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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

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