Stoke's theorem:
$\m\dis \intd{\sigma} (\nabla\times \vec V)\cdot \vec n d \sigma= \oint_{\p \sigma} \vec V\cdot d\vec r $,
here $\sigma$ is a surface in 3-d.
The following are equivalent:
- curl $\vec F = 0$;
- $\dis\oint \vec F\cdot d\vec r =0$ for any simple closed curve;
- $\vec F$ is conservative, i.e., $\dis\int_A^B \vec F\cdot d\vec r$ is independent of path;
- $\vec F = $ grad $W$ for some $W$;
We also say: $\m \vec F$ is irrotational.