格林定理

Circulation becomes a double integral

A worked verification

Area by circulation

What the theorem needs

For $\mathbf{F} = (2y, 3x)$ around the unit circle counterclockwise, what does Green's theorem give?

Which circulation computes the area of a region via Green's theorem?

The swirl field $\left(\tfrac{-y}{x^2+y^2}, \tfrac{x}{x^2+y^2}\right)$ has zero curl where defined, yet circulates $2\pi$ around the unit circle. What failed?

For $\mathbf{F} = (x^2 y,\ x y)$, give the curl density $\partial Q/\partial x - \partial P/\partial y$ as a polynomial in x and y like x^2-x.

The ellipse parametrized by $(2\cos t, \sin t)$ traversed counterclockwise has area $A$. Using the area formula, $\tfrac12\oint x\, dy - y\, dx$ evaluates to $2\pi$. Give $A$, in the form a*pi like 3*pi.