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In vector calculus, the gradient of a scalar field is a vector field which points in the direction of the greatest rate of change of the scalar field, and whose magnitude is the greatest rate of change.

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.

## Examples

• Consider a room in which the temperature is given by a scalar field φ, so at each point (x,y,z) the temperature is φ(x,y,z). We will assume that the temperature does not change in time. Then, at each point in the room, the gradient at that point will show the direction in which it gets hottest. The magnitude of the gradient will tell how fast it gets hot in that direction.
• Consider a hill whose height at a point (x,y) is H(x,y). The gradient of H at a point will show the direction of the steepest slope at that point. The magnitude of the gradient will tell how steep the slope actually is. The gradient at a point is perpendicular to the level set going through that point, that is, to the curve of constant height at that point.

## Formal definition

$\nabla \phi$

where $\nabla$ (nabla) is the vector differential operator del, and φ 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 φ is only in terms of x and y (for example, if the equation is of the form z = φ(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. 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 .