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Divergence

Divergence

In 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, for a vector field that denotes the velocity of water flowing in a draining bathtub, the divergence would have a negative value over the drain because the water vanishes there (if we only consider two dimensions); away from the drain the divergence would be zero, since there are no other sinks or sources. A vector field which has zero divergence everywhere is called solenoidal.

Definition

Let x, y, z be a system of Cartesian coordinates on a 3-dimensional Euclidean space, and let i, j, k be the corresponding basis of unit vectors. The divergence of a continuously differentiable vector field :F = F_x i + F_y j + F_z k is defined to be the scalar-valued function :

\operatorname\,\mathbf
=\frac
+\frac
+\frac.

Although expressed in terms of coordinates, the result is invariant under orthogonal transformations, as the physical interpretation suggests. Another common notation for the divergence is Image:del.gif·F, a convenient mnemonic, where the dot denotes something just reminiscent of the dot product: take the components of Image:del.gif (see del), apply them to the components of F, and sum the results.

Physical interpretation

In physical terms, the divergence of a vector field is the extent to which the vector field flow behaves like a source or a sink at a given point. Indeed, an alternative, but logically equivalent definition, gives the divergence as the derivative of the net flow of the vector field across the surface of a small sphere relative to the volume of the sphere. To wit, :

( \operatorname\,\mathbf) (p) = 
\lim_
\int_

where S(r) denotes the sphere of radius r about a point p in R³, and the integral is a surface integral taken with respect to n, the normal to that sphere. In light of the physical interpretation, a vector field with constant zero divergence is called incompressible – in this case, no net flow can occur across any closed surface. The intuition that the sum of all sources minus the sum of all sinks should give the net flow outwards of a region is made precise by the divergence theorem.

Properties

The following properties can all be derived from the ordinary differentiation rules of calculus. Most importantly, the divergence is a linear operator, i.e. :

\operatorname( a\mathbf + b\mathbf ) 
= a\;\operatorname( \mathbf ) 
+ b\;\operatorname( \mathbf )

for all vector fields F and G and all real numbers a and b. There is a product rule of the following type: if φ is a scalar valued function and F is a vector field, then :

\operatorname(\varphi \mathbf) 
= \operatorname(\varphi) \cdot \mathbf 
+ \varphi \;\operatorname(\mathbf),

or in more suggestive notation :

\nabla\cdot(\varphi \mathbf) 
= (\nabla\varphi) \cdot \mathbf 
+ \varphi \;(\nabla\cdot\mathbf).

Another product rule for the cross product of two vector fields F and G in three dimensions involves the curl and reads as follows: :

\operatorname(\mathbf\times\mathbf) 
= \operatorname(\mathbf)\cdot\mathbf 
\;-\; \mathbf \cdot \operatorname(\mathbf),

or :

\nabla\cdot(\mathbf\times\mathbf)
= (\nabla\times\mathbf)\cdot\mathbf
- \mathbf\cdot(\nabla\times\mathbf).

The Laplacian of a scalar field is the divergence of the field's gradient. The divergence of the curl of any vector field (in three dimensions) is constant and equal to zero. Conversely, if you have a vector field F with zero divergence defined on a ball in R³, say, then there exists some vector field G on the ball with F = curl(G). For regions in R³ more complicated than balls, this latter statement might not be true anymore. Indeed, the degree of failure of the truth of the statement, measured by the homology of the chain complex :

\ \;

\to\ \;

:::

\to\ \;

::::

\to\ \;

(where the first map is the gradient, the second is the curl, the third is the divergence) serves as a nice quantification of the complicatedness of the underlying region U. These are the beginnings and main motivations of de Rham cohomology.

- Gradient
- Curl
- Vector calculus
- Nabla in cylindrical and spherical coordinates
- Divergence theorem Category:Vector calculus ko:다이버전스 ja:発散

Vector field

In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a 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 the manifold's tangent bundle.

