 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. Image File history File links Vector_field. ...
Image File history File links Vector_field. ...
For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ...
Vector calculus (also called vector analysis) is a field of mathematics concerned with multivariate real analysis of vectors in two or more dimensions. ...
This article is about vectors that have a particular relation to the spatial coordinates. ...
Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...
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. A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ...
For other uses, see Force (disambiguation). ...
Magnetic field lines shown by iron filings Magnetostatics Electrodynamics Electrical Network Tensors in Relativity This box: In physics, the magnetic field is a field that permeates space and which exerts a magnetic force on moving electric charges and magnetic dipoles. ...
Gravity is a force of attraction that acts between bodies that have mass. ...
In the rigorous mathematical treatment, (tangent) vector fields are defined on manifolds as sections of a manifold's tangent bundle. They are one kind of tensor field on the manifold. On a sphere, the sum of the angles of a triangle is not equal to 180° (see spherical trigonometry). ...
In mathematics, in particular in topology, a fiber bundle is a space which locally looks like a product of two spaces but may possess a different global structure. ...
In mathematics, the tangent bundle of a differentiable manifold M, denoted by T(M) or just TM, is the disjoint union of the tangent spaces to each point of M An element of T(M) is a pair (x,v) where x ∈ M and v ∈ Tx(M), the tangent space...
In mathematics, physics and engineering, a tensor field is a very general concept of variable geometric quantity. ...
Definition
Vector fields on subsets of Euclidean space Given a subset S in Rn, a vector field is represented by a vector-valued function  in standard Euclidean coordinates (x1, ..., xn). If there is another coordinate system y on S, then  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. A graph of the vector-valued function <2Cos(t),4Sin(t),t> A vector-valued function is a mathematical function that maps real numbers onto vectors. ...
In mathematics and physics, a scalar field associates a scalar to every point in space. ...
We say V is a Ck vector field if V is k times continuously differentiable. A point p in S is called stationary if the vector at that point is zero (V(p) = 0). In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ...
In mathematics, the derivative of a function is one of the two central concepts of calculus. ...
A vector field can be visualized as a n-dimensional space with a n-dimensional vector attached to each point. Given two Ck-vector fields V, W defined on S and a real valued Ck-function f defined on S, the two operations scalar multiplication and vector addition
  define the module of Ck-vector fields over the ring of Ck-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 to lie in a field, the scalars may lie in an arbitrary ring. ...
In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ...
Vector fields on manifolds 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. That is, for each x in M, we have a tangent vector v(x) in TxM. More abstractly, a vector field is a section of the tangent bundle TM. 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. Image File history File links This is a lossless scalable vector image. ...
Image File history File links This is a lossless scalable vector image. ...
For other uses, see Sphere (disambiguation). ...
On a sphere, the sum of the angles of a triangle is not equal to 180° (see spherical trigonometry). ...
In differential geometry, one can attach to every point p of a differentiable manifold a tangent space, a real vector space which intuitively contains the possible directions in which one can pass through p. ...
In mathematics, in particular in topology, a fiber bundle is a space which locally looks like a product of two spaces but may possess a different global structure. ...
In mathematics, the tangent bundle of a differentiable manifold M, denoted by T(M) or just TM, is the disjoint union of the tangent spaces to each point of M An element of T(M) is a pair (x,v) where x ∈ M and v ∈ Tx(M), the tangent space...
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  (a fraktur "X"). The German word Fraktur (pronounced in the International Phonetic Alphabet (IPA)) refers to a specific sub-group of blackletter typefaces. ...
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). 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. In mathematics and physics, a scalar field associates a scalar to every point in space. ...
The divergence and curl are two operations on a vector field which result in a scalar field and another vector field respectively. 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 vector calculus, the divergence is an operator that measures a vector fields tendency to originate from or converge upon a given point. ...
For other uses, see Curl (disambiguation). ...
In mathematics, the exterior algebra (also known as the Grassmann algebra) of a given vector space V is a certain unital associative algebra which contains V as a subspace. ...
In mathematics, the exterior derivative operator of differential topology, extends the concept of the differential of a function to differential forms of higher degree. ...
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. - 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. 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 this case, a velocity vector is associated to each point in the fluid.
- Streamlines, Streaklines and Pathlines are 3 types of lines that can be made from vector fields. They are :
-
- streaklines — as revealed in wind tunnels using smoke.
- 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. The fieldlines can be revealed using small iron filings.
- 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.
- A gravitational field generated by any massive object is also a vector field. 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.
Image File history File links Metadata Size of this preview: 800 × 463 pixelsFull resolution‎ (2,271 × 1,313 pixels, file size: 207 KB, MIME type: image/jpeg) File historyClick on a date/time to view the file as it appeared at that time. ...
Image File history File links Metadata Size of this preview: 800 × 463 pixelsFull resolution‎ (2,271 × 1,313 pixels, file size: 207 KB, MIME type: image/jpeg) File historyClick on a date/time to view the file as it appeared at that time. ...
Look up streamline in Wiktionary, the free dictionary. ...
Wingtip vortices stream from an F-15 as it disengages from a KC-10 Extender following midair refueling. ...
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 of the class of objects to which it belongs. ...
Atmospheric pressure is the pressure caused by the weight of air above any area in the Earths atmosphere. ...
