带孔区域上的格林定理

A hole adds a boundary piece

A worked annulus computation

The punctured plane tamed

Applications of the conversion

For the annulus $1 \le r \le 2$, which orientation set is positive?

For $\mathbf{F} = (-y^3, x^3)$ on the annulus $1 \le r \le 2$, what circulation does Green's theorem report?

Around any loop enclosing the origin, the swirl field's circulation is $2\pi$. What does Green's theorem on the punctured annulus say about two such loops?

An outer boundary carries circulation $10$ and the hole boundary, correctly oriented, carries $-4$. Give the total boundary circulation, as a plain number.

The curl density is the constant $2$ over the annulus $1 \le r \le 3$. Give the total, in the form a*pi like 12*pi.