In fluid mechanics very often the density and pressure of the fluid in the design will vary with location. Mathematically, the density and pressure distribution can be described in terms of the space coordinates (x,y,z). For example, the pressure variation can be represented for  as p(x,y,z). p  is often referred to as a scalar function.  The combination of the location (x,y,z) and the value p(x,y,z) establishes a scalar field.

pressures p1, p2, and p3 all have different values as pressure increases with depth (z only).  For the same reason density of air decreases with increase in altitude. Inheat transfer,  there will be usually a temperaturedistribution that has to be solved for.
 

Why are scalars being discussed in this section ?  Good Question !! There are two weak reasons.  First, there is no section where this material can be parked. Second,Vector fields can have a similar behavior.

IMP:  One implication of scalar fields is often overlooked. A scalar function  has the same value at a point in space. It is not dependent of the type of coordinate system or the origin of the system [ this means that if the coordinates of the point change due to the new location of the origin then the scalar function must be redefined to give the same value at the point]