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# Vector field

In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. [1]

92 relations: Angular momentum, Atmospheric pressure, Calculus, Closed manifold, Conservation of energy, Conservative vector field, Covariance and contravariance of vectors, Curl (mathematics), Del, Derivation (differential algebra), Derivative, Diffeomorphism, Differentiable function, Differentiable manifold, Differential calculus over commutative algebras, Differential form, Differential geometry of curves, Divergence, Divergence theorem, Dual space, Eisenbud–Levine–Khimshiashvili signature formula, Electromagnetic field, Equivalence class, Euclidean space, Euclidean vector, Euler characteristic, Exponential map (Lie theory), Exterior derivative, Field (physics), Field line, Field strength, Flow (mathematics), Fluid, Fluid dynamics, Force, Fraktur, Fundamental theorem of calculus, Geodesic, Gradient, Gradient descent, Gravitational field, Gravity, Hairy ball theorem, Invariant (mathematics), Iron, Lie algebra, Lie derivative, Lie group, Light field, Line integral, ... Expand index (42 more) »

## Angular momentum

In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational equivalent of linear momentum.

## Atmospheric pressure

Atmospheric pressure, sometimes also called barometric pressure, is the pressure within the atmosphere of Earth (or that of another planet).

## Calculus

Calculus (from Latin calculus, literally 'small pebble', used for counting and calculations, as on an abacus), is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.

## Closed manifold

In mathematics, a closed manifold is a type of topological space, namely a compact manifold without boundary.

## Conservation of energy

In physics, the law of conservation of energy states that the total energy of an isolated system remains constant, it is said to be ''conserved'' over time.

## 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.

## Covariance and contravariance of vectors

In multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis.

## 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.

## 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;.

## Derivation (differential algebra)

In mathematics, a derivation is a function on an algebra which generalizes certain features of the derivative operator.

## 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).

## Diffeomorphism

In mathematics, a diffeomorphism is an isomorphism of smooth manifolds.

## 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.

## Differentiable manifold

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

## Differential calculus over commutative algebras

In mathematics the differential calculus over commutative algebras is a part of commutative algebra based on the observation that most concepts known from classical differential calculus can be formulated in purely algebraic terms.

## 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 geometry of curves

Differential geometry of curves is the branch of geometry that deals with smooth curves in the plane and in the Euclidean space by methods of differential and integral calculus.

## 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.

## Divergence theorem

In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a result that relates the flow (that is, flux) of a vector field through a surface to the behavior of the vector field inside the surface.

## Dual space

In mathematics, any vector space V has a corresponding dual vector space (or just dual space for short) consisting of all linear functionals on V, together with the vector space structure of pointwise addition and scalar multiplication by constants.

## Eisenbud–Levine–Khimshiashvili signature formula

In mathematics, and especially differential topology and singularity theory, the Eisenbud–Levine–Khimshiashvili signature formula gives a way of computing the Poincaré-Hopf index of a real, analytic vector field at an algebraically isolated singularity.

## Electromagnetic field

An electromagnetic field (also EMF or EM field) is a physical field produced by electrically charged objects.

## Equivalence class

In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation) defined on them, then one may naturally split the set S into equivalence classes.

## 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.

## Euler characteristic

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler&ndash;Poincaré characteristic) is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent.

## Exponential map (Lie theory)

In the theory of Lie groups, the exponential map is a map from the Lie algebra \mathfrak g of a Lie group G to the group, which allows one to recapture the local group structure from the Lie algebra.

## Exterior derivative

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

## Field (physics)

In physics, a field is a physical quantity, represented by a number or tensor, that has a value for each point in space and time.

## Field line

A field line is a locus that is defined by a vector field and a starting location within the field.

## Field strength

In physics, field strength means the magnitude of a vector-valued field (e.g., in volts per meter, V/m, for an electric field E).

## Flow (mathematics)

In mathematics, a flow formalizes the idea of the motion of particles in a fluid.

## Fluid

In physics, a fluid is a substance that continually deforms (flows) under an applied shear stress.

## Fluid dynamics

In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids - liquids and gases.

## Force

In physics, a force is any interaction that, when unopposed, will change the motion of an object.

## Fraktur

Fraktur is a calligraphic hand of the Latin alphabet and any of several blackletter typefaces derived from this hand.

## Fundamental theorem of calculus

The fundamental theorem of calculus is a theorem that links the concept of differentiating a function with the concept of integrating a function.

## Geodesic

In differential geometry, a geodesic is a generalization of the notion of a "straight line" to "curved spaces".

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

Gradient descent is a first-order iterative optimization algorithm for finding the minimum of a function.

## Gravitational field

In physics, a gravitational field is a model used to explain the influence that a massive body extends into the space around itself, producing a force on another massive body.

## Gravity

Gravity, or gravitation, is a natural phenomenon by which all things with mass or energy—including planets, stars, galaxies, and even light—are brought toward (or gravitate toward) one another.

## Hairy ball theorem

The hairy ball theorem of algebraic topology (sometimes called the hedgehog theorem in Europe) states that there is no nonvanishing continuous tangent vector field on even-dimensional ''n''-spheres.

