Gradient

From Wikinfo

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 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 <math>\phi(x,y,z)</math>: 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 <math>z(x,y)</math> -- 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.

A generalization of the gradient for functions on a Euclidean space which have values in another Euclidean space is the Jacobian. A further generalization for a function from one Banach space to another is the Fréchet derivative.

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. 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.

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.

Interpretations

For instance, consider a room in which the temperature is given by a scalar field <math>T</math>, so at each point <math>(x,y,z)</math> the temperature is <math>T(x,y,z)</math> (we will assume that the temperature does not change in time). Then, at each point in the room, the gradient of <math>T</math> at that point will show the direction in which the temperature rises most quickly. The magnitude of the gradient will determine how fast the temperature rises in that direction.

Consider a surface whose height above sea level at a point <math>(x, y)</math> is <math>H(x, y)</math>. The gradient of <math>H</math> at a point is a vector pointing in the direction of the steepest slope or grade at that point. The steepness of the slope at that point is given by the magnitude of the gradient vector.

The gradient can also be used to measure how a scalar field changes in other directions, rather than just the direction of greatest change, by taking a dot product. Consider again the example with the hill and suppose that the steepest slope on the hill is 40%. If a road goes directly up the hill, then the steepest slope on the road will also be 40%. If, instead, the road goes around the hill at an angle (the gradient vector), then it will have a shallower slope. For example, if the angle between the road and the uphill direction, projected onto the horizontal plane, is 60°, then the steepest slope along the road will be 20%, which is 40% times the cosine of 60°.

This observation can be mathematically stated as follows. If the hill height function <math>H</math> is differentiable, then the gradient of <math>H</math> dotted with a unit vector gives the slope of the hill in the direction of the vector. More precisely, when <math>H</math> is differentiable, the dot product of the gradient of <math>H</math> with a given unit vector is equal to the directional derivative of <math>H</math> in the direction of that unit vector.

Formal definition

The gradient is noted by:

<math>\nabla \phi</math>

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

In 3 dimensions, the expression expands to

<math>\begin{pmatrix}

{\partial \phi / \partial x} \\ {\partial \phi / \partial y} \\ {\partial \phi / \partial z} \end{pmatrix}</math>

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

Definition

The gradient of the function f(x,y) = −(cos²x + cos²y)² depicted as a vector field on the bottom plane

The gradient (or gradient vector field) of a scalar function <math>f(x_1, x_2, x_3, \dots, x_n)</math> is denoted <math>\nabla f</math> or <math>\vec{\nabla} f</math> where <math>\nabla</math> (the nabla symbol) denotes the vector differential operator, del. The notation <math>\operatorname{grad}(f)</math> is also used for the gradient. The gradient of f is defined to be the vector field whose components are the partial derivatives of <math>f</math>. That is:

<math> \nabla f = \left(\frac{\partial f}{\partial x_1 }, \dots, \frac{\partial f}{\partial x_n } \right). </math>

Here the gradient is written as a row vector, but it is often taken to be a column vector. When a function also depends on a parameter such as time, the gradient often refers simply to the vector of its spatial derivatives only.

See also: