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From Wikipedia

Vector space

A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied ("scaled") by numbers, called scalarsin this context. Scalars are often taken to bereal numbers, but one may also consider vector spaces with scalar multiplication by complex numbers, rational numbers, or even more general fields instead. The operations of vector addition and scalar multiplication have to satisfy certain requirements, called axioms, listedbelow. An example of a vector space is that of Euclidean vectors which are often used to represent physical quantities such as forces: any two forces (of the same type) can be added to yield a third, and the multiplication of a force vector by a real factor is another force vector. In the same vein, but in more geometric parlance, vectors representing displacements in the plane or in three-dimensional space also form vector spaces.

Vector spaces are the subject of linear algebra and are well understood from this point of view, since vector spaces are characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space. The theory is further enhanced by introducing on a vector space some additional structure, such as a norm or inner product. Such spaces arise naturally in mathematical analysis, mainly in the guise of infinite-dimensional function spaces whose vectors are functions. Analytical problems call for the ability to decide if a sequence of vectors converges to a given vector. This is accomplished by considering vector spaces with additional data, mostly spaces endowed with a suitable topology, thus allowing the consideration of proximity and continuity issues. These topological vector spaces, in particular Banach spaces and Hilbert spaces, have a richer theory.

Historically, the first ideas leading to vector spaces can be traced back as far as 17th century's analytic geometry, matrices, systems of linear equations, and Euclidean vectors. The modern, more abstract treatment, first formulated by Giuseppe Peano in the late 19th century, encompasses more general objects than Euclidean space, but much of the theory can be seen as an extension of classical geometric ideas like lines, planes and their higher-dimensional analogs.

Today, vector spaces are applied throughout mathematics, science and engineering. They are the appropriate linear-algebraic notion to deal with systems of linear equations; offer a framework for Fourier expansion, which is employed in image compression routines; or provide an environment that can be used for solution techniques for partial differential equations. Furthermore, vector spaces furnish an abstract, coordinate-free way of dealing with geometrical and physical objects such as tensors. This in turn allows the examination of local properties of manifolds by linearization techniques. Vector spaces may be generalized in several directions, leading to more advanced notions in geometry and abstract algebra.

Introduction and definition

First example: arrows in the plane

The concept of vector space will first be explained by describing two particular examples. The first example of a vector space consists of arrows in a fixed plane, starting at one fixed point. This is used in physics to describe forces or velocities. Given any two such arrows, v and w, the parallelogram spanned by these two arrows contains one diagonal arrow that starts at the origin, too. This new arrow is called the sum of the two arrows and is denoted . Another operation that can be done with arrows is scaling: given any positive real numbera, the arrow that has the same direction as v, but is dilated or shrunk by multiplying its length by a, is called multiplication of v by a. It is denoted . When a is negative, is defined as the arrow pointing in the opposite direction, instead.

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Vector field

In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a subset of Euclidean 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.

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.

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: S \to \mathbf{R}^n in standard Cartesian coordinates (x1, ..., xn). If S is an open set, then V is a continuous function provided that each component of V is continuous, and more generally V is Ck vector field if each component V is k times continuously differentiable.

A vector field can be visualized as an n-dimensional space with an 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

(fV)(p) := f(p)V(p)\,
(V+W)(p) := V(p) + W(p)\,

define the module of Ck-vector fields over the ring of Ck-functions.

Coordinate transformation law

In physics, a vector is additionally distinguished by how its coordinates change when one measures the same vector with respect to a different background coordinate system. The transformation properties of vectors distinguish a vector as a geometrically distinct entity from a simple list of scalars, or from a covector.

Thus, suppose that (x1,...,xn) is a choice of Cartesian coordinates, in terms of which the coordinates of the vector V are

V_x = (V_{1,x},\dots,V_{n,x})

and suppose that (y1,...,yn) are n functions of the xi defining a different coordinate system. Then the coordinates of the vector V in the new coordinates are required to satisfy the transformation law

Such a transformation law is called contravariant. A similar transformation law characterizes vector fields in physics: specifically, a vector field is a specification of n functions in each coordinate system subject to the transformation law () relating the different coordinate systems.

Vector fields are thus contrasted with scalar fields, which associate a number or scalar to every point in space, and are also contrasted with simple lists of scalar fields, which do not transform under coordinate changes.

Vector fields on manifolds

Given a manifoldM, a vector field on M is a continuous assignment mapping every point of M to a tangent vector to M at that point. That is, for each x in M, we have a tangent vector v(x) in TxM such that the map sending a point to the appropriate tangent vector is a continuous function from the manifold to the total space of its tangent bundle. More precisely, 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.

