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Mathematics-Online lexicon:

# 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.

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