Vector field

Vector field given by vectors of the form (−y, x)

In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a (locally) Euclidean space. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Vector calculus (also called vector analysis) is a field of Mathematics concerned with multivariable Real analysis of vectors in an Inner

Vector fields are often used in physics to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from point to point. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. In Physics, a force is whatever can cause an object with Mass to Accelerate. In Physics, a magnetic field is a Vector field that permeates space and which can exert a magnetic force on moving Electric charges Gravitation is a natural Phenomenon by which objects with Mass attract one another

In the rigorous mathematical treatment, (tangent) vector fields are defined on manifolds as sections of a manifold's tangent bundle. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be In the Mathematical field of Topology, a section (or cross section) of a Fiber bundle, &pi: E &rarr B In Mathematics, the tangent bundle of a smooth (or differentiable manifold M, denoted by T ( M) or just TM, is the They are one kind of tensor field on the manifold. In Mathematics, Physics and Engineering, a tensor field is a very general concept of variable geometric quantity

1 Definition
- 1.1 Vector fields on subsets of Euclidean space
- 1.2 Vector fields on manifolds
2 Notes
3 Examples
- 3.1 Gradient field
- 3.2 Central field
4 Line integral
5 Flow curves
- 5.1 Complete vector fields
6 Difference between scalar and vector field
- 6.1 Example 1
- 6.2 Example 2
7 See also
8 References
9 External links

Definition

Vector fields on subsets of Euclidean space

Given a subset S in Rⁿ, a vector field is represented by a vector-valued function $V_x: S \to \mathbf{R}^n$ in standard Euclidean coordinates (x₁, . A vector-valued function is a mathematical function that maps Real numbers onto vectors Vector-valued functions can be defined as \mathbf{r}(t=f(t\mathbf+g(t\mathbf . . , x_n). If there is another coordinate system y on S, then $V_y := \frac{\partial y}{\partial x} V_x$ is the expression for the same vector field in the new coordinates y. In particular, a vector field is not just a collection of scalar fields. In Mathematics and Physics, a scalar field associates a scalar value which can be either mathematical in definition or physical, to every point

We say V is a C^k vector field if V is k times continuously differentiable. In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change A point p in S is called stationary if the vector at that point is zero ( $V (p) = 0$ ).

A vector field can be visualized as a n-dimensional space with a n-dimensional vector attached to each point. Given two C^k-vector fields V, W defined on S and a real valued C^k-function f defined on S, the two operations scalar multiplication and vector addition

(f V)(p): = f (p) V (p)

(V + W)(p): = V (p) + W (p)

define the module of C^k-vector fields over the ring of C^k-functions. In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real

Vector fields on manifolds

A vector field on a sphere. "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe

Given a manifold M, a vector field on M is an assignment to every point of M a tangent vector to M at that point. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be That is, for each x in M, we have a tangent vector v(x) in T_xM. More abstractly, a vector field is a section of the tangent bundle TM. In the Mathematical field of Topology, a section (or cross section) of a Fiber bundle, &pi: E &rarr B In Mathematics, the tangent bundle of a smooth (or differentiable manifold M, denoted by T ( M) or just TM, is the If this section is continuous/differentiable/smooth/analytic, then we call the vector field continuous/differentiable/smooth/analytic. It is important to note that these properties are invariant under the change of coordinates formula, and thus can be detected by computing the local representation in any continuous/differentiable/smooth/analytic chart.

The collection of all vector fields on M is often denoted by Γ(TM) or C^∞(M,TM) (especially when thinking of vector fields as sections); the collection of all smooth vector fields is sometimes also denoted by $\mathfrak{X} (M)$ (a fraktur "X"). The German word Fraktur () refers to a specific sub-group of Blackletter Typefaces The word derives from the past participle fractus (“broken”

Notes

Vector fields should be compared to scalar fields, which associate a number or scalar to every point in space (or every point of some manifold). In Mathematics and Physics, a scalar field associates a scalar value which can be either mathematical in definition or physical, to every point Vector fields similarly associate a length or magnitude, as well as a direction to every point in space. For example, in the common (x,y,z) three-space, every point in the manifold can be associated parametrically with magnitudes of x, y and z components.

