## Derivatioin of the Continuity Equation

A conservation law is the statement that a given quantity can be neither created  nor destroyed but may merely move. That leads to the statement :

The total rate of outflow from some region must equal the rate of decrease of that  quantity located within that region.

Suppose we have a cylindrical surface, such that gas can flow in and out through the ends but not through the sides.

Suppose denotes the density of a gas at time for Thus at  any time the total mass of gas present in the region is given by  Let us denote by and the mass inflow/outflow of the gas at the ends and respectively.

The rate of change of mass of gas in the region between and is given by  and are held fixed and since By adding these we obtain In higher dimensions, we obtain This is the continuity equation. The rate of flow of mass of gas out of a surface element of area is We can write the continuity equation as The region of integration can be chosen arbitrarily, and since any continuous function with integral zero over an arbitrary region must be the zero function, hence 