This article is about a generalized derivative of a multivariate function. For another use in mathematics, seeSlope. For a similarly spelled unit of angle, seeGradian. For other uses, seeGradient (disambiguation).
The gradient, represented by the blue arrows, denotes the direction of greatest change of a scalar function. The values of the function are represented in greyscale and increase in value from white (low) to dark (high).
Invector calculus, thegradient of ascalar-valueddifferentiable function ofseveral variables is thevector field (orvector-valued function) 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 themagnitude of the gradient is the rate of increase in that direction, the greatestabsolute directional derivative.[1] Further, a point where the gradient is the zero vector is known as astationary point. The gradient thus plays a fundamental role inoptimization theory, where it is used to minimize a function bygradient 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. Thenabla symbol, written as an upside-down triangle and pronounced "del", denotes thevector differential operator.
When a coordinate system is used in which the basis vectors are not functions of position, the gradient is given by thevector[a] whose components are thepartial derivatives of at.[2] That is, for, its gradient is defined at the point inn-dimensional space as the vector[b]
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 isorthonormal. For any other basis, themetric 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.[3] In this particular example, under rotation of x-y coordinate system, the above formula for gradient fails to transform like a vector (gradient becomes dependent on choice of basis for coordinate system) 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 thetotal derivative: the value of the gradient at a point is atangent vector – a vector at each point; while the value of the derivative at a point is acotangent vector – a linear functional on vectors.[c] They are related in that thedot product of the gradient of at a point with another tangent vector equals thedirectional derivative of at of the function along; that is,. The gradient admits multiple generalizations to more general functions onmanifolds; see§ Generalizations.
Gradient of the 2D functionf(x,y) =xe−(x2 +y2) is plotted as arrows over the pseudocolor plot of the function.
Consider a room where the temperature is given by ascalar field,T, so at each point(x,y,z) the temperature isT(x,y,z), independent of time. At each point in the room, the gradient ofT at that point will show the direction in which the temperature rises most quickly, moving away from(x,y,z). 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(x,y) isH(x,y). The gradient ofH at a point is a plane vector pointing in the direction of the steepest slope orgrade 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 adot 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 (when both directions are projected onto the horizontal plane), then the slope along the road will be the dot product between the gradient vector and aunit vector along the road, as the dot product measures how much the unit vector along the road aligns with the steepest slope,[d] which is 40% times thecosine of 60°, or 20%.
More generally, if the hill height functionH isdifferentiable, then the gradient ofHdotted with aunit vector gives the slope of the hill in the direction of the vector, thedirectional derivative ofH along the unit vector.
The gradient of the functionf(x,y) = −(cos2x + cos2y)2 depicted as a projectedvector field on the bottom plane.
The gradient (or gradient vector field) of a scalar functionf(x1,x2,x3, …,xn) is denoted∇f or∇→f where∇ (nabla) denotes the vectordifferential operator,del. The notationgradf is also commonly used to represent the gradient. The gradient off is defined as the unique vector field whose dot product with anyvectorv at each pointx is the directional derivative off alongv. That is,
When a function also depends on a parameter such as time, the gradient often refers simply to the vector of its spatial derivatives only (seeSpatial gradient).
wherei,j,k are thestandard unit vectors in the directions of thex,y andz coordinates, respectively. For example, the gradient of the functionisor
In some applications it is customary to represent the gradient as arow vector orcolumn 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.
whereρ is the axial distance,φ is the azimuthal or azimuth angle,z is the axial coordinate, andeρ,eφ andez are unit vectors pointing along the coordinate directions.
wherer is the radial distance,φ is the azimuthal angle andθ is the polar angle, ander,eθ andeφ are again local unit vectors pointing in the coordinate directions (that is, the normalizedcovariant basis).
We considergeneral coordinates, which we write asx1, …,xi, …,xn, wheren is the number of dimensions of the domain. Here, the upper index refers to the position in the list of the coordinate or component, sox2 refers to the second component—not the quantityx squared. The index variablei refers to an arbitrary elementxi. UsingEinstein notation, the gradient can then be written as:
If the coordinates are orthogonal we can easily express the gradient (and thedifferential) in terms of the normalized bases, which we refer to as and, using the scale factors (also known asLamé coefficients) :
(and),
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.
The gradient is closely related to thetotal derivative (total differential): they aretranspose (dual) to each other. Using the convention that vectors in are represented bycolumn vectors, and that covectors (linear maps) are represented byrow vectors,[a] 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 acotangent vector, alinear form (or covector) which expresses how much the (scalar) output changes for a given infinitesimal change in (vector) input, while at each point, the gradient is atangent vector, which represents an infinitesimal change in (vector) 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[e] with the vector space itself, and similarly the cotangent space at each point can be naturally identified with thedual 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 bemultiplied by the derivative (as matrices), which is equal to taking thedot product with the gradient:
Much as the derivative of a function of a single variable represents theslope of thetangent to thegraph of the function,[7] the directional derivative of a function in several variables represents the slope of the tangenthyperplane in the direction of the vector.
