From the image, I believe that circulation around C1 = Flux of curl F + circulation around C2 + circulation around C3. We are given curl F = 4, circulation around C2 = 3pi and circulation around C3 = 2pi.
I parametrized C1 as: r(theta)=<5cos(theta), 5sin(theta)> and theta is between 0 and 2pi. Using Green’s Theorem and curl F = 4 (given) and dA = r dr dtheta, i found that flux of curl F is 100pi.
Therefore, flux around C1 should be 100pi + 3pi + 2pi = 105pi.
Unfortunately, I get the wrong answer but I do not know what I am doing wrong. Any leads?
Attached the working out of the problem so it’s clearer.
$\oint_{C_1}F\cdot ds - \oint_{C_2}F\cdot ds - \oint_{C_3}F\cdot ds = \iint \nabla\times F\ dA = 4(23\pi) = 92\pi$
$\oint_{C_1} = 2\pi + 3\pi + 92\pi = 97\pi$