Question about neutron flux (or maybe any particle flux for that matter)

[random_soldier1337]

Why is it that when integrating over all angles, integration is over the solid angle omega composed of theta and phi but the vector angle within flux, phi( r, E , omega), is resolved into the cartesian coordinates by cos(theta)sin(phi), sin(theta)sin(phi) and cos(phi) which is essentially the dot product of the radius vector from spherical coordinates with the cartesian coordinate vectors i.e. why is integration over theta and phi to remove angular dependence but when resolving the angle the dot product is as if using the radius vector?