Gradient

In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) ∇f whose value at a point p 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 f. If the gradient of a function is non-zero at a point p, the direction of the gradient is the direction in which the function increases most quickly from p, and the magnitude of the gradient is the rate of increase in that direction, the greatest absolute directional derivative.[1] 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 f(𝐫) may be defined by:

df=∇f⋅d𝐫

where df is the total infinitesimal change in f for an infinitesimal displacement d𝐫, and is seen to be maximal when d𝐫 is in the direction of the gradient ∇f. 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[lower-alpha 1] whose components are the partial derivatives of f at p.[2] That is, for f:ℝn->ℝ, its gradient ∇f:ℝn->ℝn is defined at the point p=(x1,…,xn) in n-dimensional space as the vector[lower-alpha 2]

∇f(p)=[∂f∂x1(p)⋮∂f∂xn(p)].

Note that the above definition for gradient is defined for the function f only if f is differentiable at p. 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 f(x,y)=x2yx2+y2 unless at origin where f(0,0)=0, 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 the total derivative df: 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.[lower-alpha 3] They are related in that the dot product of the gradient of f at a point p with another tangent vector 𝐯 equals the directional derivative of f at p of the function along 𝐯; that is, ∇f(p)⋅𝐯=∂f∂𝐯(p)=dfp(𝐯). The gradient admits multiple generalizations to more general functions on manifolds; see § Generalizations.

## Contents

* 1 Motivation
* 2 Notation
* 3 Definition
* 3.1 Cartesian coordinates
* 3.2 Cylindrical and spherical coordinates
* 3.3 General coordinates
* 4 Relationship with derivative
* 4.1 Relationship with total derivative
* 4.1.1 Differential or (exterior) derivative
* 4.1.2 Linear approximation to a function
* 4.2 Relationship with Fréchet derivative
* 5 Further properties and applications
* 5.1 Level sets
* 5.2 Conservative vector fields and the gradient theorem
* 5.3 Gradient is direction of steepest ascent
* 6 Generalizations
* 6.1 Jacobian
* 6.2 Gradient of a vector field
* 6.3 Riemannian manifolds
* 7 See also
* 8 Notes
* 9 References
* 10 Further reading
* 11 External links

## Motivation

Gradient of the 2D function f(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 a scalar field, T, so at each point (x, y, z) the temperature is T(x, y, z), independent of time. At each point in the room, the gradient of T 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) is H(x, y). The gradient of H 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 (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 a unit vector along the road, as the dot product measures how much the unit vector along the road aligns with the steepest slope,[lower-alpha 4] which is 40% times the cosine of 60°, or 20%.

More generally, 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, the directional derivative of H along the unit vector.

## Notation

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

* ∇→f(a) : to emphasize the vector nature of the result.
* grad⁡f
* ∂if and fi : Written with Einstein notation, where repeated indices (i) are summed over.

## Definition

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

The gradient (or gradient vector field) of a scalar function f(x1, x2, x3, …, xn) is denoted ∇f or ∇→f where ∇ (nabla) denotes the vector differential operator, del. The notation grad f is also commonly used to represent the gradient. The gradient of f is defined as the unique vector field whose dot product with any vector v at each point x is the directional derivative of f along v. That is,

(∇f(x))⋅𝐯^=D𝐯f(x)

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 (see Spatial gradient).

The magnitude and direction of the gradient vector are independent of the particular coordinate representation.[4][5]

### Cartesian coordinates

In the three-dimensional Cartesian coordinate system with a Euclidean metric, the gradient, if it exists, is given by

∇f=∂f∂x𝐢+∂f∂y𝐣+∂f∂z𝐤,

where i, j, k are the standard unit vectors in the directions of the x, y and z coordinates, respectively.

For example, the gradient of the function f(x,y,z)=2x+3y2−sin⁡(z) is ∇f(x,y,z)=2𝐢+6y𝐣−cos⁡(z)𝐤. or ∇f(x,y,z)=[26y−cos⁡z].

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

Main article: Del in cylindrical and spherical coordinates

In cylindrical coordinates, the gradient is given by:[6]

∇f(ρ,φ,z)=∂f∂ρ𝐞ρ+1ρ∂f∂φ𝐞φ+∂f∂z𝐞z,

where ρ is the axial distance, φ is the azimuthal or azimuth angle, z is the axial coordinate, and eρ, eφ and ez are unit vectors pointing along the coordinate directions.

