$(Chapter17,$Section $19\mathrm{.}1)$

Consider the intersection $R$ between the two circles $x^2+y^2=2$ and $(x-2{)}^2+y^2=2.$

$(a)$ Find $a2-$dimensional vector field $F_{?}=(M(x,y),N(x,y))$ such that

$\frac{\text{d}\text{e}\text{l}\text{N}}{\text{d}\text{e}\text{l}\text{x}}-\frac{\text{d}\text{e}\text{l}\text{M}}{\text{d}\text{e}\text{l}\text{y}}=1.$

$(b)$ Using this $F_{?}$ and Green's theorem $(Theorem19\mathrm{.}1\mathrm{.}1),$ write the area integral ${}_{R}1$dAy$=f(x)$ can $be$ parametrised $by{r}_t=(t,f(t)).$ This can $be$ used $to$

parametrise the linesR.