The divergence and curl are two operations on a vector field which result in a scalar field and another vector field respectively. In Vector calculus, the divergence is an Operator that measures the magnitude of a Vector field &rsquos source or sink at a given point the cURL is a Command line tool for transferring files with URL syntax. The first of these operations is defined in any number of dimensions (that is, for any value of n). The curl however, is defined only for n=3, but it can be generalized to an arbitrary dimension using the exterior product and exterior derivative. In Differential geometry, the exterior derivative extends the concept of the differential of a function which is a form of degree zero to Differential forms

Examples

The flow field around an airplane is a vector field in R³, here visualized by bubbles that follow the streamlines showing a wingtip vortex. Wingtip vortices are tubes of circulating air which are left behind by the Wing as it generates lift.

A vector field for the movement of air on Earth will associate for every point on the surface of the Earth a vector with the wind speed and direction for that point. This can be drawn using arrows to represent the wind; the length (magnitude) of the arrow will be an indication of the wind speed. The magnitude of a mathematical object is its size a property by which it can be larger or smaller than other objects of the same kind in technical terms an Ordering A "high" on the usual barometric pressure map would then act as a source (arrows pointing away), and a "low" would be a sink (arrows pointing towards), since air tends to move from high pressure areas to low pressure areas.
Velocity field of a moving fluid. In Physics, velocity is defined as the rate of change of Position. FLUID ( F ast L ight '''U'''ser '''I'''nterface D esigner is a graphical editor that is used to produce FLTK Source code In this case, a velocity vector is associated to each point in the fluid. In Physics, velocity is defined as the rate of change of Position.
Streamlines, Streaklines and Pathlines are 3 types of lines that can be made from vector fields. Fluid flow is described in general by a Vector field in three (for steady flows or four (for non-steady flows including time dimensions They are :

streaklines — as revealed in wind tunnels using smoke. A wind tunnel is a research tool developed to assist with studying the effects of air moving over or around solid objects

streamlines (or fieldlines)— as a line depicting the instantaneous field at a given time.

pathlines — showing the path that a given particle (of zero mass) would follow.

Magnetic fields. In Physics, a magnetic field is a Vector field that permeates space and which can exert a magnetic force on moving Electric charges The fieldlines can be revealed using small iron filings. Iron (ˈаɪɚn is a Chemical element with the symbol Fe (ferrum and Atomic number 26
Maxwell's equations allow us to use a given set of initial conditions to deduce, for every point in Euclidean space, a magnitude and direction for the force experienced by a charged test particle at that point; the resulting vector field is the electromagnetic field. In Classical electromagnetism, Maxwell's equations are a set of four Partial differential equations that describe the properties of the electric In Physics, a force is whatever can cause an object with Mass to Accelerate. The electromagnetic field is a physical field produced by electrically charged objects.
A gravitational field generated by any massive object is also a vector field. A gravitational field is a model used within Physics to explain how gravity exists in the universe For example, the gravitational field vectors for a spherically symmetric body would all point towards the sphere's center with the magnitude of the vectors reducing as radial distance from the body increases.

Gradient field

Vector fields can be constructed out of scalar fields using the vector operator gradient which gives rise to the following definition. In Mathematics and Physics, a scalar field associates a scalar value which can be either mathematical in definition or physical, to every point In Vector calculus, the gradient of a Scalar field is a Vector field which points in the direction of the greatest rate of increase of the scalar

A vector field V defined on a set S is called a gradient field or a conservative field if there exists a real valued function (a scalar field) f on S such that

$V = \nabla f$ .

The associated flow is called the gradient flow, and is used in the method of gradient descent. In Mathematics, a flow formalizes in mathematical terms the general idea of "a variable that depends on time" that occurs very frequently in Engineering For the analytical method called "steepest descent" see Method of steepest descent.