The gradient is related to the differential by the formulafor any, where is thedot 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 (dimension) column vectors (of real numbers), then one can regard as the row vector with componentsso that is given bymatrix multiplication. Assuming the standard Euclidean metric on, the gradient is then the corresponding column vector, that is,
The bestlinear approximation to a function can be expressed in terms of the gradient, rather than the derivative. The gradient of afunction from the Euclidean space to at any particular point in characterizes the bestlinear approximation to at. The approximation is as follows:
for close to, where is the gradient of computed at, and the dot denotes the dot product on. This equation is equivalent to the first two terms in themultivariable Taylor series expansion of at.
LetU be anopen set inRn. If the functionf :U →R is differentiable, then the differential off is theFréchet derivative off. Thus∇f is a function fromU to the spaceRn such thatwhere · is the dot product.
As a consequence, the usual properties of the derivative hold for the gradient, though the gradient is not a derivative itself, but rather dual to the derivative:
The gradient is linear in the sense that iff andg are two real-valued functions differentiable at the pointa ∈Rn, andα andβ are two constants, thenαf +βg is differentiable ata, and moreover
Suppose thatf :A →R is a real-valued function defined on a subsetA ofRn, and thatf is differentiable at a pointa. There are two forms of the chain rule applying to the gradient. First, suppose that the functiong is aparametric curve; that is, a functiong :I →Rn maps a subsetI ⊂R intoRn. Ifg is differentiable at a pointc ∈I such thatg(c) =a, then where ∘ is thecomposition operator:(f ∘ g)(x) =f(g(x)).
More generally, if insteadI ⊂Rk, then the following holds:where(Dg)T denotes the transposeJacobian matrix.
For the second form of the chain rule, suppose thath :I →R is a real valued function on a subsetI ofR, and thath is differentiable at the pointf(a) ∈I. Then
A level surface, orisosurface, is the set of all points where some function has a given value.
Iff is differentiable, then the dot product(∇f )x ⋅v of the gradient at a pointx with a vectorv gives the directional derivative off atx in the directionv. It follows that in this case the gradient off isorthogonal to thelevel sets off. For example, a level surface in three-dimensional space is defined by an equation of the formF(x,y,z) =c. The gradient ofF is then normal to the surface.
More generally, anyembeddedhypersurface in aRiemannian manifold can be cut out by an equation of the formF(P) = 0 such thatdF is nowhere zero. The gradient ofF is then normal to the hypersurface.
Similarly, anaffine algebraic hypersurface may be defined by an equationF(x1, ...,xn) = 0, whereF is a polynomial. The gradient ofF is zero at a singular point of the hypersurface (this is the definition of a singular point). At a non-singular point, it is a nonzero normal vector.
Conservative vector fields and the gradient theorem
The gradient of a function is called a gradient field. A (continuous) gradient field is always aconservative vector field: itsline integral along any path depends only on the endpoints of the path, and can be evaluated by the gradient theorem (the fundamental theorem of calculus for line integrals). Conversely, a (continuous) conservative vector field is always the gradient of a function.
Supposef :Rn →Rm is a function such that each of its first-order partial derivatives exist onℝn. Then the Jacobian matrix off is defined to be anm×n matrix, denoted by or simply. The(i,j)th entry is. Explicitly
For anysmooth functionf on a Riemannian manifold(M,g), the gradient off is the vector field∇f such that for any vector fieldX,that is,wheregx( , ) denotes theinner product of tangent vectors atx defined by the metricg and∂Xf is the function that takes any pointx ∈M to the directional derivative off in the directionX, evaluated atx. In other words, in acoordinate chartφ from an open subset ofM to an open subset ofRn,(∂Xf )(x) is given by:whereXj denotes thejth component ofX in this coordinate chart.
So, the local form of the gradient takes the form:
Generalizing the caseM =Rn, the gradient of a function is related to its exterior derivative, sinceMore precisely, the gradient∇f is the vector field associated to the differential 1-formdf using themusical isomorphism(called "sharp") defined by the metricg. The relation between the exterior derivative and the gradient of a function onRn is a special case of this in which the metric is the flat metric given by the dot product.
^abThis article uses the convention thatcolumn vectors represent vectors, androw vectors represent covectors, but the opposite convention is also common.
^Strictly speaking, the gradient is avector field, and the value of the gradient at a point is atangent vector in thetangent space at that point,, not a vector in the original space. However, all the tangent spaces can be naturally identified with the original space, so these do not need to be distinguished; see§ Definition andrelationship with the derivative.
^The value of the gradient at a point can be thought of as a vector in the original space, while the value of the derivative at a point can be thought of as a covector on the original space: a linear map.
^the dot product (the slope of the road around the hill) would be 40% if the degree between the road and the steepest slope is 0°, i.e. when they are completely aligned, and flat when the degree is 90°, i.e. when the road is perpendicular to the steepest slope.
^Informally, "naturally" identified means that this can be done without making any arbitrary choices. This can be formalized with anatural transformation.
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