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Curl and Existence of Potential Functions
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Let be a continuous differentiable vector field defined on an open set Assume that is conservative, i.e. has a potential function. Then
If is simply connected, then the converse holds.
Examples of simply connected open sets are open balls, open cubes or the entire space Open subsets with a hole are not simply connected.
In the case when is - dimensional the same statements hold, if one replaces the curl by the scalar curl.
In - space examples of simply connected sets are open rectangles, open circles or the entire plane.
|automatically generated 7/12/2005|