The path integral along any closed curve γ (γ(0) = γ(1)) in a gradient field is zero:

$\int_\gamma \langle V(x), \mathrm{d}x \rangle = \int_\gamma \langle \nabla f(x), \mathrm{d}x \rangle = f(\gamma(1)) - f(\gamma(0))$ . In Mathematics, a line integral (sometimes called a path integral or curve integral) is an Integral where the function to be integrated See also Classification of manifolds#Point-set In Mathematics, a closed manifold is type of Topological space, namely a compact

Central field

A C^∞-vector field over Rⁿ \ {0} is called a central field if

$V(T(p)) = T(V(p)) \qquad (T \in \mathrm{O}(n, \mathbf{R}))$

where O(n, R) is the orthogonal group. In Mathematics, the orthogonal group of degree n over a field F (written as O( n, F) is the group of n We say central fields are invariant under orthogonal transformations around 0. In Mathematics, an invariant is something that does not change under a set of transformations The property of being an invariant is invariance. In Matrix theory, a real orthogonal matrix is a square matrix Q whose Transpose is its inverse: Q^T

The point 0 is called the center of the field.

Since orthogonal transformations are actually rotations and reflections, the invariance conditions mean that vectors of a central field are always directed towards, or away from, 0; this is an alternate (and simpler) definition. A central field is always a gradient field, since defining it on one semiaxis and integrating gives an antigradient.

Line integral

A common technique in physics is to integrate a vector field along a curve: a line integral. This article only considers curves in Euclidean space Most of the notions presented here have analogues for curves in Riemannian and pseudo-Riemannian In Mathematics, a line integral (sometimes called a path integral or curve integral) is an Integral where the function to be integrated Given a particle in a gravitational vector field, where each vector represents the force acting on the particle at this point in space, the line integral is the work done on the particle when it travels along a certain path.

The line integral is constructed analogously to the Riemann integral and it exists if the curve is rectifiable (has finite length) and the vector field is continuous. In the branch of Mathematics known as Real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the Integral

Given a vector field V and a curve γ parametrized by [0, 1] the line integral is defined as

$\int_\gamma \langle V(x), \mathrm{d}x \rangle = \int_0^1 \langle V(\gamma(t)), \gamma'(t)\;\mathrm{d}t \rangle.$

Flow curves

Main article: Integral curve

Vector fields have a nice interpretation in terms of autonomous, first order ordinary differential equations. In Mathematics, an integral curve for a Vector field defined on a Manifold is a curve in the manifold whose tangent vector (i In Mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one Independent variable, and one or more of its

Given a vector field V defined on S, we can try to define curves γ on S such that for each t in an interval I

γ'(t) = V (γ(t))

If V is Lipschitz continuous we can find a unique C¹-curve γ_x for each point x in S so that

γ x (0) = x

$\gamma'_x(t) = V(\gamma_x(t)) \qquad ( t \in (-\epsilon, +\epsilon) \subset \mathbf{R})$

The curves γ_x are called flow curves of the vector field V and partition S into equivalence classes. In Mathematics, more specifically in Real analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a smoothness condition for functions In Mathematics, given a set X and an Equivalence relation ~ on X, the equivalence class of an element a in X It is not always possible to extend the interval (-ε, +ε) to the whole real number line. In Mathematics, the real line is simply the set R of singleton Real numbers However this term is usually used when R is to be treated as a The flow may for example reach the edge of S in a finite time. In two or three dimensions one can visualize the vector field as giving rise to a flow on S. If we drop a particle into this flow at a point p it will move along the curve γ_p in the flow depending on the initial point p. If p is a stationary point of V then the particle will remain at p.

Typical applications are streamline in fluid, geodesic flow, and one-parameter subgroups and the exponential map in Lie groups. Fluid dynamics is the sub-discipline of Fluid mechanics dealing with fluid flow: Fluids ( Liquids and Gases in motion In Mathematics, a geodesic /ˌdʒiəˈdɛsɪk -ˈdisɪk/ -dee-sik is a generalization of the notion of a " straight line " to " curved spaces In Mathematics, a one-parameter group or one-parameter subgroup usually means a continuous Group homomorphism φ: R In differential geometry the exponential map is a generalization of the ordinary Exponential function of mathematical analysis to all differentiable manifolds with an Affine In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group

Complete vector fields

A vector field is complete if its flow curve exists for all time. ^[1] In particular, compactly supported vector fields on a manifold are complete. In Mathematics, the support of a function is the set of points where the function is not zero or the closure of that set If X is a complete vector field on M, then the one-parameter group of diffeomorphisms generated by the flow along X exists for all time. In Mathematics, a one-parameter group or one-parameter subgroup usually means a continuous Group homomorphism φ: R In Mathematics, a diffeomorphism is an Isomorphism of Smooth manifolds It is an Invertible function that maps one Differentiable

Difference between scalar and vector field

The difference between a scalar and vector field is not that a scalar is just one number while a vector is several numbers. The difference is in how their coordinates respond to coordinate transformations. A scalar is a coordinate whereas a vector can be described by coordinates, but it is not the collection of its coordinates.

