line integral explained

Inside a swirling field the blue start A and the blue end B stay fixed. Drag the gold point in the middle and the path bends, and the work W along it is recomputed at once. A segment turns green where the arrow at that spot helps you along, and orange where it fights you. Bend the path with the swirl and the whole route goes green, so W rises positive. Bend it against the swirl and the whole route goes orange, so W falls negative. You still go from the same A to the same B, yet W moves up and down. The start and the end alone do not settle the answer. The gold number under the picture is that work, and it moves with the path as you drag.

Now two routes from A to B sit side by side, one arc looping high and one looping low. Press a button and the active route brightens while the other fades, and the work of each route and the gap between them appear as numbers. The field is a swirl, so the upper route and the lower route disagree even though they join the same two points. The faint arrows behind the routes are that swirl, the same kind of field as in the first picture. That gap equals the circulation once around the closed loop the two routes make together. In a field that rotates, the path changes the answer, and the two numbers show it.

Look at one point at a time to see how a line integral actually piles up. This field is a steady wind blowing north, and the path is an arc that rises and then comes back down. Move the slider t and two arrows appear together at the chosen point. The wind arrow is gold, the unit tangent in the direction of travel is blue, and their dot product F·t̂ is the contribution at that instant. On the way up the wind pushes your back and F·t̂ adds a positive amount. Past the crest, on the way down, you walk against the wind and F·t̂ subtracts. Slide along and the running sum peaks at the crest, then shrinks again. The line integral is every one of these small F·t̂ values added along the path.

Here the two kinds of field split apart. Press a button and the difference shows at once. A conservative field is the gradient of some potential, so the work along the gold straight route from A to B equals the work along the blue detour. The answer is only the difference between the start and the end, and the path does not matter. Switch to the swirl and the two values split. Same two points, same two routes, and one property of the field decides everything. The line under the two numbers turns green and says the values are equal, or turns orange and says they differ. That line is the mark of a conservative field.

Last, walk the path and watch the work accumulate one segment at a time. In the same north wind as before, an arc that rises and falls is cut into short segments. Each press of the step button advances one segment. The part you have walked turns gold, and the part still ahead stays faint. On the left, going up, the wind pushes your back and adds positive work. On the right, past the crest, you walk against the wind and the work turns negative. The sign of the segment you just stepped on is shown, and the bar below is green when the total is positive and orange when it is negative. Reset and you walk it again from the start. The count under the bar shows how far along the path you are, and the step button stops when you arrive. A line integral is this process of adding one segment after another.