Potential Field

(also conservative field), a vector field whose circulation around any closed curve is equal to zero. If the potential field is a force field, this implies that the work done by the forces in going around a closed path is equal to zero. For a potential field a(M) there exists a single-valued function u(M), called the potential of the field, such as a = grad u. If the potential field is given in a simply connected region ft, then the potential of this field can be found from the formula

u = ∫_AM (a,t) dl

in which AM is any smooth curve connecting the fixed point A of Ω with point M, t is the unit vector tangent to the curve AM, and l is the length of path AM as measured from point A. If a(M) is the potential field, then curl a = 0 (seeCURL). Conversely, if curl a = 0 and the field is given in a simply connected region and is differentiable, then a(M) is a potential field. Examples of potential fields are electrostatic fields, gravitational fields and velocity fields of irrotational flow.