Directional derivative, the Glossary

Table of Contents

1. 50 relations: Abelian group, Cartan subalgebra, Chain rule, Coordinate system, Covariant derivative, Curvilinear coordinates, Derivative, Differentiable function, Differentiable manifold, Differentiation rules, Dot product, Euclidean space, Exterior derivative, Function (mathematics), Gateaux derivative, General relativity, Gradient, Group representation, Hilbert space, Hypersurface, Lie algebra, Lie derivative, Lie group, Limit (mathematics), MathWorld, Metric space, Multivariable calculus, Neighbourhood (mathematics), Neumann boundary condition, Normal (geometry), Orthogonality, Partial derivative, PlanetMath, Poincaré group, Power series, Product rule, Riemann curvature tensor, Rotation operator (quantum mechanics), Scalar field, Self-adjoint operator, Sign convention, Structure constants, Subgroup, Tangent vector, Tensor, Total derivative, Unit vector, Unitary operator, Vector (mathematics and physics), 3D rotation group.

2. Generalizations of the derivative
3. Scalars

Abelian group

In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written.

See Directional derivative and Abelian group

Cartan subalgebra

In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra \mathfrak of a Lie algebra \mathfrak that is self-normalising (if \in \mathfrak for all X \in \mathfrak, then Y \in \mathfrak).

See Directional derivative and Cartan subalgebra

Chain rule

In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and.

See Directional derivative and Chain rule

Coordinate system

In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space.

See Directional derivative and Coordinate system

Covariant derivative

In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Directional derivative and covariant derivative are differential geometry.

See Directional derivative and Covariant derivative

Curvilinear coordinates

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

See Directional derivative and Curvilinear coordinates

Derivative

The derivative is a fundamental tool of calculus that quantifies the sensitivity of change of a function's output with respect to its input. Directional derivative and derivative are differential calculus and rates.

See Directional derivative and Derivative

Differentiable function

In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. Directional derivative and differentiable function are differential calculus and Multivariable calculus.

See Directional derivative and Differentiable function

Differentiable manifold

In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus.

See Directional derivative and Differentiable manifold

Differentiation rules

This is a summary of differentiation rules, that is, rules for computing the derivative of a function in calculus. Directional derivative and differentiation rules are differential calculus.

See Directional derivative and Differentiation rules

Dot product

In mathematics, the dot product or scalar productThe term scalar product means literally "product with a scalar as a result". Directional derivative and dot product are scalars.

See Directional derivative and Dot product

Euclidean space

Euclidean space is the fundamental space of geometry, intended to represent physical space.

See Directional derivative and Euclidean space

Exterior derivative

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

See Directional derivative and Exterior derivative

Function (mathematics)

In mathematics, a function from a set to a set assigns to each element of exactly one element of.

See Directional derivative and Function (mathematics)

Gateaux derivative

In mathematics, the Gateaux differential or Gateaux derivative is a generalization of the concept of directional derivative in differential calculus. Directional derivative and Gateaux derivative are generalizations of the derivative.

See Directional derivative and Gateaux derivative

General relativity

General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. Directional derivative and general relativity are differential geometry.

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Gradient

In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p gives the direction and the rate of fastest increase. Directional derivative and gradient are differential calculus, generalizations of the derivative and rates.

See Directional derivative and Gradient

Group representation

In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used to represent group elements as invertible matrices so that the group operation can be represented by matrix multiplication.

See Directional derivative and Group representation

Hilbert space

In mathematics, Hilbert spaces (named after David Hilbert) allow the methods of linear algebra and calculus to be generalized from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional.

See Directional derivative and Hilbert space

Hypersurface

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

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Lie algebra

In mathematics, a Lie algebra (pronounced) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identity.

See Directional derivative and Lie algebra

Lie derivative

In differential geometry, the Lie derivative, named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector field. Directional derivative and Lie derivative are differential geometry and generalizations of the derivative.

See Directional derivative and Lie derivative

Lie group

In mathematics, a Lie group (pronounced) is a group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable.

See Directional derivative and Lie group

Limit (mathematics)

In mathematics, a limit is the value that a function (or sequence) approaches as the input (or index) approaches some value. Directional derivative and limit (mathematics) are differential calculus.

See Directional derivative and Limit (mathematics)

MathWorld

MathWorld is an online mathematics reference work, created and largely written by Eric W. Weisstein.

See Directional derivative and MathWorld

Metric space

In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points.

See Directional derivative and Metric space

Multivariable calculus

Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving multiple variables (multivariate), rather than just one.

See Directional derivative and Multivariable calculus

Neighbourhood (mathematics)

In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space.

