Kontinuitätsgleichung

The Kontinuitätsgleichung, or continuity equation in English, is a fundamental principle in physics that expresses the conservation of a certain quantity. It states that in any region of space, the rate of accumulation of this quantity is equal to the net rate at which it flows into that region. Mathematically, it can be expressed in differential or integral form.

The differential form typically involves the divergence of a flux vector and the time rate of change of a density. If the quantity is conserved, its divergence over a volume plus the time derivative of its density within that volume equals zero. This means that any decrease in density within a volume must be accounted for by a net outflow of the quantity from that volume.

In fluid dynamics, the continuity equation is used to describe the conservation of mass. For an incompressible fluid, where density is constant, the equation simplifies to state that the volume flow rate is constant throughout a pipe, regardless of its cross-sectional area. This explains why fluid speed increases when a hose is squeezed.

The continuity equation also finds applications in electromagnetism, describing the conservation of electric charge, and in quantum mechanics, relating to the conservation of probability. Its general applicability makes it a cornerstone of many physical theories.