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. Vector calculus (also called vector analysis) is a field of Mathematics concerned with multivariable Real analysis of vectors in an Inner In Mathematics and Physics, a scalar field associates a scalar value which can be either mathematical in definition or physical, to every point In Mathematics a vector field is a construction in Vector calculus which associates a vector to every point in a (locally Euclidean space. The magnitude of a mathematical object is its size a property by which it can be larger or smaller than other objects of the same kind in technical terms an Ordering

A generalization of the gradient, for functions on a Banach space which have vectorial values, is the Jacobian. In Mathematics, Banach spaces (ˈbanax named after Polish Mathematician Stefan Banach) are one of the central objects of study in Functional analysis In Vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its Determinant, the Jacobian determinant.

Interpretations of the gradient

Consider a room in which the temperature is given by a scalar field T, so at each point (x,y,z) the temperature is T(x,y,z) (we will assume that the temperature does not change in time). Then, at each point in the room, the gradient of T 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 hill whose height above sea level at a point (x,y) is H(x,y). The gradient of H at a point is a vector pointing in the direction of the steepest slope or grade at that point. Slope is used to describe the steepness incline gradient or grade of a straight line. The grade (or gradient or pitch or slope) of any physical feature such as a Hill, Stream, Roof, railroad, or 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. In Mathematics, the dot product, also known as the scalar product, is an operation which takes two vectors over the Real numbers R 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 with the uphill direction (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 H is differentiable, then the gradient of H dotted with a unit vector gives the slope of the hill in the direction of the vector. In Mathematics, the dot product, also known as the scalar product, is an operation which takes two vectors over the Real numbers R More precisely, when H is differentiable the dot product of the gradient of H with a given unit vector is equal to the directional derivative of H in the direction of that unit vector. In Mathematics, the directional derivative of a multivariate Differentiable function along a given vector V at a given point P intuitively represents the

Formal definition

The gradient (or gradient vector field) of a scalar function f(x) with respect to a vector variable is denoted by or where (the nabla symbol) denotes the vector differential operator, del. Nabla is the Symbol \nabla The name comes from the Greek word for a Hebrew Harp, which had a similar shape In Mathematics, a differential operator is an Operator defined as a function of the differentiation operator &nablaDel The notation is also used for the gradient.

By definition, the gradient is a vector field whose components are the partial derivatives of f. In Mathematics a vector field is a construction in Vector calculus which associates a vector to every point in a (locally Euclidean space. In Mathematics, a partial derivative of a function of several variables is its Derivative with respect to one of those variables with the others held constant That is:

(Here the gradient is written as a row vector, but it is often taken to be a column vector; note also that when a function has a time component, the gradient often refers simply to the vector of its spatial derivatives only. )

The dot product of the gradient at a point x with a vector v gives the directional derivative of f at x in the direction v. In Mathematics, the dot product, also known as the scalar product, is an operation which takes two vectors over the Real numbers R In Mathematics, the directional derivative of a multivariate Differentiable function along a given vector V at a given point P intuitively represents the It follows that the gradient of f is orthogonal to the level sets of f. In Mathematics, two Vectors are orthogonal if they are Perpendicular, i In Mathematics, a level set of a real -valued function f of n variables is a set of the form { ( x 1 This also shows that, although the gradient was defined in terms of coordinates, it is actually invariant under orthogonal transformations, as it should be, in view of the geometric interpretation given above. In Matrix theory, a real orthogonal matrix is a square matrix Q whose Transpose is its inverse: Q^T

Because the gradient is orthogonal to level sets, it can be used to construct a vector normal to a surface. Consider any manifold that is one dimension less than the space it is in (i. e. , a surface in 3D, a curve in 2D, etc. ). Let this manifold be defined by an equation e. g. F(x, y, z) = 0 (i. e. , move everything to one side of the equation). We have now turned the manifold into a level set. To find a normal vector, we simply need to find the gradient of the function F at the desired point.

The gradient is an irrotational vector field and line integrals through a gradient field are path independent and can be evaluated with the gradient theorem. In Vector calculus a conservative vector field is a Vector field which is the Gradient of a Scalar potential. The gradient theorem, sometimes also known as the fundamental theorem of calculus for line integrals, says that a Line integral through a Gradient field Conversely, an irrotational vector field in a simply connected region is always the gradient of a function. In Topology, a geometrical object or space is called simply connected (or 1-connected) if it is Path-connected and every path between two points can be

Expressions for the gradient in 3 dimensions

The form of the gradient depends on the coordinate system used.

