This illustration implies that if the vector field is tangent to "S" at each point, then the flux is zero, because the fluid just flows in parallel to "S", and neither in nor out. This also implies that if v does not just flow along "S", that is, if v has both a tangential and a normal component, then only the normal component contributes to the flux. Based on this reasoning, to find the flux, we need to take the dot product of v with the unit surface normal to "S" at each point, which will give us a scalar field, and integrate the obtained field as above. This gives the formula