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 also denoted by \scriptstyle \mathfrak{X} (M) (a fraktur "X").

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 (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 :

Addition theorem

In mathematics, an addition theorem is a formula such as that for the exponential function

ex + y = ex·ey

that expresses, for a particular function f, f(x + y) in terms of f(x) and f(y). Slightly more generally, as is the case with the trigonometric functions sin and cos, several functions may be involved; this is more apparent than real, in that case, since there cos is an algebraic function of sin (in other words, we usually take their functions both as defined on the unit circle).

The scope of the idea of an addition theorem was fully explored in the nineteenth century, prompted by the discovery of the addition theorem for elliptic functions. To 'classify' addition theorems it is necessary to put some restriction on the type of function G admitted, such that

F(x + y) = G(F(x), F(y)).

In this identity one can assume that F and G are vector-valued (have several components). An algebraic addition theorem is one in which G can be taken to be a vector of polynomials, in some set of variables. The conclusion of the mathematicians of the time was that the theory of abelian functions essentially exhausted the interesting possibilities: considered as a functional equation to be solved with polynomials, or indeed rational functions or algebraic functions, there were no further types of solution.

In more contemporary language this appears as part of the theory of algebraic groups, dealing with commutative groups. The connected, projective variety examples are indeed exhausted by abelian functions, as is shown by a number of results characterising an abelian variety by rather weak conditions on its group law. The so-called quasi-abelian functions are all known to come from extensions of abelian varieties by commutative affine group varieties. Therefore the old conclusions about the scope of global algebraic addition theorems can be said to hold. A more modern aspect is the theory of formal groups.



From Yahoo Answers

Question:

Answers:let us take two vectors 'a' and 'b' inclined at an angle 'x' taking X-axis along the direction of vector 'a' and Y-axis normal to it, we see that for vector 'a' ax = a and ay = 0, while for 'b' bx = b cos(x) and by = b sin(x) then resultant X-component Rx = a + b cos(x) and resultant Y-component Ry = b sin(x) hence, resultant magnitude R = square root of [ Rx^2 + Ry^2] =squre root of [ (a+bcosx)^2+(b six)^2] =square root of [ a^2 +b^2 + 2 a b cos(x)] that's it parallalogram law of vectors!!!!

Question:Please don't tell me the things written in book......give me the appropriate reason

Answers:They are essentially the same, because the third side of the triangle used in that method is just the diagonal of the parallelogram used in that method.

Question:I can understand that why these laws r used for displacement and velocity as they are units of length and its clear to me that why these two quantities follow the above laws but my question is that why resultant FORCE also can be calculated by these laws...i am aware of this fact that force is a vector quantity...........please...help me...answer me that why these laws are applicable for forces too

Answers:because the "laws" that you're talking about simply use magnitude and direction to figure out the sum or difference. since FORCE is a magnitude and direction, you can use the same "laws". i don't understand what you don't understand about this. can you clarify?

Question:When verifying the vector law of addition (where there is equlibrium and there is no net force) using hangers with weights on them and If all hangers have the same mass, is it possible to neglect their weights because their contributions cancel out when the vector forces representing the string tensions are added together? Explain.

Answers:yes, you are correct

From Youtube

Addition and Subtraction Of Vectors :Check us out at www.tutorvista.com Addition and Subtraction of Vectors Geometrical method To find a + b , shift vector b such that its initial point coincides with the terminal point of vector a. Now, the vector whose initial point coincides with the initial point of vector a , and terminal point coincides with the terminal point of vector b represents (a +b ) as shown in the above figure. To find (b +a ), shift a such that its initial point coincides with the terminal point b . A vector whose initial point coincides with the initial point of b and terminal point coincides with the terminal point of a represents (b +a ). Law of Parallelogram of Vectors The addition of two vectors may also be understood by the law of parallelogram. According to this law if two vectors P and Q are represented by two adjacent sides of a parallelogram both pointing outwards as shown in the figure below , then the diagonal drawn through the intersection of the two vectors represents the resultant (ie vector sum of P and Q). If Q is displacement from position AD to BC by displacing it parallel to itself, this method becomes equivalent to the triangle method. In case of addition of two vectors by parallelogram method as shown in figure, the magnitude of resultant will be given by, (AC)2 = (AE)2 + (EC)2 or R2 = (P + Q cos )2 (Q sin )2

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