Continuity Equation

If $D$ is a region in space bounded by a closed oriented surface $S$ and $v(x,y,z)$, the velocity field of a fluid flowing through $D$ with $\delta =\delta (t,x,y,z)$ being its density at point and time and $F=\delta v$ then the continuity equation states that

$$\nabla \cdot F+\frac{\partial \delta }{\partial t}=0$$

When applied to the Divergence Theorem the continuity equation is a statement about conservation of mass by claiming that the net decrease of the fluid in region $D$ is accounted for by the mass transported across the surface $S$

Derivation

The instantaneous rate at which mass leaves $D$ across $S$ is

$$\frac{dm}{dt}=\iint \_S\delta v\cdot \vec{n}d\sigma$$

Radius of Cylinder going to 0 causes

$$\nabla \cdot F=-\frac{\partial \delta }{\partial t}$$