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In mathematics, the gradient is a multi-variable generalization of the derivative. [1]

72 relations: Affine variety, Atlas (topology), Banach space, Cartesian coordinate system, Chain rule, Christoffel symbols, Composition operator, Conservative vector field, Curl (mathematics), Curvilinear coordinates, Del, Derivative, Differentiable function, Differential (mathematics), Differential form, Differential operator, Directional derivative, Divergence, Dot product, Dyadics, Einstein notation, Euclidean distance, Euclidean space, Euclidean vector, Exterior derivative, Four-gradient, Fréchet derivative, Function (mathematics), Grade (slope), Gradient theorem, Graduate Texts in Mathematics, Graph of a function, Hessian matrix, Hypersurface, Inner product space, Isosurface, Jacobian matrix and determinant, Khan Academy, Level set, Levi-Civita connection, Line integral, Linear approximation, Linear map, Linearity, Magnitude (mathematics), Manifold, Mathematics, Metric tensor, Musical isomorphism, Nabla symbol, ... Expand index (22 more) »

## Affine variety

In algebraic geometry, an affine variety over an algebraically closed field k is the zero-locus in the affine ''n''-space k^n of some finite family of polynomials of n variables with coefficients in k that generate a prime ideal.

## Atlas (topology)

In mathematics, particularly topology, one describes a manifold using an atlas.

## Banach space

In mathematics, more specifically in functional analysis, a Banach space (pronounced) is a complete normed vector space.

## Cartesian coordinate system

A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length.

## Chain rule

In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions.

## Christoffel symbols

In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection.

## Composition operator

In mathematics, the composition operator C_\phi with symbol \phi is a linear operator defined by the rule where f \circ\phi denotes function composition.

## Conservative vector field

In vector calculus, a conservative vector field is a vector field that is the gradient of some function, known in this context as a scalar potential.

## Curl (mathematics)

In vector calculus, the curl is a vector operator that describes the infinitesimal rotation of a vector field in three-dimensional Euclidean space.

## Curvilinear coordinates

In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved.

## Del

Del, or nabla, is an operator used in mathematics, in particular in vector calculus, as a vector differential operator, usually represented by the nabla symbol &nabla;.

New!!: Gradient and Del · See more »

## Derivative

The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value).

## Differentiable function

In calculus (a branch of mathematics), a differentiable function of one real variable is a function whose derivative exists at each point in its domain.

## Differential (mathematics)

In mathematics, differential refers to infinitesimal differences or to the derivatives of functions.

## Differential form

In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates.

## Differential operator

In mathematics, a differential operator is an operator defined as a function of the differentiation operator.

## Directional derivative

In mathematics, the directional derivative of a multivariate differentiable function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity specified by v. It therefore generalizes the notion of a partial derivative, in which the rate of change is taken along one of the curvilinear coordinate curves, all other coordinates being constant.

## Divergence

In vector calculus, divergence is a vector operator that produces a scalar field, giving the quantity of a vector field's source at each point.

## Dot product

In mathematics, the dot product or scalar productThe term scalar product is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space.

In mathematics, specifically multilinear algebra, a dyadic or dyadic tensor is a second order tensor, written in a notation that fits in with vector algebra.

## Einstein notation

In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention that implies summation over a set of indexed terms in a formula, thus achieving notational brevity.

## Euclidean distance

In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" straight-line distance between two points in Euclidean space.

## Euclidean space

In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.

## Euclidean vector

In mathematics, physics, and engineering, a Euclidean vector (sometimes called a geometric or spatial vector, or—as here—simply a vector) is a geometric object that has magnitude (or length) and direction.

## Exterior derivative

On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree.

In differential geometry, the four-gradient (or 4-gradient) \mathbf is the four-vector analogue of the gradient \vec from Gibbs–Heaviside vector calculus.

## Fréchet derivative

In mathematics, the Fréchet derivative is a derivative defined on Banach spaces.

## Function (mathematics)

In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.

The grade (also called slope, incline, gradient, mainfall, pitch or rise) of a physical feature, landform or constructed line refers to the tangent of the angle of that surface to the horizontal.

The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve.

## Graph of a function

In mathematics, the graph of a function f is, formally, the set of all ordered pairs, and, in practice, the graphical representation of this set.

## Hessian matrix

In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field.

## Hypersurface

In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface.

## Inner product space

In linear algebra, an inner product space is a vector space with an additional structure called an inner product.

## Isosurface

An isosurface is a three-dimensional analog of an isoline.

## Jacobian matrix and determinant

In vector calculus, the Jacobian matrix is the matrix of all first-order partial derivatives of a vector-valued function.