In Cartesian coordinates, the above expression expands to

In cylindrical coordinates:

(where θ is the azimuthal angle and z is the axial coordinate). In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane The cylindrical coordinate system is a three-dimensional Coordinate system which essentially extends circular polar coordinates by adding a third coordinate (usually

In spherical coordinates:

(where φ is the azimuth angle and θ is the zenith angle). In Mathematics, the spherical coordinate system is a Coordinate system for representing geometric figures in three dimensions using three coordinates the radial Azimuth ( is a mathematical concept defined as the angle usually measured in degrees (° between a reference plane and a point. In broad terms the zenith is the direction pointing directly above a particular location ( Perpendicular, Orthogonal)

Example

For example, the gradient of the function in Cartesian coordinates

is:

The gradient and the derivative or differential

Linear approximation to a function

The gradient of a function f from the Euclidean space to at any particular point x₀ in characterizes the best linear approximation to f at x₀. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, a linear approximation is an approximation of a general function using a Linear function (more precisely an Affine function The approximation is as follows:

for x close to x₀, where is the gradient of f computed at x₀, and the dot denotes the dot product on . In Mathematics, the dot product, also known as the scalar product, is an operation which takes two vectors over the Real numbers R This equation is equivalent to the first two terms in the multi-variable Taylor Series expansion of f at x₀. In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives

The differential or (exterior) derivative

The best linear approximation to a function at a point x in is a linear map from to which is often denoted by df_x or Df(x) and called the differential or (total) derivative of f at x. In mathematics and more specifically in Differential calculus, the term differential has several interrelated meanings In the mathematical field of Differential calculus, the term total derivative has a number of closely related meanings The gradient is therefore related to the differential by the formula

for any . The function df, which maps x to df_x, is called the differential or exterior derivative of f and is an example of a differential 1-form. In Differential geometry, the exterior derivative extends the concept of the differential of a function which is a form of degree zero to Differential forms In the mathematical fields of Differential geometry and Tensor calculus, differential forms are an approach to Multivariable calculus which is

If is viewed as the space of (length n) column vectors (of real numbers), then one can regard df as the row vector

so that df_x(v) is given by matrix multiplication. The gradient is then the corresponding column vector, i. e. , .

The covariance of the gradient

The differential is more natural than the gradient because it is invariant under all coordinate transformations (or diffeomorphisms), whereas the gradient is only invariant under orthogonal transformations (because of the implicit use of the dot product in its definition). In Mathematics, a diffeomorphism is an Isomorphism of Smooth manifolds It is an Invertible function that maps one Differentiable Because of this, it is common to blur the distinction between the two concepts using the notion of covariant and contravariant vectors. For other uses of "covariant" or "contravariant" see Covariance and contravariance. From this point of view, the components of the gradient transform covariantly under changes of coordinates, so it is called a covariant vector field, whereas the components of a vector field in the usual sense transform contravariantly. In this language the gradient is the differential, as a covariant vector field is the same thing as a differential 1-form. ^[1]

^ Unfortunately this confusing language is confused further by differing conventions. Although the components of a differential 1-form transform covariantly under coordinate transformations, differential 1-forms themselves transform contravariantly (by pullback) under diffeomorphism. For this reason differential 1-forms are sometimes said to be contravariant rather than covariant, in which case vector fields are covariant rather than contravariant.

The gradient on Riemannian manifolds

For any smooth function f on a Riemannian manifold (M,g), the gradient of f is the vector field such that for any vector field X,

where denotes the inner product of tangent vectors at x defined by the metric g and (sometimes denoted X(f)) is the function that takes any point x∈M to the directional derivative of f in the direction X, evaluated at x. In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M In Mathematics a vector field is a construction in Vector calculus which associates a vector to every point in a (locally Euclidean space. In Mathematics, an inner product space is a Vector space with the additional Structure of inner product. In Mathematics, the directional derivative of a multivariate Differentiable function along a given vector V at a given point P intuitively represents the In other words, in a coordinate chart from an open subset of M to an open subset of Rⁿ, is given by:

where X^j denotes the jth component of X in this coordinate chart. For other uses of "atlas" see Atlas (disambiguation. In Mathematics, particularly topology an atlas describes how

So, the local form of the gradient takes the form:

Generalizing the case M=Rⁿ, the gradient of a function is related to its exterior derivative, since . In Differential geometry, the exterior derivative extends the concept of the differential of a function which is a form of degree zero to Differential forms More precisely, the gradient is the vector field associated to the differential 1-form df using the musical isomorphism (called "sharp") defined by the metric g. In Mathematics, the musical isomorphism (or canonical isomorphism is an Isomorphism between the Tangent bundle TM and the Cotangent The relation between the exterior derivative and the gradient of a function on Rⁿ is a special case of this in which the metric is the flat metric given by the dot product.

See also

References

1. Theresa M. For the analytical method called "steepest descent" see Method of steepest descent. cURL is a Command line tool for transferring files with URL syntax. In Vector calculus, the divergence is an Operator that measures the magnitude of a Vector field &rsquos source or sink at a given point the In Mathematics and Physics, the Laplace operator or Laplacian, denoted by \Delta\  or \nabla^2  and named after In Cellular biology, an electrochemical gradient is a spatial variation of both Electrical potential and chemical Concentration across a membrane In Mathematics, a level set of a real -valued function f of n variables is a set of the form { ( x 1 In Mathematics, the musical isomorphism (or canonical isomorphism is an Isomorphism between the Tangent bundle TM and the Cotangent &nablaDel The Sobel operator is used in Image processing, particularly within Edge detection algorithms The grade (or gradient or pitch or slope) of any physical feature such as a Hill, Stream, Roof, railroad, or Slope is used to describe the steepness incline gradient or grade of a straight line. In Vector calculus, the surface gradient is a vector Differential operator that is similar to the conventional Gradient. Korn; Korn, Granino Arthur. Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review. New York: Dover Publications, 157-160. ISBN 0-486-41147-8.