Conservative Field

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

1. For any oriented simple closed curve C, the line integral ∮_CF·ds=0.

2. For any two oriented simple curves C_1 and C_2 with the same endpoints, int_(C_1)F·ds=int_(C_2)F·ds.

3. There exists a scalar potential function f such that F=del f, where del is the gradient.

4. If D is simply connected, then curl del xF=0.

The domain D 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 A and B but are not conservative in their union A union B.

Note that conditions 1, 2, and 3 are equivalent for any vector field F defined in any open set D, with the understanding that the curves C, C_1, and C_2 are contained in D and that F=del f holds at every point of D.

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 F are continuous, then these conditions do imply 4. In order for condition 4 to imply the others, D must be simply connected.

See also

Curl, Gradient, Line Integral, Poincaré's Theorem, Potential Function, Simply Connected, Vector Field

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Cite this as:

Weisstein, Eric W. "Conservative Field." From MathWorld--A Wolfram Web Resource.

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