Referring to Figure 25, suppose that
\[
\oint_{C_{2}} \mathbf{F} \cdot d \mathbf{r}=3 \pi, \quad \oint_{C_{3}} \mathbf{F} \cdot d \mathbf{r}=4 \pi
\]
Use Green's Theorem to determine the circulation of \(\mathbf{F}\) around \(C_{1}\), assuming that \(\frac{\partial F_{2}}{\partial x}-\frac{\partial F_{1}}{\partial y}=9\) on the shaded region.

Transcribed Image Text:

THEOREM 1 Green's Theorem Let D be a domain whose boundary JD is a simple closed curve, oriented counterclockwise. If F and F2 have continuous partial deriva- tives in an open region containing D, then aD a F F - SS (31-3F) DA d ay F dx + F dy = F2 2