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Work done by force F moving along curve C: W = ∫_C F · dr. For a closed curve, ∮_C F · dr is circulation (net rotation). Circulation = 0 for conservative fields.
From line integrals over vector fields you know that ∫_C F · dr accumulates the dot product of a vector field F with the tangent direction of a curve C. Physically, when F is a force field, this integral measures work — the total energy transferred by the force as a particle moves along C. The dot product F · dr captures the key idea: only the component of force *along* the direction of motion contributes to work. A force perpendicular to motion does zero work; a force opposing motion does negative work.
To evaluate ∫_C F · dr, parametrize the curve as r(t) for t ∈ [a, b]. Then dr = r'(t) dt and the integral becomes ∫_ab F(r(t)) · r'(t) dt — a standard single-variable integral. The result depends in general on the curve C, not just its endpoints. If you take a different path from the same start to the same end, you may get a different value of work. This path-dependence is the generic situation.
Circulation is the line integral around a *closed* curve: ∮_C F · dr. Think of C as a loop. Circulation measures the net tendency of the field to push fluid (or a particle) around the loop — the net "spinning" effect. Imagine a water wheel placed in a stream: if the current flows preferentially around the wheel in one direction, the circulation around a loop encircling the wheel will be nonzero. Circulation has a sign: positive if the field tends to push counterclockwise around C (by convention), negative for clockwise.
The crucial special case is conservative fields. A field F is conservative if F = ∇f for some scalar potential f. For conservative fields, the work integral depends only on the endpoints: ∫_C F · dr = f(end) − f(start). This is the multivariable Fundamental Theorem of Calculus. As an immediate consequence, circulation around any closed loop is zero — you return to the starting point and the potential difference is f(start) − f(start) = 0. Non-zero circulation is therefore a signature of a non-conservative field. Green's theorem, which you will study next, quantifies this precisely: it relates circulation around a closed curve to the "curl" of the field over the enclosed region.