Conservative Field

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:

1. For any oriented simple closed curve, the line integral of the vector field along the curve is equal to zero.
2. 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.
3. F is the gradient of some scalar field ƒ.
4. The curl of F is equal to zero