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Curl and Existence of Potential Functions

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Let $ \Phi : D \longrightarrow \mathbb{R}^3 $ be a continuous differentiable vector field defined on an open set $ D .$ Assume that $ \Phi $ is conservative, i.e. $ \Phi $ has a potential function. Then

$\displaystyle \operatorname{curl} \Phi = 0 \ . $

If $ D$ is simply connected, then the converse holds.

Examples of simply connected open sets are open balls, open cubes or the entire space $ \mathbb{R}^3 .$ Open subsets with a hole are not simply connected.

In the case when $ \Phi $ is $ 2$ - dimensional the same statements hold, if one replaces the curl by the scalar curl.

In $ 2$ - space examples of simply connected sets are open rectangles, open circles or the entire plane.

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  automatically generated 7/12/2005