## Invariant (mathematics)

In mathematics, an invariant is a property, held by a class of mathematical objects, which remains unchanged when transformations of a certain type are applied to the objects.

## Iron

Iron is a chemical element with symbol Fe (from ferrum) and atomic number 26.

## Lie algebra

In mathematics, a Lie algebra (pronounced "Lee") is a vector space \mathfrak g together with a non-associative, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g; (x, y) \mapsto, called the Lie bracket, satisfying the Jacobi identity.

## Lie derivative

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

## Lie group

In mathematics, a Lie group (pronounced "Lee") is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure.

## Light field

The light field is a vector function that describes the amount of light flowing in every direction through every point in space.

## Line integral

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

## Line of force

A line of force in Faraday's extended sense is synonymous with Maxwell's line of induction.

## Linear form

In linear algebra, a linear functional or linear form (also called a one-form or covector) is a linear map from a vector space to its field of scalars.

## Lipschitz continuity

In mathematical analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a strong form of uniform continuity for functions.

## Magnetic field

A magnetic field is a vector field that describes the magnetic influence of electrical currents and magnetized materials.

## 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.

## Map (mathematics)

In mathematics, the term mapping, sometimes shortened to map, refers to either a function, often with some sort of special structure, or a morphism in category theory, which generalizes the idea of a function.

## MathWorld

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

## Maxwell's equations

Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.

Michael Faraday FRS (22 September 1791 – 25 August 1867) was an English scientist who contributed to the study of electromagnetism and electrochemistry.

## Module (mathematics)

In mathematics, a module is one of the fundamental algebraic structures used in abstract algebra.

## Multivector

A multivector is the result of a product defined for elements in a vector space V. A vector space with a linear product operation between elements of the space is called an algebra; examples are matrix algebra and vector algebra.

## One-parameter group

In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphism from the real line \mathbb (as an additive group) to some other topological group G. That means that it is not in fact a group, strictly speaking; if \varphi is injective then \varphi(\mathbb), the image, will be a subgroup of G that is isomorphic to \mathbb as additive group.

## Open set

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

## Orthogonal group

In mathematics, the orthogonal group in dimension, denoted, is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations.

## Orthogonal matrix

In linear algebra, an orthogonal matrix is a square matrix whose columns and rows are orthogonal unit vectors (i.e., orthonormal vectors), i.e. where I is the identity matrix.

## Parametric equation

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

## Picard–Lindelöf theorem

In mathematics, in the study of differential equations, the Picard–Lindelöf theorem, Picard's existence theorem or Cauchy–Lipschitz theorem is an important theorem on existence and uniqueness of solutions to first-order equations with given initial conditions.

## PlanetMath

PlanetMath is a free, collaborative, online mathematics encyclopedia.

## Poincaré–Hopf theorem

In mathematics, the Poincaré–Hopf theorem (also known as the Poincaré–Hopf index formula, Poincaré–Hopf index theorem, or Hopf index theorem) is an important theorem that is used in differential topology.

## Real line

In mathematics, the real line, or real number line is the line whose points are the real numbers.

## Real number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.

## Riemann integral

In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval.

## 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.

## Ring (mathematics)

In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra.

## Scalar field

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

## Section (fiber bundle)

In the mathematical field of topology, a section (or cross section) of a fiber bundle E is a continuous right inverse of the projection function \pi.

## Smoothness

In mathematical analysis, the smoothness of a function is a property measured by the number of derivatives it has that are continuous.

## Space (mathematics)

In mathematics, a space is a set (sometimes called a universe) with some added structure.

## Stokes' theorem

In vector calculus, and more generally differential geometry, Stokes' theorem (also called the generalized Stokes theorem or the Stokes–Cartan theorem) is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus.

## Streamlines, streaklines, and pathlines

Streamlines, streaklines and pathlines are field lines in a fluid flow.

## Support (mathematics)

In mathematics, the support of a real-valued function f is the subset of the domain containing those elements which are not mapped to zero.

## Surface (topology)

In topology and differential geometry, a surface is a two-dimensional manifold, and, as such, may be an "abstract surface" not embedded in any Euclidean space.

## Tangent bundle

In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M. As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector.

## 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 field

In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold).

## Time dependent vector field

In mathematics, a time dependent vector field is a construction in vector calculus which generalizes the concept of vector fields.

## Vector calculus

Vector calculus, or vector analysis, is a branch of mathematics concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3.

## Vector fields in cylindrical and spherical coordinates

NOTE: This page uses common physics notation for spherical coordinates, in which \theta is the angle between the z axis and the radius vector connecting the origin to the point in question, while \phi is the angle between the projection of the radius vector onto the x-y plane and the x axis.

## 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.

## Velocity

The velocity of an object is the rate of change of its position with respect to a frame of reference, and is a function of time.

## Wind tunnel

A wind tunnel is a tool used in aerodynamic research to study the effects of air moving past solid objects.

## Work (physics)

In physics, a force is said to do work if, when acting, there is a displacement of the point of application in the direction of the force.

## References

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