Gradient

In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field whose value at a point gives the direction and the rate of fastest increase. The gradient transforms like a vector under change of basis of the space of variables of. If the gradient of a function is non-zero at a point, the direction of the gradient is the direction in which the function increases most quickly from, and the magnitude of the gradient is the rate of increase in that direction, the greatest absolute directional derivative. Further, a point where the gradient is the zero vector is known as a stationary point. The gradient thus plays a fundamental role in optimization theory, machine learning, and artificial intelligence, where it is used to minimize a function by gradient descent. In coordinate-free terms, the gradient of a function may be defined by:
where is the total infinitesimal change in for an infinitesimal displacement , and is seen to be maximal when is in the direction of the gradient. The nabla symbol, written as an upside-down triangle and pronounced "del", denotes the vector differential operator.
When a coordinate system is used in which the basis vectors are not functions of position, the gradient is given by the vector whose components are the partial derivatives of at. That is, for, its gradient is defined at the point in n-dimensional space as the vector
Note that the above definition for gradient is defined for the function only if is differentiable at. There can be functions for which partial derivatives exist in every direction but fail to be differentiable. Furthermore, this definition as the vector of partial derivatives is only valid when the basis of the coordinate system is orthonormal. For any other basis, the metric tensor at that point needs to be taken into account.
For example, the function unless at origin where, is not differentiable at the origin as it does not have a well defined tangent plane despite having well defined partial derivatives in every direction at the origin. In this particular example, under rotation of x-y coordinate system, the above formula for gradient fails to transform like a vector and also fails to point towards the 'steepest ascent' in some orientations. For differentiable functions where the formula for gradient holds, it can be shown to always transform as a vector under transformation of the basis so as to always point towards the fastest increase.
The gradient is dual to the total derivative : the value of the gradient at a point is a tangent vector – a vector at each point; while the value of the derivative at a point is a cotangent vector – a linear functional on vectors. They are related in that the dot product of the gradient of at a point with another tangent vector equals the directional derivative of at of the function along ; that is,.
The gradient admits multiple generalizations to more general functions on manifolds; see.

Motivation

Consider a room where the temperature is given by a scalar field,, so at each point the temperature is, independent of time. At each point in the room, the gradient of at that point will show the direction in which the temperature rises most quickly, moving away from. The magnitude of the gradient will determine how fast the temperature rises in that direction.
Consider a surface whose height above sea level at point is. The gradient of at a point is a plane 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. Suppose that the steepest slope on a hill is 40%. A road going directly uphill has slope 40%, but a road going around the hill at an angle will have a shallower slope. For example, if the road is at a 60° angle from the uphill direction, then the slope along the road will be the dot product between the gradient vector and a unit vector along the road, as the dot product measures how much the unit vector along the road aligns with the steepest slope, which is 40% times the cosine of 60°, or 20%.
More generally, if the hill height function is differentiable, then the gradient of dotted with a unit vector gives the slope of the hill in the direction of the vector, the directional derivative of along the unit vector.

Notation

The gradient of a function at point is usually written as. It may also be denoted by any of the following:

: to emphasize the vector nature of the result.
and : Written with Einstein notation, where repeated indices are summed over.
Definition

The gradient of a scalar function is denoted or where denotes the vector differential operator, del. The notation is also commonly used to represent the gradient. The gradient of is defined as the unique vector field whose dot product with any vector at each point is the directional derivative of along. That is,
where the right-hand side is the directional derivative and there are many ways to represent it. Formally, the derivative is dual to the gradient; see [|relationship with derivative].
When a function also depends on a parameter such as time, the gradient often refers simply to the vector of its spatial derivatives only.
The magnitude and direction of the gradient vector are independent of the particular coordinate representation.

Cartesian coordinates

In the three-dimensional Cartesian coordinate system with a Euclidean metric, the gradient, if it exists, is given by
where,, are the standard unit vectors in the directions of the, and coordinates, respectively.
For example, the gradient of the function
is
or
In some applications it is customary to represent the gradient as a row vector or column vector of its components in a rectangular coordinate system; this article follows the convention of the gradient being a column vector, while the derivative is a row vector.

Cylindrical and spherical coordinates

In cylindrical coordinates, the gradient is given by:
where is the axial distance, is the azimuthal or azimuth angle, is the axial coordinate, and, and are unit vectors pointing along the coordinate directions.
In spherical coordinates with a Euclidean metric, the gradient is given by:
where is the radial distance, is the azimuthal angle and is the polar angle, and, and are again local unit vectors pointing in the coordinate directions.
For the gradient in other orthogonal coordinate systems, see Orthogonal coordinates.

General coordinates

We consider general coordinates, which we write as, where is the number of dimensions of the domain. Here, the upper index refers to the position in the list of the coordinate or component, so refers to the second component—not the quantity squared. The index variable refers to an arbitrary element. Using Einstein notation, the gradient can then be written as:
,
where and refer to the unnormalized local covariant and contravariant bases respectively, is the inverse metric tensor, and the Einstein summation convention implies summation over i and j.
If the coordinates are orthogonal we can easily express the gradient in terms of the normalized bases, which we refer to as and , using the scale factors :
,
where we cannot use Einstein notation, since it is impossible to avoid the repetition of more than two indices. Despite the use of upper and lower indices,,, and are neither contravariant nor covariant.
The latter expression evaluates to the expressions given above for cylindrical and spherical coordinates.

Relationship with derivative

Relationship with total derivative

The gradient is closely related to the total derivative : they are transpose to each other. Using the convention that vectors in are represented by column vectors, and that covectors are represented by row vectors, the gradient and the derivative are expressed as a column and row vector, respectively, with the same components, but transpose of each other:
While these both have the same components, they differ in what kind of mathematical object they represent: at each point, the derivative is a cotangent vector, a linear form which expresses how much the output changes for a given infinitesimal change in input, while at each point, the gradient is a tangent vector, which represents an infinitesimal change in input. In symbols, the gradient is an element of the tangent space at a point,, while the derivative is a map from the tangent space to the real numbers,. The tangent spaces at each point of can be "naturally" identified with the vector space itself, and similarly the cotangent space at each point can be naturally identified with the dual vector space of covectors; thus the value of the gradient at a point can be thought of a vector in the original, not just as a tangent vector.
Computationally, given a tangent vector, the vector can be multiplied by the derivative, which is equal to taking the dot product with the gradient:

Differential or (exterior) derivative

The best linear approximation to a differentiable function
at a point in is a linear map from to which is often denoted by or and called the differential or total derivative of at. The function, which maps to, is called the total differential or exterior derivative of and is an example of a differential 1-form.
Much as the derivative of a function of a single variable represents the slope of the tangent to the graph of the function, the directional derivative of a function in several variables represents the slope of the tangent hyperplane in the direction of the vector.
The gradient is related to the differential by the formula
for any, where is the dot product: taking the dot product of a vector with the gradient is the same as taking the directional derivative along the vector.
If is viewed as the space of column vectors, then one can regard as the row vector with components
so that is given by matrix multiplication. Assuming the standard Euclidean metric on, the gradient is then the corresponding column vector, that is,