Intuitive

The time derivative can be understood as the accumulation (or loss) of mass in the system, while the divergence term represents the difference in flow in versus flow out.

The continuity equation states that the $\color{red}{\text{total amount of a quantity (like water) that is produced (or destroyed) inside some volume}}$ is proportional to the $\color{blue}{\text{change of the quantity}}$ plus the $\color{magenta}{\text{total amount that flows in minus the amount that flows out of the volume}}$.

Or formulated differently, the total $\color{blue}{\text{change of some quantity}}$ is equal to the $\color{red}{\text{amount that gets produced}}$ plus the amount that $\color{magenta}{\text{flows in minus the amount that flows out of the volume}}$.

If we are dealing with a conserved quantity, like energy or electric charge, the total amount that is produced or destroyed is exactly zero.

Concrete

• The continuity equation in hydrodynamics describes the flow of mass. Here $ρ$ is fluid density and $ \vec v$ the fluid flow velocity.
• The continuity equation in electrodynamics describes the flow of electric charge. Here $ρ$ is the charge density and $ \vec v$ the electric flow velocity, such that $ρ \vec v = j$ is the electric current.
• The continuity equation in quantum mechanics describes the flow of probability. Here $ρ = \Psi^\dagger \Psi$ is the probability density and $ \vec v$ the probability flow velocity, such that $ρ \vec v = j = \frac { \hbar } { 2m i } [ \Psi ^ { * } ( \nabla \Psi ) - \Psi ( \nabla \Psi ^ { * } )]$ is the probability current .

Derivation of the continuity equation in hydrodynamics

Derivation of the continuity equation in electrodynamics

Derivation of the continuity equation in quantum mechanics

Abstract

Why is it interesting?

A continuity equation is important whenever we are dealing with a system where some quantity is conserved. The continuity helps us to keep track how the conserved quantity moves through the system.

Important examples are

• the continuity equation in electrodynamics that encodes the conservation of electric charge,
• the continuity equation in hydrodynamics that encodes the conservation of mass.