##  [Conservative Vector Field](/conservative-vector-field-0) 

 Definition

A vector field F on a region is conservative when there exists a scalar potential φ such that F equals the vector of partial derivatives of φ; equivalently, line integrals of F depend only on endpoints and not on path within the region.

 

 

 

 

 

 





## Principle

Principle

Conservativity characterizes exact differential vector fields whose circulation along closed loops vanishes on simply connected domains, reflecting that local differentials integrate to a global potential.

 

 

 

 

 





## Demonstration

Demonstration

In electrostatics in a static region, the electric field (neglecting time variation) is conservative: it is given by the spatial derivative of an electrostatic potential and its line integral between two points is path independent.

 

 

 

 

## Misapplication

Misapplication

Concluding a field is conservative from the local condition curl F = 0 without checking domain topology: curl-free does not imply a global potential on domains with holes or nontrivial topology.

 

 

 

 

 





## Consequence

Consequence

Correct identification yields a scalar potential, path-independent work calculations, and simplifications of boundary-value problems by reducing vector equations to a scalar equation for the potential.

 

 

 

 

## Reversal

Reversal

A non-conservative (rotational) field, where circulation around closed loops may be nonzero and work depends on the path, such as the Lorentz force from a magnetic field on a moving charge.

 

 

 

 

 





## Boundary

Boundary

Requires sufficient smoothness and consideration of domain connectivity; in regions that are not simply connected additional tests are needed and some curl-free fields may fail to be conservative globally.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Conservative versus irrotational: 'irrotational' often denotes local vanishing of curl, while 'conservative' requires the existence of a global potential; the two coincide under appropriate topological hypotheses.

 

 

 

 

 





## Synthesis

Synthesis

A conservative vector field is an exact differential field that globally arises from a scalar potential so that integrals depend only on endpoints and closed-loop circulation vanishes in the permitted domain.