Exercise 1910Calculus › Multivariable calculus › Vector calculusM1Added 3 Nov 2026

Recovering a field from its curl (2D)

#Green#curl#reconstruction

Statement

Given $\vec F=(P,Q)$ with $Q_x-P_y=f$ , find $P,Q$ via Green: express the circulation $\oint_C$ as $\iint_D f$ .

Hint

Green's theorem already relates the two.

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

Application. If $f=1$ and $D$ is the unit disk, circulation is $\pi$ .

Conclusion. The circulation of any field with a given 2D curl on a region equals the integral of the curl.