Consider a scalar S, the value of which is dependent upon its position in space. That is, a field of the scalar S exists such that S is a function of x, y and z or, symbolically
The scalar in question might represent, for example, the mean salinity at the point x, y, z in the ocean. The distribution of this scalar quantity can be described by means of equiscalar surfaces in the region for which S is defined. That is, for a given equiscalar surface
This defines a surface in space presuming that the function S(x,y,z) is known. By assigning different values of S at some regular interval of values, a systematic set of surfaces are defined which describe the distribution of the variable S.
The geometrical spacings of these surfaces, which have presumably
been drawn for equal increments of S (i.e. 
),
reflect the space rate of change of the values of S.
The ascendant of S is defined as
This represents a vector which has a direction normal to the
equiscalar surface at a given point x, y, z and points in
the direction of ascending values of S. Since the
differential operation indicated above occurs so often,
it is given a special symbol 
, defined by
Thus
The negative of this quantity will be referred to as the gradient of S (or grad S); i.e.,
The gradient of S represents a vector directed towards lower values of S.
It is readily shown that 
is perpendicular to the equiscalar
surface at the point x, y, z. Suppose 
represents
a differential vector which is tangent to the equiscalar
surface at the point in question. That is
where dx, dy and dz are related through the equation of the surface S(x,y,z) = constant, which in differential form is
The expression on the left is actually
; consequently
But the above statement is tantamount to the condition that
is perpendicular to and since
was defined as parallel to the equiscalar
surface, then is normal to the equiscalar surface at the
point in question.
The magnitude of , namely
represents the maximum rate of change of S with distance which exists at the point x, y, z. If n represents distance measured normal to the equiscalar surface then
The above figure illustrates the vectors and
graphically for the case of two dimensions.
The magnitude of can be approximated by
where 
is the interval
of S for which the equiscalar surfaces (or isolines
of S in the two dimensional case) are constructed and
is the geometrical spacing of the surfaces (normal
distance between the surfaces) at the point in question.