Relationship between rate of change of a physical quantity and its divergence

For a conserved quantity (like mass, momentum, or energy), the local rate of change within a volume is directly linked to the divergence of its flux. In particular, conservation laws often take the form:

where is the density of the conserved quantity and is its flux. Integrating over a volume and applying the divergence theorem:

shows that the rate of change of in that volume (the first term) is exactly balanced by the net flux across the boundary (the second term). Thus, encapsulates the source or sink behavior within the volume and is intimately connected to how changes over time.