In Cartesian coordinates the advection operator is
where  is the velocity field, and  is the del operator (note that Cartesian coordinates are used here).
The advection equation for a conserved quantity described by a scalar field  is expressed mathematically by a continuity equation:
where  is the divergence operator and again  is the velocity vector field. Frequently, it is assumed that the flow is incompressible, that is, the velocity field satisfies
In this case,  is said to be solenoidal. If this is so, the above equation can be rewritten as
In particular, if the flow is steady, then
which shows that  is constant along a streamline. 
If a vector quantity  (such as a magnetic field) is being advected by the solenoidal velocity field , the advection equation above becomes:
Here,  is a vector field instead of the scalar field .