Exercise 1912Calculus › Multivariable calculus › Vector calculusL1Added 3 Nov 2026

Green's theorem on an annulus

#Green#annulus#inner boundary

Statement

Compute $\oint_C\vec F\cdot d\vec r$ for $\vec F=(-y,x)$ around the annulus $1\le x^2+y^2\le4$ , outer boundary counterclockwise, inner clockwise.

Hint

Green on the annulus: $\iint_D(Q_x-P_y)=2\cdot\text{Area}(D)$ .

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Curl. $Q_x-P_y=1+1=2$ .

Area. $\pi(4-1)=3\pi$ .

Green. $\oint=2\cdot3\pi=6\pi$ .

Conclusion. The circulation around the annulus is $6\pi$ .