gradient

A mathematical field's gradient at some point is essentially the maximum differential found along any line that passes through the point, expressed as a vector along that line and magnitude indicating the differential along that line. For the gradient to exist, the field must be such that the point's differentials are defined. A mathematical field may have a gradient at each of its points, the gradients forming an associated gradient field. The gradient of function f is indicated by ∇ f. The delta symbol (Δ) turned upside down is called a nabla.

A function of two variables has an analogous gradient, and an easy-to-imagine example of a gradient is that of such a function of two dimensions that represents the altitudes of some region of the ground, i.e., the function represents a topographic map. Such a function's gradient at a point within its domain represents the corresponding ground's incline, the specific direction of the gradient-vector being that in which the altitude is rising. Functions of four or more variables can also have analogous gradients.

Typical in the physical sciences are pairs of associated mathematical fields, one a scalar field, and the other its associated gradient field. The latter can be derived from the former, and if a single value is known, the former can be derived from the latter, making the two a pair of mathematical representations of the same physical field. Such a scaler field is sometimes described as a potential field. Examples:

• A gravitational field throughout a region of space can be described by a scalar field (gravitational potential) or by its associated gradient field, which consists of vectors indicating gravity's force and direction at each point.
• Electric fields can be analogously described with two such related mathematical fields.

Having the two representations offers additional mathematical techniques for calculations relevant to the field.

When the term gradient is modified with a direction, e.g., horizontal gradient or vertical gradient, it refers to the differential at a point across that direction. This usage is often in cases where some particular set of points within the field have the identical such differential.

The term gradient is also used simply to mean incline or slope along some given line of interest. A phrase such as temperature gradient might be referring to the gradient specified as above, or it could be along some specified line (in some specified direction) irrespective of whether it is the steepest slope.