Path Independence

Calculus 3 · Axiom Academy

In the right kind of field, the line integral doesn't care which road you take — only where you start and where you finish. A line integral usually depends on the whole road C — bend it, lengthen it, and the total changes. But for one special kind of field, something remarkable happens: reshape the path all you like, and the total refuses to budge. Watch it happen before you take it on faith. Two travelers leave the same start A and reach the same finish B through the field — one takes the straight diagonal, the other rounds the corner. As each moves, the running total climbs. Watch where they land. Different roads, different journeys — but the running totals meet at exactly the same number. Move the finish — the two totals stay locked together Drag the finish point B anywhere you like. The straight path and the corner path are completely different roads — yet their integrals recompute in real time and never disagree. Both always equal f(B)-f(A) , the value at the finish minus the value at the start. Two different roads, and yet a single value — set entirely by the endpoints. Go all the way around, gain nothing If the road back always undoes the road out, then a round trip should total zero. Drag the marker around the loop and watch the running : whatever it gains on one side it gives back on another, and it lands on exactly 0 the moment you return to the start. Any closed loop in a conservative field integrates to zero — the defining fingerprint.

This is the written version of the interactive lesson above. See the full Calculus 3 course.