We refer to the integrand that occurs in Green's Theorem and that appears as

$${\mathrm{curl}}_z\left(F\right)=\frac{\partial F_2}{\partial x}-\frac{\partial F_1}{\partial y}$$

Let $F$ be a velocity field. Estimate the circulation of $F$ around a circle of radius $R=0.05$ with center $P$, assuming that ${\mathrm{curl}}_z\left(F\right)\left(P\right)=-3$. In which direction would a small paddle placed at $P$ rotate? How fast would it rotate (in radians per second) if $F$ is expressed in meters per second?

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THEOREM 1 Green's Theorem Let D be a domain whose boundary 3D is a simple closed curve, oriented counterclockwise. If F and F2 have continuous partial deriva- tives in an open region containing D, then aD F aF - 11. (af_art) x y F dx + F dy = d A 2