Calculus
Line Integrals
Ride a bead along a curve through a field and keep one running account: at every step, how much of the field points the way you are going. That is the whole of a line integral, and the shaded area under the little graph is the number itself.
Collected so far—
Whole path—
Straight path, for comparison—
Adding right now—
The bead adds up F·T̂ step by step along the curve you see, and you can drag along the graph to move it by hand.
What to observe
- Look at the panel on the right. The field arrow points wherever the field points, the grey arrow points the way you are travelling, and the only thing that gets added is the shadow of one on the other. Side on, the bead travels a long way and collects nothing at all.
- The graph under the scene is that shadow, plotted as you go. The shaded area is the integral: teal above the line adds, orange below it takes away, and the running total is just the area so far.
- With uniform or central, note the total on the straight path, then switch to detour and swing the slider anywhere you like. The bead travels much further and spends time fighting the field, yet arrives withexactly the same total. Only the endpoints mattered, which is what conservative means.
- Now switch to swirl and try it again. The detour no longer matches the straight line, so the path itself matters, no potential can exist for this field, and there is nothing to call height.
- Take the closed loop. In the conservative fields you come back to zero however you go round, because you finish where you started. In the swirl you do not, and what is left over is the circulation: the spin trapped inside the loop.
Shortcuts: space run/pause · s step · r reset · f fullscreen