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

Conservative Field (idea)