In spherical coordinates with a Euclidean metric, the gradient is given by:[6]

∇f(r,θ,φ)=∂f∂r𝐞r+1r∂f∂θ𝐞θ+1rsin⁡θ∂f∂φ𝐞φ,

where r is the radial distance, φ is the azimuthal angle and θ is the polar angle, and er, eθ and eφ are again local unit vectors pointing in the coordinate directions (that is, the normalized covariant basis).

For the gradient in other orthogonal coordinate systems, see Orthogonal coordinates (Differential operators in three dimensions).

### General coordinates

We consider general coordinates, which we write as x1, …, xi, …, xn, where n 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 x2 refers to the second component—not the quantity x squared. The index variable i refers to an arbitrary element xi. Using Einstein notation, the gradient can then be written as:

∇f=∂f∂xigij𝐞j (Note that its dual is df=∂f∂xi𝐞i),

where 𝐞i=dxi and 𝐞i=∂𝐱/∂xi refer to the unnormalized local covariant and contravariant bases respectively, gij 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 (and the differential) in terms of the normalized bases, which we refer to as 𝐞^i and 𝐞^i, using the scale factors (also known as Lamé coefficients) hi=‖𝐞i‖=gii=1/‖𝐞i‖ :

∇f=∂f∂xigij𝐞^jgjj=∑i=1n∂f∂xi1hie^i (and df=∑i=1n∂f∂xi1hie^i),

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, e^i, e^i, and hi are neither contravariant nor covariant.

The latter expression evaluates to the expressions given above for cylindrical and spherical coordinates.

## Relationship with derivative

Part of a series of articles about

Calculus

* Fundamental theorem

* Limits of functions
* Continuity

* Mean value theorem
* Rolle's theorem

Differential

|  Definitions

* Derivative (generalizations)

* Differential
* infinitesimal
* of a function
* total

Concepts

* Differentiation notation
* Second derivative
* Third derivative
* Change of variables
* Implicit differentiation
* Related rates
* Taylor's theorem

Rules and identities

* Sum
* Product
* Chain
* Power
* Quotient
* Inverse
* General Leibniz
* Faà di Bruno's formula

Integral

* Lists of integrals

Definitions

* Antiderivative
* Integral (improper)
* Riemann integral
* Lebesgue integration
* Contour integration

Integration by

* Parts
* Discs
* Cylindrical shells
* Substitution (trigonometric)
* Partial fractions
* Order
* Reduction formulae

Series

* Geometric (arithmetico-geometric)
* Harmonic
* Alternating
* Power
* Binomial
* Taylor

Convergence tests

* Summand limit (term test)
* Ratio
* Root
* Integral
* Direct comparison

Limit comparison
* Alternating series
* Cauchy condensation
* Dirichlet
* Abel

Vector

* Gradient
* Divergence
* Curl
* Laplacian
* Directional derivative
* Identities

Theorems

* Divergence
* Gradient
* Green's
* Kelvin–Stokes
* Stokes

Multivariable

|  Formalisms

* Matrix
* Tensor
* Exterior
* Geometric

Definitions

* Partial derivative
* Multiple integral
* Line integral
* Surface integral
* Volume integral
* Jacobian
* Hessian

Specialized

* Fractional
* Malliavin
* Stochastic
* Variations

Glossary of calculus

* Glossary of calculus

* v
* t
* e

### Relationship with total derivative

The gradient is closely related to the total derivative (total differential) df: they are transpose (dual) to each other. Using the convention that vectors in ℝn are represented by column vectors, and that covectors (linear maps ℝn->ℝ) are represented by row vectors,[lower-alpha 1] the gradient ∇f and the derivative df are expressed as a column and row vector, respectively, with the same components, but transpose of each other:

∇f(p)=[∂f∂x1(p)⋮∂f∂xn(p)]; dfp=[∂f∂x1(p)⋯∂f∂xn(p)].

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 (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 a tangent vector, which represents an infinitesimal change in (vector) input. In symbols, the gradient is an element of the tangent space at a point, ∇f(p)∈Tpℝn, while the derivative is a map from the tangent space to the real numbers, dfp:Tpℝn->ℝ. The tangent spaces at each point of ℝn can be "naturally" identified[lower-alpha 5] with the vector space ℝn itself, and similarly the cotangent space at each point can be naturally identified with the dual vector space (ℝn)* of covectors; thus the value of the gradient at a point can be thought of a vector in the original ℝn, not just as a tangent vector.

