Worked example - work done by a force field along a path. F = (y, -x, 0) (a circulating field), path: unit circle r(t) = (cos t, sin t, 0), t: 0 -> 2 pi. Step 1: dr = (-sin t, cos t, 0) dt. Step 2: F along path = (sin t, -cos t, 0). Step 3: F.dr = [sin t x (-sin t) + (-cos t)(cos t)] dt = -(sin^2 t + cos^2 t) dt = -dt. Step 4: W = integral(0 to 2 pi) -dt = -2 pi. Negative: the field opposes the (counterclockwise) motion. Check via Stokes: curl F = (0, 0, -2), flux through unit disk = -2 x pi = -2 pi. Matches - and shows the field is non-conservative (closed-loop work nonzero, so no potential V with F = -grad V exists).