A vector field satisfying any of the following conditions is called a conservative vector field.
Each of the following conditions implies the other conditions, meaning that if one is true, then all are true.
Let F be a continuously differentiable vector field defined on R3 except possibly for a finite number of points:
- For any oriented simple closed curve, the line integral of the vector field along the curve is equal to zero.
- For any two oriented curves C1 and C2 which have the same endpoints, the line integral of the vector field along each path is equal to the other.
- F is the gradient of some scalar field ƒ.
- The curl of F is equal to zero