Computationally, given a tangent vector, the vector can be multiplied by the derivative (as matrices), which is equal to taking the dot product with the gradient: (dfp)(v)=[∂f∂x1(p)⋯∂f∂xn(p)][v1⋮vn]=∑i=1n∂f∂xi(p)vi=[∂f∂x1(p)⋮∂f∂xn(p)]⋅[v1⋮vn]=∇f(p)⋅v

#### Differential or (exterior) derivative

The best linear approximation to a differentiable function f:ℝn->ℝ at a point x in ℝn is a linear map from ℝn to ℝ which is often denoted by dfx or Df(x) and called the differential or total derivative of f at x. The function df, which maps x to dfx, is called the total differential or exterior derivative of f 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,[7] 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 (∇f)x⋅v=dfx(v) for any v∈ℝn, 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 ℝn is viewed as the space of (dimension n) column vectors (of real numbers), then one can regard df as the row vector with components (∂f∂x1,…,∂f∂xn), so that dfx(v) is given by matrix multiplication. Assuming the standard Euclidean metric on ℝn, the gradient is then the corresponding column vector, that is, (∇f)i=dfiT.

#### Linear approximation to a function

The best linear approximation to a function can be expressed in terms of the gradient, rather than the derivative. The gradient of a function f from the Euclidean space ℝn to ℝ at any particular point x0 in ℝn characterizes the best linear approximation to f at x0. The approximation is as follows:

f(x)≈f(x0)+(∇f)x0⋅(x−x0)

for x close to x0, where (∇f)x0 is the gradient of f computed at x0, and the dot denotes the dot product on ℝn. This equation is equivalent to the first two terms in the multivariable Taylor series expansion of f at x0.

### Relationship with Fréchet derivative

Let U be an open set in Rn. If the function f : U → R is differentiable, then the differential of f is the Fréchet derivative of f. Thus ∇f is a function from U to the space Rn such that limh->0|f(x+h)−f(x)−∇f(x)⋅h|‖h‖=0, where · 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:

Linearity
The gradient is linear in the sense that if f and g are two real-valued functions differentiable at the point a ∈ Rn, and α and β are two constants, then αf \+ βg is differentiable at a, and moreover ∇(αf+βg)(a)=α∇f(a)+β∇g(a).
Product rule
If f and g are real-valued functions differentiable at a point a ∈ Rn, then the product rule asserts that the product fg is differentiable at a, and ∇(fg)(a)=f(a)∇g(a)+g(a)∇f(a).
Chain rule
Suppose that f : A → R is a real-valued function defined on a subset A of Rn, and that f is differentiable at a point a. There are two forms of the chain rule applying to the gradient. First, suppose that the function g is a parametric curve; that is, a function g : I → Rn maps a subset I ⊂ R into Rn. If g is differentiable at a point c ∈ I such that g(c) = a, then (f∘g)′(c)=∇f(a)⋅g′(c), where ∘ is the composition operator: (f ∘ g)(x) = f(g(x)).

More generally, if instead I ⊂ Rk, then the following holds: ∇(f∘g)(c)=(Dg(c))T(∇f(a)), where (Dg)T denotes the transpose Jacobian matrix.

For the second form of the chain rule, suppose that h : I → R is a real valued function on a subset I of R, and that h is differentiable at the point f(a) ∈ I. Then ∇(h∘f)(a)=h′(f(a))∇f(a).

## Further properties and applications

### Level sets

A level surface, or isosurface, is the set of all points where some function has a given value.

If f is differentiable, then the dot product (∇f )x ⋅ v of the gradient at a point x with a vector v gives the directional derivative of f at x in the direction v. It follows that in this case the gradient of f is orthogonal to the level sets of f. For example, a level surface in three-dimensional space is defined by an equation of the form F(x, y, z) = c. The gradient of F is then normal to the surface.

More generally, any embedded hypersurface in a Riemannian manifold can be cut out by an equation of the form F(P) = 0 such that dF is nowhere zero. The gradient of F is then normal to the hypersurface.

Similarly, an affine algebraic hypersurface may be defined by an equation F(x1, ..., xn) = 0, where F is a polynomial. The gradient of F 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

Main article: Gradient theorem

The gradient of a function is called a gradient field. A (continuous) gradient field is always a conservative vector field: its line 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.

