Theorem for Curl-less Fields

When one of the following is true, all are true.

• $\nabla \times F=0$ everywhere
• $\int \_a^{b}F\cdot dl$ is independent of path for any given endpoints
• $\oint F\cdot dl=0$ for any closed loop
• $F$ is the gradient of some scalar function: $F=-\nabla V$