wind arrow (gold) and the unit tangent that points the way you travel (blue) appear together, and their dot product F·t̂ is the contribution at that instant. On the way up the wind is at your back, so F·t̂ adds positively; once you crest the top and head down you move against the wind, so it subtracts. Push the slider and watch the running sum swell to its largest at the crest, then shrink back down. A line integral is, in the end, all these tiny F·t̂ added up along the path.
Here comes the key contrast. Toggle the kind of field. A conservative field is the gradient of some potential, so the work along a straight route from A to B and along a detour arc come out exactly equal. The answer is only the difference of the start and end values; the path is irrelevant (path independent). But switch to the swirling field and the two routes split apart (path dependent). Same two points, same two routes, yet one property of the field decides everything. Watch the "path independent?" indicator switch on and off. That is the very signature of a conservative field.
Finally, walk it yourself to see how the work piles up one segment at a time. In the same north wind as before, an arc that rises and falls is split into short segments. Each press of the step button advances you one notch to the next segment; the path you have traversed brightens to gold while the rest stays dim. Going up the left side the wind is at your back and adds positive work; once you crest the top and descend the right side you move against the wind and it subtracts. A sign shows whether the segment you just stepped over added or subtracted, and the bar below builds up the cumulative work and then eases back down. The reset button lets you walk it again from the start. You confirm by hand that a line integral is, in the end, this process of adding up one segment at a time.