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Topic: Vector field

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  Vector field   (Site not responding. Last check: 2007-10-22)
Vector fields are often used in physics, for instance to indicate 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.
In the rigorous mathematical treatment, vector fields are defined on manifolds: a vector field is a section of the manifold's tangent bundle.
The gradient of a scalar field is a vector field.
bopedia.com /en/wikipedia/v/ve/vector_field.html   (335 words)

 Vector field - Wikipedia, the free encyclopedia
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.
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).
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.
en.wikipedia.org /wiki/Vector_field   (1384 words)

 Vector calculus - LearnThis.Info Enclyclopedia   (Site not responding. Last check: 2007-10-22)
Vector calculus is a field of mathematics concerned with multivariate real analysis of vectorss in 2 or more dimensions.
We consider vector fields, which associate a vector to every point in space, and scalar fields, which associate a scalar to every point in space.
divergence: measures a vector field's tendency to originate from or converge upon certain points; the divergence of a vector field is a scalar field.
encyclopedia.learnthis.info /v/ve/vector_calculus_1.html   (342 words)

 PlanetMath: vector field   (Site not responding. Last check: 2007-10-22)
Vector fields on a smooth manifold support an operation called the Lie bracket, making them into a Lie algebra; this construction produces an intimate link between Lie algebras and Lie groups, which are of great interest to physicists and mathematicians alike.
This viewpoint on vector fields emphasizes the machinery of modern geometry, namely sheaves, local rings, and bundles; this machinery is useful in differential geometry, important in complex analtyic geometry, and foundational in algebraic geometry -- schemes cannot be described without it.
This is version 11 of vector field, born on 2001-11-16, modified 2005-08-26.
www.planetmath.org /encyclopedia/VectorField.html   (770 words)

 Vector field   (Site not responding. Last check: 2007-10-22)
In the rigorous mathematical treatment, vector fields are defined on manifolds as a section of the manifold's tangent bundle.
The derivatives of a vector field, resulting in a scalar field or another vector field, are called the divergence and curl respectively.
Given a vector field F(x) and a curve γ(t) from a to b the curve integral is defined as
www.sciencedaily.com /encyclopedia/vector_field   (863 words)

 Path-independent or conservative vector fields*
the path integral of a vector field can be viewed as the total work performed by the force field on an object moving along the path.
The vector field appears to be path-independent, as promised.
By comparing the paths to the vector field, you should be able to explain the behavior of the path integral to intermediate points.
www.math.umn.edu /~nykamp/m2374/readings/conservfield   (941 words)

 Vector fields
If this vector field F(x, y) = < x, 3y > is the gradient for some function z = f(x, y) then F(x, y) is said to be conservative and the function f(x, y) is its potential function.
For vector fields of three variables the situation is a bit more complex as you would imagine.
Line integrals may be evaluated over a vector field along a path determined by a vector-valued function.
math.stcc.edu /CalculusIII/171.html   (846 words)

 What is a magnetic field?
A magnet produces a vector field, the magnetic field, at all points in the space around it.
A magnetic field can also be created by the spin magnetic dipole moment, and by the orbital magnetic dipole moment of an electron within an atom.
By convention, we state that the magnetic field has a direction associated with it, such that the field exits the North end of a magnet, flows through the air or other materials nearby, and re-enters the South end of the magnet.
my.execpc.com /~rhoadley/magfield.htm   (718 words)

 Vectors, A Vector Field Plotting Utility
A common problem with using discrete vector arrows to represent a vector field is that as the size of the dataset increases, the vectors become too crowded and the rendering ceases to be intelligible.
The length of each vector arrow is determined in relation to a vector reference magnitude that may be specified by the parameter VRM and a vector reference length that may be specified, as a fraction of the viewport width, by the parameter VRL.
Vectors belonging to the dataset whose magnitude is equal to the reference magnitude are drawn at the reference length.
ngwww.ucar.edu /ngdoc/ng4.3/supplements/vectors   (18529 words)

 A normalized-cut algorithm for hierarchical vector field data segmentation
In the context of vector field data visualization, it is often desirable to construct a hierarchical data representation.
A modified normalized cut (NC) method is used to obtain a near-optimal clustering of a given discrete vector field data set.
To obtain a hierarchical representation, the NC method is applied to simple, analytically defined vector fields as well as discrete vector field data generated by turbulent flow simulation.
repositories.cdlib.org /lbnl/LBNL-52025   (232 words)

 [No title]
The time used for conversion depends on the complexity of the formulas and on the resolution of the vector field (which is equal to the resolution of the visualization windows).
The vector field resolution can be specified by the user by changing the values of the "Width" and "Height" fields (Attention: If a visualization window is already open, changing these values and displaying another visualization confuses the original window).
This field indicates the number of circles that are used for the approximation of the OLIC image.
www.cg.tuwien.ac.at /research/vis/dynsys/frolic   (1486 words)

 Path integral (or line integral) of a vector field   (Site not responding. Last check: 2007-10-22)
(And the fourth involves surface integrals of vector fields, which are closely related to path integrals of vector fields.) I cannot emphasize too strongly the importance of these integrals.
Since the bead is magnetized, the magnetic field exerts a force on the bead.
Recall that the dot product is zero if the two vectors are orthogonal, is negative if their angle is greater than 90 degrees, and is positive if their angle is less than 90 degrees.
www.math.umn.edu /~nykamp/m2374/readings/pathintvec   (585 words)

 Vector (Java 2 Platform SE v1.4.2)
of this vector to be the specified object.
The capacity of the vector is the length of this array buffer, and is at least large enough to contain all the vector's elements.
If the object is found in this vector, each component in the vector with an index greater or equal to the object's index is shifted downward to have an index one smaller than the value it had previously.
java.sun.com /j2se/1.4.2/docs/api/java/util/Vector.html   (3241 words)

