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

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

For the vector fields (A)-(D) in Figure 27, state whether \(\operatorname{curl}_{z}\) at the origin appears to be positive, negative, or zero.

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