For $a\in(0,1)$, calculate without use of the divergence theorem the flux of $\underline{v}(x,y) = g(y/x)(-1/x,1/y)$ across the boundary of the sector
$ S_a := \{(x,y)\in \Omega : 1\leqslant x^2+y^2 \leqslant 4, a\leqslant y/x\leqslant 1/a \} $ where $\Omega:=\{(x,y): x>0, y>0\} $.
Initial progress:
I've drawn out what I think $S_a$ looks like and it's a sector of an annulus in the upper right quadrant of the $x,y$ plane. I've worked out the flux across each of the straight sides of the boundary and got $(a+1/a)g(1/a)ln(2)$ and $-(a+1/a)g(a)ln(2)$ for the other.
The issue I'm having is with the two curved sides of the boundary because I'm unsure how to parametrise them.
Is the work I've done so far correct? And how would I finish it off?
New progress:
So I used that parametrisation and worked out the flux across both of the curved sides of the boundary to be $0$. Is this correct? Does this mean the answer is $Flux= (a+1/a)ln(2)(g(1/a)-g(a))$?