Circulation and Flux

Calculus 3 · Axiom Academy

A vector field pushes on a curve in two independent ways — along it, and across it. Those two measurements are the two most important line integrals in vector calculus. One field, two questions: does it flow along the curve, or across it? Picture a river's current as a vector field — an arrow at every point telling you which way the water moves and how fast. Drop a closed loop into that current. Two completely different things can happen: the water can swirl around your loop, or it can pour through it. Vector calculus gives you one number for each. Watch a point travel around the loop. At every spot, the field arrow splits into two pieces: the part along the curve (its tangent direction ) and the part across it (the outward normal ). Add up the along-part all the way around and you get circulation ; add up the across-part and you get flux . The same arrow, resolved two ways. Along-the-curve builds circulation; across-the-curve builds flux. A field that only flows along: Here is a field that swirls counter-clockwise. Drag the point around the unit circle and watch the split: the along-part (green) is always full, the across-part (orange) is always nothing. Every step adds to circulation and adds zero to flux — so the total winds all the way up to while flux never leaves 0 . Purely tangential: at every point, . Circulation , flux 0 . A field that only flows across:

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