Khan Academy is a non-profit educational organization created in 2006 by educator Salman Khan with a goal of creating a set of online tools that help educate students.

## Level set

In mathematics, a level set of a real-valued function ''f'' of ''n'' real variables is a set of the form that is, a set where the function takes on a given constant value c. When the number of variables is two, a level set is generically a curve, called a level curve, contour line, or isoline.

## Levi-Civita connection

In Riemannian geometry, the Levi-Civita connection is a specific connection on the tangent bundle of a manifold.

## Line integral

In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve.

## Linear approximation

In mathematics, a linear approximation is an approximation of a general function using a linear function (more precisely, an affine function).

## Linear map

In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication.

## Linearity

Linearity is the property of a mathematical relationship or function which means that it can be graphically represented as a straight line.

## Magnitude (mathematics)

In mathematics, magnitude is the size of a mathematical object, a property which determines whether the object is larger or smaller than other objects of the same kind.

## Manifold

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.

## Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

## Metric tensor

In the mathematical field of differential geometry, a metric tensor is a type of function which takes as input a pair of tangent vectors and at a point of a surface (or higher dimensional differentiable manifold) and produces a real number scalar in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space.

## Musical isomorphism

In mathematics, the musical isomorphism (or canonical isomorphism) is an isomorphism between the tangent bundle TM and the cotangent bundle T^* M of a pseudo-Riemannian manifold induced by its metric tensor.

## Nabla symbol

∇ The nabla symbol The nabla is a triangular symbol like an inverted Greek delta:Indeed, it is called anadelta (ανάδελτα) in Modern Greek.

## Open set

In topology, an open set is an abstract concept generalizing the idea of an open interval in the real line.

## Orthogonal coordinates

In mathematics, orthogonal coordinates are defined as a set of d coordinates q.

## Orthogonality

In mathematics, orthogonality is the generalization of the notion of perpendicularity to the linear algebra of bilinear forms.

## Parametric equation

In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters.

## Product rule

In calculus, the product rule is a formula used to find the derivatives of products of two or more functions.

## Riemannian manifold

In differential geometry, a (smooth) Riemannian manifold or (smooth) Riemannian space (M,g) is a real, smooth manifold M equipped with an inner product g_p on the tangent space T_pM at each point p that varies smoothly from point to point in the sense that if X and Y are differentiable vector fields on M, then p \mapsto g_p(X(p),Y(p)) is a smooth function.

## Row and column vectors

In linear algebra, a column vector or column matrix is an m &times; 1 matrix, that is, a matrix consisting of a single column of m elements, Similarly, a row vector or row matrix is a 1 &times; m matrix, that is, a matrix consisting of a single row of m elements Throughout, boldface is used for the row and column vectors.

## Scalar field

In mathematics and physics, a scalar field associates a scalar value to every point in a space – possibly physical space.

In mathematics, a skew gradient of a harmonic function over a simply connected domain with two real dimensions is a vector field that is everywhere orthogonal to the gradient of the function and that has the same magnitude as the gradient.

## Slope

In mathematics, the slope or gradient of a line is a number that describes both the direction and the steepness of the line.

New!!: Gradient and Slope · See more »

A spatial gradient is a gradient whose components are spatial derivatives, i.e., rate of change of a given scalar physical quantity with respect to the position coordinates.

## Standard basis

In mathematics, the standard basis (also called natural basis) for a Euclidean space is the set of unit vectors pointing in the direction of the axes of a Cartesian coordinate system.

## Submanifold

In mathematics, a submanifold of a manifold M is a subset S which itself has the structure of a manifold, and for which the inclusion map S → M satisfies certain properties.

## Tangent

In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point.

## Tangent space

In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector that gives the displacement of the one point from the other.

## Tensor

In mathematics, tensors are geometric objects that describe linear relations between geometric vectors, scalars, and other tensors.

## Tensor product

In mathematics, the tensor product of two vector spaces and (over the same field) is itself a vector space, together with an operation of bilinear composition, denoted by, from ordered pairs in the Cartesian product into, in a way that generalizes the outer product.

## Total derivative

In the mathematical field of differential calculus, a total derivative or full derivative of a function f of several variables, e.g., t, x, y, etc., with respect to an exogenous argument, e.g., t, is the limiting ratio of the change in the function's value to the change in the exogenous argument's value (for arbitrarily small changes), taking into account the exogenous argument's direct effect as well as indirect effects via the other arguments of the function.

## Trigonometric functions

In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are functions of an angle.

## Unit vector

In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1.

## Vector field

In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space.

## Vector-valued function

A vector-valued function, also referred to as a vector function, is a mathematical function of one or more variables whose range is a set of multidimensional vectors or infinite-dimensional vectors.

## References

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