Question on Solid Angles and Linking Number

Question

In Spivak's Calculus on Manifolds there is a question about solid angles and linking number that confuses me. Suppose $f([0,1])=\partial M$ where $f$ is a closed curve and $M$ is a compact oriented two-dimensional manifold-with-boundary. Define $\Omega(a,b,c)$ to be the solid angle subtended by $M$ from $(a,b,c)$. If $g$ is another closed curve which doesn't intersect $f$, the problem claims that $$-\frac{1}{4\pi}\int_g d\Omega$$ is the linking number between $f$ and $g$. My question is, if $\Omega(a,b,c)$ is undefined when $(a,b,c)\in M$, how can the above integral be well defined when $g$ intersects $M$?

Answer 1

Indeed, $\Omega$ will have jump discontinuities at points $p_i =g(t_i)$ where $g$ intersects $M$, of size $\pm 4\pi$; I think the intended interpretation of the integral is as sum of integrals over (finite number) of open segments $I_i=(p_{i-1}, p_i)$ where $g(I_i)$ doesn't intersect $M$. Then one can apply Stokes to closed subsegments of each segment and take limits.