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In vector calculus, gradient is a vector-valued operator that acts on a scalar field. The gradient of a scalar field is a vector field called gradient field which shows its rate and direction of change.

For example, consider a room. This is a 3-dimensional space, and the temperature of the air at any point is a scalar field : a number associated to each point vector (we are considering the temperature as unchanging, so there is no time variable). At any given point, the gradient is a vector that points in the direction of the greatest rate of change and has a magnitude equal to that rate.

A good two-dimensional example is a hill. The contour map of the terrain is, in effect, a scalar function -- the height z defined by the coordinates of the given point. The gradient of z at a point is a two-dimensional vector which points in the direction of the greatest slope, which is perpendicular to the contour lines. The magnitude indicates how steep the slope is.

In multivariable calculus, the gradient of a function F : Rⁿ → R¹ is the 1 by n matrix of the best linear approximation to that function. In this sense, it is just a special case of the Jacobian.

1 Spatial representation of gradient

Given a scalar field, the gradient of the field is a vector field, where all vectors point towards the higher values, with magnitude equal to the rate of change of values.

2 Formal definition

The gradient is noted by:

where ( nabla) is the vector differential operator del, and is a scalar function. It is sometimes also written grad(φ).

In 3 dimensions, the expression expands to

in cartesian coordinates. If is only in terms of x and y (for example, if the equation is of the form ), just use the first two components.

Note: The gradient does not necessarily exist at all points - for example it may not exist at discontinuities or where the function or its partial derivative is undefined. Normally in vector calculus, one studies scalar fields whose gradient is defined at all points except for certain singularities. In functional analysis, one might study a space of scalar fields on which the gradient operator is only densely defined .

3 See also

• Jacobian
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• vector calculus
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