Gradient

For other uses, see Gradient (disambiguation).

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.

Missing image
Grad1.jpg

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.

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.

Contents

1 Examples

2 Formal definition

2.1 Example

3 The gradient on manifolds

4 See also

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

The gradient of a scalar function $φ$ is denoted by:

$\nabla \phi$

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

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.)

Example

For example, the gradient of the function $φ = 2 x + 3 y 2 - sin(z)$ is:

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

The gradient on manifolds

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 $ξ f$ 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 $\varphi$ , $ξ f (p)$ 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 $ξ f (p) = d f (ξ)$ . 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.

Gradient

Examples

Formal definition

Example

The gradient on manifolds

See also

See also: Gradient, Cartesian coordinates, Coordinate chart, Curl, Del, Divergence, Euclidean space, Exterior derivative, Function (mathematics)