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

\[
\operatorname{curl}_{z}(\mathbf{F})=\frac{\partial F_{2}}{\partial x}-\frac{\partial F_{1}}{\partial y}
\]

Estimate ${\oint }_{C}F\cdot dr$, where $F(x,y)=⟨x+0.1y^2,y-0.1x^2⟩$ and $C$ encloses a small region of area 0.25 containing the point $P=(1,1)$.

Transcribed Image Text:

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