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The continuity equation is a fundamental principle in physics that expresses the conservation of a physical quantity. It states that the rate at which the quantity within a given region changes is equal to the net flux of that quantity into the region. In simpler terms, if a quantity is conserved, it cannot be created or destroyed, only moved around.

The most common application of the continuity equation is in fluid dynamics, where it describes the conservation of mass. For an incompressible fluid, this simplifies to the statement that the volume flow rate is constant. If a pipe narrows, the fluid must speed up to maintain the same volume flow rate. Another important application is in electromagnetism, where it expresses the conservation of electric charge. This means that electric charge cannot be created or destroyed, only transferred from one place to another.

Mathematically, the continuity equation is often expressed in differential form as the divergence of the flux density of the conserved quantity being equal to the negative time derivative of the density of that quantity. Integral forms also exist, relating the change in the total amount of the quantity within a region to the net flux through the boundary of that region. The equation is a cornerstone for understanding phenomena ranging from the flow of water in rivers to the behavior of elementary particles.