Topics: Vector Field


A conservative vector field is a vector field that has one (or several) potential functions.

All conservative fields are irrotational (i.e. their curl is ).

Line Integral

Path Independence


The line integral of a conservative vector field is path-independent.

That means that given a conservative vector field , its line integral will only depend on the endpoints of the path, regardless of the actual path chosen. That is, if and are any two paths that have the same endpoints:


The above expression is equivalent to:

…where is a closed path (i.e. its endpoints are one same point).

Fundamental Theorem of Calculus for Line Integrals


Given a conservative vector field on a region , then the integral of on a curve in is equal to:

…where is a potential function of , and and are the extreme points of the curve.

Notice that this is basically the fundamental theorem of calculus for line integrals.