 Learning by Simulations: Vector Fields
A vector field is a field which associates a vector to every point in the field space.
Vector fields are often used in physics to model observations which include a direction for each point of the observed space.
Examples are movement of a fluid, or the force generated by a magnetic of gravitational field, or athmospheric models, where both the strength (speed) and the direction of winds are recorded.
www.vias.org /simulations/simusoft_vectorfields.html   (251 words)

 Visualizing a Tangent Vector Field   (Site not responding. Last check: 2007-10-22)
A common approach is to draw integral curves through the vector field, either as curvilinear segments or a s a texture applied to the surface.
An alternative approach is to reveal the vector field indirectly, using it only to govern the reflective properties of the surface.
A new illumination model uses the vector field to produce anisotropic reflections from the surface.
www.icase.edu /docs/hilites/banks/tanVector.html   (230 words)

 Vector Magnetic Field Observations of the Solar South Pole   (Site not responding. Last check: 2007-10-22)
The structure and dynamics of the magnetic field near the poles of the sun, and its relationship to the coronal temperature and density structure is not well understood.
Vector magnetic field observations are uniquely well suited to study the polar field.
First, the polar field is expected to be mostly perpendicular to the line-of-sight, and second, the observed direction of the field allows a direct comparison with the observed structure of the corona.
www.aas.org /publications/baas/v28n2/aas188/abs/S002005.html   (235 words)

 Vector Field Diagram   (Site not responding. Last check: 2007-10-22)
This applet is designed to allow you to explore both the vector field diagram concept and the field line concept.
To represent an electric field with a vector field diagram we calculate the field on a mesh of points.
A field line is constructed starting at any point and tracing out a line which everywhere parallel to the local field.
web.mit.edu /jbelcher/www/java/vecnodyncirc/vecnodyncirc.html   (232 words)

A vector field on a domain in the plane or in space is a function that assigns a vector to each point in the domain.
At the center, we have the vector 25k, at the edges, the vectors are near 0 in length.
Note that the curl of a vector field is itself a vector function.
www.ac.cc.md.us /~donr/CalcIII/unit5/lesson1/u5l1.html   (1130 words)

 Vector fields and Line integrals, Part 3
As you draw, the curve appears in green, and arrows representing the outputs of the vector field at points on the curve appear in yellow.
The purple number that appears in the upper left hand corner of the applet gives the amount of work that has been done so far, if the force is represented by the vector field and the path that the object is moved along is the green curve you have traced out.
For two of the curves that you have found, explain why the shape of the curve and the arrangement of vectors in the vector field combine to give the amount of work (large positive, small positive, small negative, or large negative) that you predict.
www.math.duke.edu /education/ccp/materials/mvcalc/vfield/vfield3.html   (396 words)

 Vector field format (OVF)
Vector field files specify vector quantities (e.g., magnetization or magnetic flux density) as a function of spatial position.
This may be changed in the future to allow for multiple vector fields per file.
For irregular meshes, each data element is a 6-tuple, consisting of the x, y and z components of the node position, followed by the x, y and z components of the field at that position.
math.nist.gov /oommf/doc/userguide12a1/userguide/vectorfieldformat.html   (1799 words)

 New Transformation Equations and the Electric Field Four-vector
A change in the velocity of a particle causes a change in the rotation of its field, and therefore, a change in the components of that field relative to the unprimed frame.
The proper magnitude of the charge, q', at rest in the primed frame is greater than its magnitude, q, in motion in the unprimed frame, but the radial distance, r', in the primed frame is also greater by the same factor, so observers in both frames measure the same potential.
The decreased reaction of a moving charged particle to an electric or "magnetic" field is due, at least in part, to a decrease in the magnitude of its charge rather than to an increase in its mass.
www.softcom.net /users/der555/prl.html   (862 words)

 Java Microscope   (Site not responding. Last check: 2007-10-22)
This program produces a vector field where M is the i component and N is the j component (i.e.
It may take some time for the graph to load after pressing the "Draw Vector Field" button.
To create a magnifying lens, drag your mouse across the vector field.
www.eas.asu.edu /~asufc/microscope   (166 words)

 Vector Field Methods - Research - Scientific Computing and Imaging Institute
Visualizing vector field data is challenging because no existing natural representation can visually convey large amounts of three-dimensional directional information.
One result has been a set of graphical icons such as arrows, motion particles, stream lines, stream ribbons, and stream tubes that act as three-dimensional depth cues.
SCI Institute researchers have developed local and global visualization techniques to explore three-dimensional vector field data.
www.sci.utah.edu /research/vector-field-meth.html   (117 words)

 VECTOR FIELDS AND LINE INTEGRALS   (Site not responding. Last check: 2007-10-22)
We can think of the vector field F as representing a force field, or as representing a fluid velocity flow field.
Our sign convention is that work that is done by the vector field, and hence that we can use, will be positive.
Well, you need to be given the three things listed at the beginning, a vector field F, a path C, and you have to be told whether you are to do a tangential line integral, as you would in a work computation, or a normal line integral, as in a flux computation.
www.math.gatech.edu /~carlen/2507/notes/vectorCalc/vectorfields.html   (1026 words)

 PlanetMath: Hamiltonian vector field
is the Hamiltonian, then the flow of the Hamiltonian vector field
Cross-references: flow, manifold, 1-form, Hamiltonian, symplectic vector field, vector field, smooth function, cotangent bundle, tangent bundle, isomorphism, symplectic manifold
This is version 2 of Hamiltonian vector field, born on 2002-12-09, modified 2004-01-23.
planetmath.org /encyclopedia/HamiltonianVectorField.html   (102 words)

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