Many physical problems can be cast in the form of conservation law PDEs. For simplicity, we begin by discussing conservation law PDEs involving only one space variable.
Suppose that some material quantity 
, such as fluid or heat, moves parallel
to the 
-axis and that the flow field is independent of 
and 
.
Under this assumption of one dimensional flow, it is sufficient to examine
the flow in a cylinder of cross-sectional area 
whose generators are parallel to the -axis.
Let 
be the linear density of material quantity (amount of
per unit length per unit area) and let 
denote the
flux of material in the -direction.
The flux is the amount of material that crosses a
unit planar area perpendicular to the direction of flow per unit time.
Consider flow of through the portion of the cylinder defined by
.
If there
are no additions (sources) or removals (sinks) of material from within the
interval 
,
then the following conservation principle holds.
The time rate of change of the amount of materialminus the flux of material
.
In mathematical terms this statement of conservation of material can be
written as follows.
Interchanging the time derivative with the integral on the left side and using the fundamental theorem of calculus to express the right side as an equivalent integral, permits Eq. 1 to be written as
If the integrands in the above equation are continuous, then, since the
interval is arbitrary, we obtain the conservation law PDE
