Gill's theorem

Gill's theorem states that, for a certain class of *divergence-free vector fields* (which can describe, for example, incompressible fluid flow), the integral of the vector field dotted with the normal vector over the boundary of a domain equals the divergence of the field integrated over that domain. In simpler terms, it connects how a fluid's flow across a boundary relates to the behavior of the flow within the region. This theorem generalizes the divergence theorem and is useful for understanding flow properties, conservation, and analyzing vector fields in physics and engineering contexts.