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# Conservative Field

The following conditions are equivalent for a conservative vector field on a particular domain :

1. For any oriented simple closed curve , the line integral .

2. For any two oriented simple curves and with the same endpoints, .

3. There exists a scalar potential function such that , where is the gradient.

4. If is simply connected, then curl .

The domain is commonly assumed to be the entire two-dimensional plane or three-dimensional space. However, there are examples of fields that are conservative in two finite domains and but are not conservative in their union .

Note that conditions 1, 2, and 3 are equivalent for any vector field defined in any open set , with the understanding that the curves , , and are contained in and that holds at every point of .

In general, condition 4 is not equivalent to conditions 1, 2 and 3 (and counterexamples are known in which 4 does not imply the others and vice versa), although if the first derivatives of the components of are continuous, then these conditions do imply 4. In order for condition 4 to imply the others, must be simply connected.