Continuity Equation

$Q=Av$

• $Q$ is volumetric flow rate (this is conserved)
• $A$ is cross sectional area of the pipe
• $v$ is the linear flow velocity

$\frac{\partial \rho }{\partial t}+\nabla \cdot j=\sigma$

• Fluid mechanics version
• $\rho$ is the fluid density
• $j$ is the flux of the fluid
• $\sigma$ is the amount of fluid generated

$\frac{\partial \rho }{\partial t}+\nabla \cdot \vec{J}=0$

• Electromagnetism version
• $J$ is the current (e.g. $\frac{\mathrm{d}Q}{\mathrm{d}t}$) of a quantity
• $\rho$ is the quantity density (e..g charge density) per unit volume
• Can be interpreted as saying that the time rate of change in a quantity is equal to the negative of the amount leaving it
• Can be derived by divergence theorem
• Can also be derived by taking the divergence of Ampere’s law and applying Gauss’s law to $\nabla \cdot \vec{E}$

$\frac{\partial u}{\partial t}+\nabla \cdot \vec{S}=0$

• Conservation of Energy version
• $u$ is electromagnetic field energy density
• $\vec{S}$ is the Poynting vector

$\frac{\partial \vec{g}}{\partial t}+\nabla \cdot \stackrel{\Rightarrow }{T}=0$

• Conservation of Momentum version
• $\vec{g}$ is the momentum density
• $\stackrel{\Rightarrow }{T}$ is the Maxwell’s stress tensor

$\frac{\partial P}{\partial t}+\nabla \cdot J=0$

• Quantum mechanics version
• $P$ is the probability density function of a particle
• $J$ is the probability current, or how much of a particle is flowing