Non User
Index: > A B C D E F G H I J K L M N O P Q R S T U V W X Y Z	Business Non User Industries Finance Tax

Home > Vector field

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.

In the rigorous mathematical treatment, vector fields are defined on manifolds as a section of the manifold's tangent bundle.

1 Definition

Given an open and connected subset X in Rⁿ a vector field is a vector valued function

We say F is a C^k vector field if F is k times continuously differentiable in X.

A point x in X is called stationary if

A vector field can be visualized as a n-dimensional space with a n-dimensional vector attached to each point in X.

Given two C^k-vector fields F,G defined over X and a real valued C^k-function f defined over X

defines the moduleAlgebra In abstract algebra, a module is a generalization of a vector space. In a vector space the set of scalars forms a field whereas in a module the scalars just form a ring. Much of the theory of modules consists of recovering desirable properties of of C^k-vector fields over the ringIn ring theory, a branch of abstract algebra, a ring is an algebraic structure in which addition and multiplication are defined and have similar properties to those familiar from the integers. History See Ring theory Definition and notation A ring is an a of C^k-functions.

2 Notes

Vector fields should be compared to scalar fieldA scalar field associates a single number (or scalar to every point in space. Scalar fields are often used in physics, for instance to indicate the temperature distribution throughout space, or the air pressure. Definition A scalar field is a function : os, which associate a number or scalar to every point in space (or every point of some manifold).

The derivativeCalculus In mathematics, the derivative of a function is one of the two central concepts of calculus. The inverse of a derivative is called the antiderivative, or indefinite integral. The derivative of a function at a certain point is a measure of the rats of a vector field, resulting in a scalar field or another vector field, are called the divergenceIn vector calculus, the divergence is an operator that measures a vector field's tendency to originate from or converge upon a given point. For instance, in a vector field that denotes the velocity of water flowing in a draining bathtub, the divergence wo and curlThis article is about curl in mathematics, see also Curl programming language and cURL, the Unix command line tool for transferring files. In vector calculus, curl is a vector operator that shows a vector field's tendency to rotate about a point. A vector respectively.

3 Examples

• 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 ( magnitudeIn science, magnitude refers to the numerical size of something: see orders of magnitude. In mathematics, the magnitude of an object is a non-negative real number, which in simple terms is its length. In astronomy, magnitude refers to the logarithmic meas) 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. In wind tunnels, the fieldlines can be revealed using smoke.
• 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.

Politics	Business	Finance

Topics: Vector Field Point Fields Curve Integral Particle Called Space...

---

Vector spaceThe fundamental concept in linear algebra is that of a vector space or linear space . This is a generalization of the set of all geometrical vectors and is used throughout modern mathematics. Formal definition A set V is a vector space over a field F (for Vector space example 1Example I Let V be the set of all n-tuples, [v,v,v,. v] where v, for i {1,2,3,. n} is a member of R { real numbers}. Let the field be R as well. Define Vector Addition: For all v, w, in V define v+w [v+w,v+w,v+w,. Define Scalar Multiplication: For all a i Vector space example 2Example II Let M be the set of all (m×n) matrices, with complex numbers as entries. Let C be the field of complex numbers. Then if : where p is in C . Define vector addition in M : Define scalar multiplication: : Then M is a vector space over C and we den Vector space (example 3)In analysis, many function sets have the structure of a vector space; these are often called linear spaces instead of vector spaces''. This third example is one such set of functions. Example III Consider the set C[a,b] of all continuous functions f defin Vector graphicsVector graphics or geometric modeling describes the use of geometrical primitives such as points, lines, curves, and polygons to represent images in computer graphics. It is used by contrast to the term raster graphics, which is the representation of imag Vector (spatial)A vector in physics and engineering typically refers to a quantity that has close relationship to the spatial coordinates, informally described as an object with a "magnitude" and a "direction". The word vector is also now used for more general concepts ( Vector calculus Vector graphics editor Vector quantization Vector (biology) Vector processor Vector field Vector (band) Vector bundle Vector