Line integral

A line integral ∫F·dℓ adds up the component of a vector field F along a curve. At every point along the path, you take the projection of F onto the tangent direction dℓ and integrate. If F is a force and the curve is the trajectory of a particle, the line integral is the work done by the force. If F is an electric field and the curve runs from point A to point B, the line integral equals the potential difference V(A) − V(B).

Line integrals can depend on the path — the work done by friction dragging a box across a room depends on which route you take. But for electric fields produced by static charges, the integral is path-independent: it only depends on the endpoints. That is exactly what it means to say a field is conservative, and it is the condition that lets us define a potential function in the first place. Give the same endpoints, get the same answer, no matter how convoluted the route.

The closed-loop version of this, ∮F·dℓ, is the circulation of the field around a loop. For static electric fields the circulation is always zero — another way of saying that electrostatic fields are conservative. When the field is not static (in time-varying electromagnetism) the circulation can be nonzero, and that is precisely Faraday's law of induction.