1. For a curve in a vector space defined by x = x(t ), and a vector function V defined on this curve, the line integral of V along the curve is the integral over t of the scalar product of V[x(t )] and dx/dt; this is written ∫ V.dx.

2. For a curve which is defined by x = x(t ), y = y(t ), and a scalar function f depending on x and y, the line integral of f along the curve is the integral over t of ƒ [x(t ),y(t )]. √((dx /dt )² + (dy /dt )²); this is written ∫ fds, where ds = √((dx )² + (dy )²) is an infinitesimal element of length along the curve.

3. For a curve in the complex plane defined by z = z(t ), and a function f depending on z, the line integral of f along the curve is the integral over t of ƒ [z(t )] (dz / dt ); this is written ∫ fdz.