Let $V=\{(x,y,z)\in\mathbb{R}^3: 1\leq x^2+y^2+z^2 \leq 2, \ z^2 \geq x^2+y^2 \}.$
Now, I would picture this $V$ as being the intersection between a cone and the surface $S$ given by a 3D ball of radius 2 that has an empty space inside which is given by the ball of radius 1. In other words we have two 3D balls with the same center, the little inside the big, and $S$ should be the space between the inside of the big and the outside of the small one.
Now, suppose one wants to determine the flux of some vector field $F$ through the portion of $\partial V$ lying on the cone $z^2 \leq x^2+y^2.$ Let $\Phi$ be the flux of $F$ through all of $\partial V,$ and let $\Phi_{S_2},\Phi_{S_1}$ be the flux of $F$ through the sphere of radius $2$ and $1.$
I was told that one way to compute the flux of $F$ the portion of $\partial V$ lying on the cone $C,$ call it $\Phi_C,$ is as $$\Phi_C= \Phi-\Phi_{S_2}-\Phi_{S_1.}$$
In other words, the total flux should be the sum of the flux through the boundaries of the three surfaces.
Now I do not understand why it should be so.
Does it depend also on the vector field $F?$ Does it depend on the normal? Is it a geometric reason?
If it matters, the vector field is $F=(x-y,x+y,z).$