How to evaluate an integral in a multiple connect region a while ago I was faced by this problem:

Let $D$ be a multiple connected region interior to $C$ and exterior to
$C_{1}$ and $C_{2}$ (See figure). We know that $Q_{x} - P_{y} = 0$.

*

*What is the value of $\oint_{C} F. dr$?

*If $\oint_{C_{1}} P dx + Q dy = 2$ and $\oint_{C_{2}} P dx + Q dy = 3$, what is the value of $\oint_{C} P dx + Q dy$?


This is the region:

My attempt:
Well, if $Q_{x} - P_{y} = 0$, then $F$ is conservative, so, $\oint_{C} F. dr = 0$.
For the second part, I don't know how to proceed, I was thinking in sum both of the $C_{i}$ integrals.
Can someone help me with this?
Thanks in advance!
 A: If ${\bf F} = \langle P,Q\rangle$, Green's theorem  (which remains true for multiply connected regions) gives
$$ \int_{\partial D} {\bf F}\cdot d{\bf r} = \int_{\partial D}P\,dx+Q\,dy =\iint_D (Q_x-P_y)\,dx\,dy = 0.$$
With the given orientations, we have $\partial D = C+C_1+C_2$ (keep in mind the inner boundary orientations need to clockwise rather than counterclockwise, which we have in this case). Thus
$$ \int_{\partial D} {\bf F}\cdot d{\bf r} = \oint_{C} {\bf F}\cdot d{\bf r}+\oint_{C_1} {\bf F}\cdot d{\bf r}+\oint_{C_2} {\bf F}\cdot d{\bf r}.$$
It follows that
$$\oint_{C} {\bf F}\cdot d{\bf r} = -\oint_{C_1} {\bf F}\cdot d{\bf r}-\oint_{C_2} {\bf F}\cdot d{\bf r}.$$
So there isn't a numerical answer to the first problem, since it depends on the value of the integrals along the inner boundaries. I imagine you thought ${\bf F}$ was conservative because $Q_x = P_y$ in $D$, but that implication only works in a simply-connected region (and $D$ is not). You would be able to conclude the integral around $C$ is $0$ if $Q_x = P_y$ everywhere inside $C$, but I don't think this is what they meant (since in this case, the situation of $2$ would be impossible; the integrals around the inner boundaries could not be nonzero numbers).
Anyway, for 2, the above formula implies
$$\oint_C P\,dx+Q\,dy = -2-3 = -5.$$