See Directional derivative and Neighbourhood (mathematics)

Neumann boundary condition

In mathematics, the Neumann (or second-type) boundary condition is a type of boundary condition, named after Carl Neumann.

See Directional derivative and Neumann boundary condition

Normal (geometry)

In geometry, a normal is an object (e.g. a line, ray, or vector) that is perpendicular to a given object.

See Directional derivative and Normal (geometry)

Orthogonality

In mathematics, orthogonality is the generalization of the geometric notion of perpendicularity.

See Directional derivative and Orthogonality

Partial derivative

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 (as opposed to the total derivative, in which all variables are allowed to vary). Directional derivative and partial derivative are Multivariable calculus.

See Directional derivative and Partial derivative

PlanetMath

PlanetMath is a free, collaborative, mathematics online encyclopedia.

See Directional derivative and PlanetMath

Poincaré group

The Poincaré group, named after Henri Poincaré (1906), was first defined by Hermann Minkowski (1908) as the isometry group of Minkowski spacetime.

See Directional derivative and Poincaré group

Power series

In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n. Directional derivative and power series are Multivariable calculus.

See Directional derivative and Power series

Product rule

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

See Directional derivative and Product rule

Riemann curvature tensor

In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. Directional derivative and Riemann curvature tensor are differential geometry.

See Directional derivative and Riemann curvature tensor

Rotation operator (quantum mechanics)

This article concerns the rotation operator, as it appears in quantum mechanics.

See Directional derivative and Rotation operator (quantum mechanics)

Scalar field

In mathematics and physics, a scalar field is a function associating a single number to every point in a space – possibly physical space. Directional derivative and scalar field are Multivariable calculus.

See Directional derivative and Scalar field

Self-adjoint operator

In mathematics, a self-adjoint operator on a complex vector space V with inner product \langle\cdot,\cdot\rangle is a linear map A (from V to itself) that is its own adjoint.

See Directional derivative and Self-adjoint operator

Sign convention

In physics, a sign convention is a choice of the physical significance of signs (plus or minus) for a set of quantities, in a case where the choice of sign is arbitrary.

See Directional derivative and Sign convention

Structure constants

In mathematics, the structure constants or structure coefficients of an algebra over a field are the coefficients of the basis expansion (into linear combination of basis vectors) of the products of basis vectors.

See Directional derivative and Structure constants

Subgroup

In group theory, a branch of mathematics, given a group under a binary operation ∗, a subset of is called a subgroup of if also forms a group under the operation ∗.

See Directional derivative and Subgroup

Tangent vector

In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point.

See Directional derivative and Tangent vector

Tensor

In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space.

See Directional derivative and Tensor

Total derivative

In mathematics, the total derivative of a function at a point is the best linear approximation near this point of the function with respect to its arguments. Directional derivative and total derivative are differential calculus and Multivariable calculus.

See Directional derivative and Total derivative

Unit vector

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

See Directional derivative and Unit vector

Unitary operator

In functional analysis, a unitary operator is a surjective bounded operator on a Hilbert space that preserves the inner product.

See Directional derivative and Unitary operator

Vector (mathematics and physics)

In mathematics and physics, vector is a term that refers informally to some quantities that cannot be expressed by a single number (a scalar), or to elements of some vector spaces.

See Directional derivative and Vector (mathematics and physics)

3D rotation group

In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space \R^3 under the operation of composition.

See Directional derivative and 3D rotation group

See also

Generalizations of the derivative

• Arithmetic derivative
• Clarke generalized derivative
• Differentiable vector–valued functions from Euclidean space
• Differential of a function
• Differentiation in Fréchet spaces
• Differintegral
• Dini derivative
• Directional derivative
• Distribution (mathematics)
• Exterior calculus identities
• Exterior derivative
• Fréchet derivative
• Gateaux derivative
• Generalizations of the derivative
• Gradient
• H-derivative
• Hadamard derivative
• Image derivative
• Jacobian matrix and determinant
• Lie algebroid
• Lie derivative
• Malliavin derivative
• Material derivative
• P-derivation
• Pushforward (differential)
• Q-derivative
• Quasi-derivative
• Radon–Nikodym theorem
• Skew gradient
• Spaces of test functions and distributions
• Spatial gradient
• Subderivative
• Weak derivative

Scalars

• Determinants
• Directional derivative
• Dot product
• Lorentz scalar
• Pseudoscalar
• Relative scalar
• Scalar (mathematics)
• Scalar field theory

References

[1] https://en.wikipedia.org/wiki/Directional_derivative

Also known as Normal derivative.