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Existence of Potential Functions and Path Independence

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A continuous vector field $ \Phi $ on a region $ D$ has a potential function $ u$ if, and only if,it is conservative, i.e.that line integrals of $ \Phi $ in $ D$ are independ of the path. In this case

$\displaystyle u(P) = u(P_0)+ \int\limits_{C_P} \Phi \ dx $

where $ {C_P}$ denotes an arbitrary curve in $ D$ joining a fixed chosen point $ P_0\in D$ with a point $ P .$ In particular the potential function $ u$ is uniquely determined up to a constant.

Example:

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  automatically generated 6/15/2005