### Gradient is direction of steepest ascent

The gradient of a function f:ℝn->ℝ at point x is also the direction of its steepest ascent, i.e. it maximizes its directional derivative:

Let v∈ℝn be an arbitrary unit vector. With the directional derivative defined as

∇vf(x)=limh->0f(x+vh)−f(x)h,

we get, by substituting the function f(x+vh) with its Taylor series,

∇vf(x)=limh->0(f(x)+∇f⋅vh+R)−f(x)h,

where R denotes higher order terms in vh.

Dividing by h, and taking the limit yields a term which is bounded from above by the Cauchy–Schwarz inequality[8]

|∇vf(x)|=|∇f⋅v|≤|∇f||v|=|∇f|.

Choosing v*=∇f/|∇f| maximizes the directional derivative, and equals the upper bound

|∇v*f(x)|=|(∇f)2/|∇f||=|∇f|.

## Generalizations

### Jacobian

Main article: Jacobian matrix and determinant

The Jacobian matrix is the generalization of the gradient for vector-valued functions of several variables and differentiable maps between Euclidean spaces or, more generally, manifolds.[9][10] A further generalization for a function between Banach spaces is the Fréchet derivative.

Suppose f : Rn → Rm is a function such that each of its first-order partial derivatives exist on ℝn. Then the Jacobian matrix of f is defined to be an m×n matrix, denoted by 𝐉𝕗(𝕩) or simply 𝐉. The (i,j)th entry is 𝐉ij=∂fi/∂xj. Explicitly 𝐉=[∂𝐟∂x1⋯∂𝐟∂xn]=[∇Tf1⋮∇Tfm]=[∂f1∂x1⋯∂f1∂xn⋮⋱⋮∂fm∂x1⋯∂fm∂xn].

### Gradient of a vector field

See also: Covariant derivative

Since the total derivative of a vector field is a linear mapping from vectors to vectors, it is a tensor quantity.

In rectangular coordinates, the gradient of a vector field f = ( f1, f2, f3) is defined by:

∇𝐟=gjk∂fi∂xj𝐞i⊗𝐞k,

(where the Einstein summation notation is used and the tensor product of the vectors ei and ek is a dyadic tensor of type (2,0)). Overall, this expression equals the transpose of the Jacobian matrix:

∂fi∂xj=∂(f1,f2,f3)∂(x1,x2,x3).

In curvilinear coordinates, or more generally on a curved manifold, the gradient involves Christoffel symbols:

∇𝐟=gjk(∂fi∂xj+Γijlfl)𝐞i⊗𝐞k,

where gjk are the components of the inverse metric tensor and the ei are the coordinate basis vectors.

Expressed more invariantly, the gradient of a vector field f can be defined by the Levi-Civita connection and metric tensor:[11]

∇afb=gac∇cfb,

where ∇c is the connection.

### Riemannian manifolds

For any smooth function f on a Riemannian manifold (M, g), the gradient of f is the vector field ∇f such that for any vector field X, g(∇f,X)=∂Xf, that is, gx((∇f)x,Xx)=(∂Xf)(x), where gx( , ) denotes the inner product of tangent vectors at x defined by the metric g and ∂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 other words, in a coordinate chart φ from an open subset of M to an open subset of Rn, (∂X f )(x) is given by: ∑j=1nXj(φ(x))∂∂xj(f∘φ−1)|φ(x), where Xj denotes the jth component of X in this coordinate chart.

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

∇f=gik∂f∂xkei.

Generalizing the case M = Rn, the gradient of a function is related to its exterior derivative, since (∂Xf)(x)=(df)x(Xx). More precisely, the gradient ∇f is the vector field associated to the differential 1-form df using the musical isomorphism ♯=♯g:T*M->TM (called "sharp") defined by the metric g. 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

* Curl - Circulation density in a vector field
* Divergence
* Four-gradient
* Hessian matrix - Matrix of second derivatives
* Skew gradient
* Spatial gradient - Gradient whose components are spatial derivatives

## Notes

1. ↑ 1.0 1.1 This article uses the convention that column vectors represent vectors, and row vectors represent covectors, but the opposite convention is also common.
2. ↑ Strictly speaking, the gradient is a vector field f:ℝn->Tℝn, and the value of the gradient at a point is a tangent vector in the tangent space at that point, Tpℝn, not a vector in the original space ℝn. However, all the tangent spaces can be naturally identified with the original space ℝn, so these do not need to be distinguished; see § Definition and relationship with the derivative.
3. ↑ The value of the gradient at a point can be thought of as a vector in the original space ℝn, while the value of the derivative at a point can be thought of as a covector on the original space: a linear map ℝn->ℝ.
4. ↑ 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.
5. ↑ Informally, "naturally" identified means that this can be done without making any arbitrary choices. This can be formalized with a natural transformation.

