旋度检验与单连通区域

Two derivatives decide, when the domain cooperates

The test on two fields

Simply connected, in pictures

A decision procedure for line integrals

For $\mathbf{F} = (e^x \sin y,\ e^x \cos y)$ on the whole plane, what does the curl test conclude?

The swirl field $\left(\tfrac{-y}{x^2+y^2}, \tfrac{x}{x^2+y^2}\right)$ has zero curl where defined. Why is it not conservative?

A curve integral has curl vanishing identically, and the curve winds once around the field's single singular point. What is the computation route?

For $\mathbf{F} = (2y,\ 3x)$, give the scalar curl $\partial Q/\partial x - \partial P/\partial y$, as a plain number.

A curl-free field has a single singular point carrying circulation $2\pi$. A loop winding twice counterclockwise around it carries circulation $c \cdot \pi$. Give $c$, as a plain number.