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In vector calculus, gradient is a vector operator[?] which acts on a scalar field. The gradient of a scalar is a vector 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 $\phi(x,y,z)$: 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 $z(x,y)$ -- the height z defined by the co-ordinates 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. The magnitude indicates how steep the slope is.

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. in the following two images, the scalar field is in black and white, black representing higher values, and its corresponding gradient is represented by blue arrows.

 Formal definition

$\nabla \phi$

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

In 3 dimensions, the expression expands to

$\begin{pmatrix} {\partial \phi / \partial x} \\ {\partial \phi / \partial y} \\ {\partial \phi / \partial z} \end{pmatrix}$

in cartesian coordinates. If $\phi$ is only in terms of x and y (for example, if the equation is of the form $z = \phi(x,y)$), 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.