Exercise 1913Calculus › Multivariable calculus › Vector calculusL1Added 3 Nov 2026

Green's theorem

#Green theorem#curl#region

Statement

Use Green's theorem to compute $\oint_C(-y\,dx+x\,dy)$ where $C$ is the unit circle oriented counterclockwise.

Hint

$\oint_C(P\,dx+Q\,dy)=\iint_D(Q_x-P_y)\,dx\,dy$ .

Show solutionHide solution

Identify. $P=-y$ , $Q=x$ ; $Q_x=1$ , $P_y=-1$ .

Green. $\oint=\iint_D(1-(-1))\,dx\,dy=2\,\text{Area}(D)=2\pi$ .

Conclusion. $\oint_C(-y\,dx+x\,dy)=2\pi$ .