If a vector function V(x,y,z) is defined between two points A and B on a curve, the curve is approximately given by a series of equal directed chords Δ1l,Δ2l,…Δnl. In each segment i of the curve it is possible to define the scalar product Vi·Δil. If the sum of the scalar products∑in= Vi·Δilis considered, the line integral is defined by∫ABV·dl=limn→∞ ∑i=1n V·Δ il.An example of a line integral is given by the case of a force F acting on a particle in the field of the force. The line integral ∫ABF·dlis the work done on the particle as it moves from A to B because of the force.
If the line integral is taken round a closed path (loop), the line integral is denoted by
A line integral for a scalar function φ(x,y,z) is defined in a similar way and is denoted ∫ABφdl. It is also possible to define another type of line integral for a vector function Vby∫ABV×dl..