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Conservative Field

The following conditions are equivalent for a conservative Vector Field:

1. For any oriented simple closed curve $C$, the Line Integral $\oint_C {\bf F}\cdot d{\bf s} = 0$.

2. For any two oriented simple curves $C_1$ and $C_2$ with the same endpoints, $\int_{C_1} {\bf F}\cdot d{\bf s} =
\int_{C_2} {\bf F}\cdot d{\bf s}$.

3. There exists a Scalar Potential Function $f$ such that ${\bf F} = \nabla f$, where $\nabla$ is the Gradient.

4. The Curl $\nabla \times {\bf F} = {\bf0}$.

See also Curl, Gradient, Line Integral, Potential Function, Vector Field

© 1996-9 Eric W. Weisstein