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
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.