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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, which form an associated gradient field. The gradient of function f is indicated by ∇ f. (The upside-down delta 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 a region, i.e., 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 the former can be derived from the latter (if a single value of the former is known), 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:
Having the two representations offers additional mathematical techniques for calculations relevant to the field.
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.