PathIndependent

Pathindependent refers to a property of line integrals in a vector field F on a region D: the work done by F along any curve from point a to b depends only on the endpoints a and b, not on the chosen path. It is a key concept in vector calculus and relates closely to conservative fields and potential functions.

It is equivalent to F being conservative or admitting a potential function φ such that F = ∇φ. In such a case the line integral ∫_C F · dr = φ(b) − φ(a). This means the integral depends only on the endpoints, not on the path taken between them.

For a vector field F = (P, Q, R) in R^3 on a simply connected region, path independence is equivalent to curl F = ∇ × F = 0. In two dimensions, for F = (P, Q), path independence requires ∂P/∂y = ∂Q/∂x and the domain to be simply connected. If the domain is not simply connected, curl F may vanish yet the line integral around a hole may depend on the path; a classic example is the field F = (-y/(x^2+y^2), x/(x^2+y^2)) on R^2 \ {(0,0)}, which has curl zero but is not conservative.

Path independence is tied to the concept of an exact differential form: there exists a scalar potential φ with dφ = P dx + Q dy (and R dz in 3D) on the region.

Applications include computing work in physics, energy differences, and simplifying line integrals in engineering and mathematics.