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