Continuity Equation

#Physics

$\displaystyle Q=Av$

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

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

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

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

Electromagnetism version
$\displaystyle J$ is the current (e.g. $\displaystyle \frac{\mathrm{d}Q }{ \mathrm{d}t}$) of a quantity
$\displaystyle \rho$ 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 $\displaystyle \nabla \cdot \vec{E}$

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

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

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

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

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

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