This article is about velocity in physics. ...
A fluid is defined as a substance that continually deforms (flows) under an applied shear stress regardless of the magnitude of the applied stress. ...
This article is about velocity in physics. ...
Solid blue lines and broken grey lines represent the streamlines. ...
NASA wind tunnel with the model of a plane A wind tunnel is a research tool developed to assist with studying the effects of air moving over or around solid objects. ...
Magnetic field lines shown by iron filings Magnetostatics Electrodynamics Electrical Network Tensors in Relativity This box: In physics, the magnetic field is a field that permeates space and which exerts a magnetic force on moving electric charges and magnetic dipoles. ...
For other uses, see Iron (disambiguation). ...
For thermodynamic relations, see Maxwell relations. ...
Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...
For other uses, see Force (disambiguation). ...
The electromagnetic field is a physical field that is produced by electrically charged objects and which affects the behaviour of charged objects in the vicinity of the field. ...
A gravitational field is a model used within physics to explain how gravity exists in the universe. ...
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 to every point in space. ...
For other uses, see Gradient (disambiguation). ...
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
 . The path integral along any closed curve γ (γ(0) = γ(1)) in a gradient field is zero:  .
Central field A C∞-vector field over Rn {0} is called a central field if  where O(n, R) is the orthogonal group. We say central fields are invariant under orthogonal transformations around 0. In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ...
In mathematics, an invariant is something that does not change under a set of transformations. ...
In matrix theory, a real orthogonal matrix is a square matrix Q whose transpose is its inverse: // Overview An orthogonal matrix is the real specialization of a unitary matrix, and thus always a normal matrix. ...
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.
Curve integral A common technique in physics is to integrate a vector field along a curve: a line integral. Given a particle in a gravitational vector field, where each vector represents the force acting on the particle at this point in space, the curve integral is the work done on the particle when it travels along a certain path. In mathematics, the differential geometry of curves provides definitions and methods to analyze smooth curves in Riemannian manifolds and Pseudo-Riemannian manifolds (and in particular in Euclidean space) using differential and integral calculus. ...
This article is about path integrals in the general mathematical sense, and not the path integral formulation of physics which was studied by Richard Feynman. ...
The curve 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 â„›, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. ...
Given a vector field V and a curve γ parametrized by [0, 1] the curve integral is defined as
Flow curves Vector fields have a nice interpretation in terms of autonomous, first order ordinary differential equations. 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 derivatives with respect to that variable. ...
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 γx for each point x in S so that In mathematics, more specifically in real analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a smoothness condition for functions which is stronger than regular continuity. ...
- γx(0) = x
 The curves γx are called flow curves of the vector field V and partition S into equivalence classes. It is not always possible to extend the interval (-ε, +ε) to the whole real number line. The flow may for example reach the edge of S in a finite time. In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x ∈ X | x ~ a } The notion of equivalence classes is useful for constructing sets out...
In mathematics, the real line is simply the set of real numbers. ...
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 γp in the flow depending on the initial point p. If p is a stationary point of V then the particle will remain at p. Look up flow in Wiktionary, the free dictionary. ...
Typical applications are streamline in fluid, geodesic flow, and one-parameter subgroups and the exponential map in Lie groups. Look up streamline in Wiktionary, the free dictionary. ...
This article or section should be merged with Fluid mechanics Fluid dynamics is the study of fluids (liquids and gases) in motion, and the effect of the fluid motion on fluid boundaries, such as solid containers or other fluids. ...
This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesnt cover the terminology of differential topology. ...
In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphism φ : R → G from the real line R (as an additive group) to some other topological group G. That means that it is not in fact a group, strictly speaking; if φ is...
There are two different (but closely related) notions of an exponential map in differential geometry, both of which generalize the ordinary exponential function of mathematical analysis. ...
In mathematics, a Lie group, named after Norwegian mathematician Sophus Lie (IPA pronunciation: , sounds like Lee), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. ...
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). 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 This article describes some of the common coordinate systems that appear in elementary mathematics. ...
 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  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,  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
 which are different.
See also In mathematics and physics, a scalar field associates a scalar to every point in space. ...
Equipotential surfaces are surfaces of constant scalar potential. ...
Vector calculus (also called vector analysis) is a field of mathematics concerned with multivariate real analysis of vectors in two or more dimensions. ...
In mathematics, a Lie derivative, named after Sophus Lie, is a derivation on the algebra of tensor fields over a manifold M. The vector space of all Lie derivatives on M forms an infinite dimensional Lie algebra with respect to the Lie bracket defined by The Lie derivatives are represented...
In mathematics, the differential geometry of curves provides definitions and methods to analyze smooth curves in Riemannian manifolds and Pseudo-Riemannian manifolds (and in particular in Euclidean space) using differential and integral calculus. ...
In mathematics, a time dependent vector field is a construction in vector calculus which generalizes the concept of vector fields. ...
// Vector fields in cylindrical coordinates Vectors are defined in cylindrical coordinates by (Ï,φ,z), where Ï is the length of the vector projected onto the X-Y-plane, φ is the angle of the projected vector with the positive X-axis (0 ≤ φ < 2Ï€), z is the regular z-coordinate. ...
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