Example 1

This example is about 2-dimensional Euclidean space (R²) where we examine Euclidean (x, y) and polar (r, θ) coordinates (which are undefined at the origin). In Mathematics, the polar coordinate system is a two-dimensional Coordinate system in which each point on a plane is determined by Thus x = r cos θ and y = r sin θ and also r² = x² + y², cos θ = x/(x² + y²)^1/2 and sin θ = y/(x² + y²)^1/2. Suppose we have a scalar field which is given by the constant function 1, and a vector field which attaches a vector in the r-direction with length 1 to each point. More precisely, they are given by the functions

$s_{\mathrm{polar}}:(r, \theta) \mapsto 1, \quad v_{\mathrm{polar}}:(r, \theta) \mapsto (1, 0).$

Let us convert these fields to Euclidean coordinates. The vector of length 1 in the r-direction has the x coordinate cos θ and the y coordinate sin θ. Thus in Euclidean coordinates the same fields are described by the functions

$s_{\mathrm{Euclidean}}:(x, y) \mapsto 1,$

$v_{\mathrm{Euclidean}}:(x, y) \mapsto (\cos \theta, \sin \theta) = \left(\frac{x}\sqrt{{x^2 + y^2}}, \frac{y}{\sqrt{x^2 + y^2}}\right).$

We see that while the scalar field remains the same, the vector field now looks different. The same holds even in the 1-dimensional case, as illustrated by the next example.

Example 2

Consider the 1-dimensional Euclidean space R with its standard Euclidean coordinate x. Suppose we have a scalar field and a vector field which are both given in the x coordinate by the constant function 1,

$s_{\mathrm{Euclidean}}:x \mapsto 1, \quad v_{\mathrm{Euclidean}}:x \mapsto 1.$

Thus, we have a scalar field which has the value 1 everywhere and a vector field which attaches a vector in the x-direction with magnitude 1 unit of x to each point.

Now consider the coordinate ξ := 2x. If x changes one unit then ξ changes 2 units. Thus this vector field has a magnitude of 2 in units of ξ. Therefore, in the ξ coordinate the scalar field and the vector field are described by the functions

$s_{\mathrm{unusual}}:\xi \mapsto 1, \quad v_{\mathrm{unusual}}:\xi \mapsto 2.$

which are different.

References

^ Sharpe, R. A field line is a Locus that is defined by a Vector field and a starting location within the field In Mathematics, the Lie derivative, named after Sophus Lie by Władysław Ślebodziński, evaluates the change of one Vector field along the In Mathematics, a time dependent vector field is a construction in Vector calculus which generalizes the concept of vector fields. Cylindrical coordinate system Vector fields Vectors are defined in Cylindrical coordinates by (ρφz where ρ is the length of the vector projected In Mathematics and Physics, a scalar field associates a scalar value which can be either mathematical in definition or physical, to every point In Mathematics, Physics and Engineering, a tensor field is a very general concept of variable geometric quantity (1997). Differential geometry. Springer-Verlag. ISBN 0-387-94732-9.

External links

Vector field -- Mathworld
Vector field -- PlanetMath
3D Magnetic field viewer
Vector Field Simulation Java applet illustrating vectors fields
Vector fields and field lines
Vector field simulation An interactive application to show the effects of vector fields

MathWorld is an online Mathematics reference work created and largely written by Eric W PlanetMath is a free, collaborative online Mathematics Encyclopedia.

Dictionary

-noun

(mathematics) A construction in which each point in a Euclidean space is associated with a vector; a function whose range is a vector space.
(differential geometry) A function which associates, to each point on a surface, a vector in the tangent plane of that point.

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