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

In vector calculus, the gradient of a scalar field is a vector field which points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change. More rigorously, the gradient of a function from the Euclidean space Rn to R is the best linear approximation to that function at any particular point in Rn. To that extent, the gradient is a particular case of the Jacobian.

In the case of a real-valued function of a single variable, the gradient is simply the derivative, or, for a linear function, the slope of the line.

The word gradient is sometimes used synonymously with grade, meaning the inclination of a surface along a given direction. One can obtain the grade by taking the dot product of the vector gradient with the unit vector in the direction of interest. The magnitude of the gradient is also sometimes referred to as just the gradient.

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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 hot most quickly. 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 is in the direction of the steepest slope/grade at that point. The magnitude of the gradient tells how steep the slope actually is.

The gradient can also be used to tell how things change in other directions rather than the direction of largest change. Consider again the example with the hill. One can have a road which goes right uphill where the slope is largest and then its slope is the magnitude of the gradient. Or one can have a road which goes under an angle with the uphill direction, say for example an angle of 60° when projected onto the horizontal plane. Then, if the steepest slope on the hill is 40%, the road will make a shallower slope of 20% which is 40% times the cosine of 60°.

This observation can be mathematically stated as follows. The gradient of the hill height function H dotted with a unit vector gives the slope of the surface in the direction of the vector. This is called the directional derivative.

Formal definition

The gradient of a scalar function Template:Phisymbol is denoted by:

$\nabla \phi$

where $\nabla$ (nabla) denotes the vector differential operator del. The gradient of Template:Phisymbol is sometimes also written as grad(Template:Phisymbol).

In 3 dimensions, the expression expands to

$\nabla \phi = \begin{pmatrix} {\frac{\partial \phi}{\partial x}}, {\frac{\partial \phi}{\partial y}}, {\frac{\partial \phi}{\partial z}} \end{pmatrix}$

in Cartesian coordinates. (See partial derivative and vector.)

Although expressed in terms of coordinates, the result is invariant under orthogonal transformations, as it should, in view of the geometric definition.

Example

The gradient of the function Template:Phisymbol = 2x + 3y2 − sin(z) is:

$\nabla \phi = \begin{pmatrix} {\frac{\partial \phi}{\partial x}}, {\frac{\partial \phi}{\partial y}}, {\frac{\partial \phi}{\partial z}} \end{pmatrix} = \begin{pmatrix} {2}, {6y}, {-\cos(z)} \end{pmatrix}.$

For any differentiable function f on a Riemannian manifold M, the gradient of f is the vector field such that for any vector ξ,

$\langle \nabla f(x), \xi \rangle := \xi f$

where $\langle \cdot, \cdot \rangle$ denotes the inner product on M (the metric) and ξf is the function that takes any point p to the directional derivative of f in the direction ξ evaluated at p. In other words, under some coordinate chart$\varphi$, ξf(p) will be:

$\sum \xi_{x_{j}} (\partial_{j}f \mid_{p}) := \sum \xi_{x_{j}} (\frac{\partial}{\partial x_{j} }(f \circ \varphi^{-1}) \mid_{\varphi(p)}).$

The gradient of a function is related to the exterior derivative, since ξf(p) = df(ξ). Indeed, the metric allows one to associate canonically the 1-form df to the vector field $\nabla f$. In Rn the flat metric is implicit and the gradient can be identified with the exterior derivative.