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.

the field pushes you alongit holds you backthe field at the bead

Drag along the graph to move the bead by hand

Collected so far—

—

Field

Path from A to B

Show the field

whole path—

straight path, for comparison—

adding right now—

What to observe

Run the bead. The card shows the field and the way you are travelling, and only the shadow of one on the other is added. The graph plots it: the shaded area is the integral.
On Central, take the detour and swing its slider. The bead goes further and fights the field, yet collects exactly what the straight path does. Only the endpoints matter: the field is conservative.
The same detour in Swirl no longer matches the straight line. The path matters, so no potential exists for this field.
On the closed loop a conservative field gives zero. Swirl leaves something over: the circulation, the spin trapped inside the loop.

Shortcuts:space run/pause · s step · r reset · f fullscreen