## References

1. ↑
* (Bachman 2007)
* (Downing 2010)
* (Kreyszig 1972)
* (McGraw-Hill 2007)
* (Moise 1967)
* (Protter Morrey)
* (Swokowski et al. 1994)
2. ↑
* (Bachman 2007)
* (Beauregard Fraleigh)
* (Downing 2010)
* (Harper 1976)
* (Kreyszig 1972)
* (McGraw-Hill 2007)
* (Moise 1967)
* (Protter Morrey)
* (Swokowski et al. 1994)
3. ↑ "Non-differentiable functions must have discontinuous partial derivatives - Math Insight". https://mathinsight.org/nondifferentiable_discontinuous_partial_derivatives.
4. ↑ (Kreyszig 1972)
5. ↑ (Stoker 1969)
6. ↑ 6.0 6.1 Schey 1992, pp. 139–142.
7. ↑ (Protter Morrey)
8. ↑ T. Arens (2022). Mathematik (5th ed.). Springer Spektrum Berlin. doi:10.1007/978-3-662-64389-1. ISBN 978-3-662-64388-4. https://doi.org/10.1007/978-3-662-64389-1.
9. ↑ (Beauregard Fraleigh)
10. ↑ (Kreyszig 1972)
11. ↑ Dubrovin, Fomenko & Novikov 1991, pp. 348–349.

* Bachman, David (2007), Advanced Calculus Demystified, New York: McGraw-Hill, ISBN 978-0-07-148121-2
* Beauregard, Raymond A.; Fraleigh, John B. (1973), A First Course In Linear Algebra: with Optional Introduction to Groups, Rings, and Fields, Boston: Houghton Mifflin Company, ISBN 0-395-14017-X, https://archive.org/details/firstcourseinlin0000beau
* Downing, Douglas, Ph.D. (2010), Barron's E-Z Calculus, New York: Barron's, ISBN 978-0-7641-4461-5
* Dubrovin, B. A.; Fomenko, A. T.; Novikov, S. P. (1991). Modern Geometry—Methods and Applications: Part I: The Geometry of Surfaces, Transformation Groups, and Fields. Graduate Texts in Mathematics (2nd ed.). Springer. ISBN 978-0-387-97663-1.
* Harper, Charlie (1976), Introduction to Mathematical Physics, New Jersey: Prentice-Hall, ISBN 0-13-487538-9
* Kreyszig, Erwin (1972), Advanced Engineering Mathematics (3rd ed.), New York: Wiley, ISBN 0-471-50728-8, https://archive.org/details/advancedengineer00krey
* "McGraw Hill Encyclopedia of Science & Technology". McGraw-Hill Encyclopedia of Science & Technology (10th ed.). New York: McGraw-Hill. 2007. ISBN 978-0-07-144143-8.
* Moise, Edwin E. (1967), Calculus: Complete, Reading: Addison-Wesley
* Protter, Murray H.; Morrey, Charles B. Jr. (1970), College Calculus with Analytic Geometry (2nd ed.), Reading: Addison-Wesley
* Schey, H. M. (1992). Div, Grad, Curl, and All That (2nd ed.). W. W. Norton. ISBN 0-393-96251-2. OCLC 25048561. https://archive.org/details/divgradcurlall00sche.
* Stoker, J. J. (1969), Differential Geometry, New York: Wiley, ISBN 0-471-82825-4
* Swokowski, Earl W.; Olinick, Michael; Pence, Dennis; Cole, Jeffery A. (1994), Calculus (6th ed.), Boston: PWS Publishing Company, ISBN 0-534-93624-5, https://archive.org/details/calculus00swok
* Arens, T.; Hettlich, F.; Karpfinger, C.; Kockelkorn, U.; Lichtenegger, K.; Stachel, H. (2022), Mathematik (5th ed.), Springer Spektrum Berlin, doi:10.1007/978-3-662-64389-1, ISBN 978-3-662-64388-4, https://doi.org/10.1007/978-3-662-64389-1

## Further reading

* Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review. Dover Publications. 2000. pp. 157–160. ISBN 0-486-41147-8. OCLC 43864234.

## External links

* "Gradient". Khan Academy. https://www.khanacademy.org/math/multivariable-calculus/multivariable-derivatives/gradient-and-directional-derivatives/v/gradient.
* Hazewinkel, Michiel, ed. (2001), "Gradient", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=G/g044680 .
* Weisstein, Eric W.. "Gradient". http://mathworld.wolfram.com/Gradient.html.

* v
* t
* e

Calculus

Precalculus|

* Binomial theorem
* Concave function
* Continuous function
* Factorial
* Finite difference
* Free variables and bound variables
* Fundamental theorem
* Graph of a function
* Linear function
* Mean value theorem
* Radian
* Rolle's theorem
* Secant
* Slope
* Tangent

Limits|

* Indeterminate form
* Limit of a function
* One-sided limit
* Limit of a sequence
* Orders of approximation
* (ε, δ)-definition of limit

Differential calculus|

* Chain rule
* Derivative
* Differential
* Differential equation
* Differential operator
* Implicit differentiation
* Inverse functions and differentiation
* L'Hôpital's rule
* Leibniz's rule
* Logarithmic derivative
* Mean value theorem
* Newton's method
* Notation
* Leibniz's notation
* Newton's notation
* Product rule
* Quotient rule
* Regiomontanus' angle maximization problem
* Related rates
* Simplest rules
* Constant factor rule in differentiation
* Linearity of differentiation
* Power rule
* Sum rule in differentiation
* Stationary point
* First derivative test
* Second derivative test
* Extreme value theorem
* Maxima and minima
* Taylor's theorem

Integral calculus|

* Antiderivative
* Arc length
* Constant of integration
* Differentiation under the integral sign
* Fundamental theorem of calculus
* Integral of secant cubed
* Integral of the secant function
* Integration by parts
* Integration by substitution
* Tangent half-angle substitution
* Partial fractions in integration
* Quadratic integral
* Proof that 22/7 exceeds π
* Simplest rules
* Constant factor rule in integration
* Linearity of integration
* Sum rule in integration
* Trapezoidal rule
* Trigonometric substitution

Vector calculus|

* Curl
* Directional derivative
* Divergence
* Divergence theorem
* Gradient
* Gradient theorem
* Green's theorem
* Laplacian
* Stokes' theorem

Multivariable calculus|

* Curvature
* Disc integration
* Divergence theorem
* Exterior
* Gabriel's Horn
* Geometric
* Hessian matrix
* Jacobian matrix and determinant
* Line integral
* Matrix
* Multiple integral
* Partial derivative
* Shell integration
* Surface integral
* Tensor
* Volume integral

Series|

* Abel's test
* Alternating
* Alternating series test
* Arithmetico-geometric sequence
* Binomial
* Cauchy condensation test
* Direct comparison test
* Dirichlet's test
* Euler–Maclaurin formula
* Fourier
* Geometric
* Harmonic
* Infinite
* Integral test for convergence
* Limit comparison test
* Maclaurin
* Power
* Ratio test
* Root test
* Taylor
* Term test

Special functions and numbers|

* Bernoulli numbers
* e (mathematical constant)
* Exponential function
* Natural logarithm
* Stirling's approximation

History of calculus|

* Adequality
* Brook Taylor
* Colin Maclaurin
* Generality of algebra
* Gottfried Wilhelm Leibniz
* Infinitesimal
* Infinitesimal calculus
* Isaac Newton
* Fluxion
* Law of Continuity
* Leonhard Euler
* Method of Fluxions
* The Method of Mechanical Theorems

Lists|

* Differentiation rules
* List of integrals of exponential functions
* List of integrals of hyperbolic functions
* List of integrals of inverse hyperbolic functions
* List of integrals of inverse trigonometric functions
* List of integrals of irrational functions
* List of integrals of logarithmic functions
* List of integrals of rational functions
* List of integrals of trigonometric functions
* List of limits
* List of mathematical symbols
* Lists of integrals

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Gradient. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Gradient&oldid=4226747"
*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template