Definition

Given a subset S in Rⁿ a vector field is represented by a vector-valued function

V: S \to \mathbf^n

in standard Euclidean coordinates (x₁, ..., x_n). If there is another coordinate system y, then

V_y := \frac V

is the expression for the same vector field in the new coordinates. In particular a vector field is not a bunch of scalar fields. We say V is a C^k 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

). 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 :

(fV)(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.

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). 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.

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.
- There are 3 types of lines that can be made from vector fields. They are : ::streaklines — as revealed in wind tunnels using smoke. ::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.

Gradient field

Vector fields can be constructed out of scalar fields using the vector operator gradient which gives rise to the following definition. A vector field V over S is called a gradient field or a conservative field if there exists a real valued function f on X(a scalar field) such that

V = \nabla f.

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

\int_\gamma \langle V(x), \mathrmx \rangle = \int_\gamma \langle \nabla f(x), \mathrmx \rangle = f((\gamma)(1)) - f((\gamma)(0))

Central field

A C^∞-vector field over Rⁿ \ is called a central field if :

V(T(p)) = T(V(p)) \qquad (T \in \mathrm(n, \mathbf))

where O(n, R) is the orthogonal group. We say central fields are invariant under orthogonal transformations around 0. 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 path 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. 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. Given a vector field V and a curve γ parametrized by [0, 1] the curve integral is defined as :

\int_\gamma \langle V(x), \mathrmx \rangle = \int_0^1 \langle V(\gamma(t)), \gamma'(t)\;\mathrmt \rangle

Flow curves

Vector fields have a nice interpretation in terms of autonomous, first order ordinary differential equations. 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 :

\gamma'(t) = V(\gamma(t))

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

\gamma_x(0) = x

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

The curves γ_x 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 two or three dimensions one can visualize the vector field as given 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.

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) and sin θ = y/(x² + y²). 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_:(r, \theta) \mapsto 1, \quad v_:(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_:(x, y) \mapsto 1, \quad v_:(x, y) \mapsto (\cos \theta, \sin \theta) = \left(\frac, \frac\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. Also consider the coordinate ξ := 2x. Suppose we have a scalar field and a vector field which are both given in the ξ coordinates by the constant function 1, :

s_:\xi \mapsto 1, \quad v_:\xi \mapsto 1.

Thus, we have a scalar field which has the value 1 everywhere and a vector field which attaches a vector in the ξ-direction with magnitude 1 unit of ξ to each point. But if ξ changes one unit then x changes 2 units. Then, this vector field has a magnitude of 2 in units of x. Therefore, in the x coordinate the scalar field and the vector field are described by the functions :

s_:x \mapsto 1, \quad v_:x \mapsto 2,

which are different.

Example 3

In 1D, an example of a scalar field is the electric potential V, which is e.g. 20 volt at a particular point. This is a scalar, not depending on the coordinate system. An electric field at that point of 5 volt/metre in some coordinate system is −5 volt/metre in an inverse coordinate system. Since a physical quantity is not just a number, but a number times a unit, there is no change of coordinate system that gives any other than one of these two values for the electric field at the point.

External links

- [http://mathworld.wolfram.com/VectorField.html Vector field] -- Mathworld
- [http://planetmath.org/encyclopedia/VectorField.html Vector field] -- PlanetMath
- [http://www.amasci.com/electrom/statbotl.html 3D Magnetic field viewer]
- [http://www-solar.mcs.st-and.ac.uk/~alan/MT3601/Fundamentals/node2.html Vector fields and field lines]
- [http://www.vias.org/simulations/simusoft_vectorfields.html Vector field simulation] An interactive application to show the effects of vector fields Category:Differential topology Category:Vector calculus Category:Infographics

Bathtub

:For the foundations of the World Trade Center, please see The Bathtub The Bathtub A bathtub (in the UK simply bath) is a plumbing fixture used for bathing. Most modern bathtubs are made of acrylic or fibreglass, but alternatives are available in the form of porcelain-coated steel or increasingly wood. Older western bathtubs are usually made of galvanized steel or iron. Traditional Japanese bathtubs were usually of a wooden construction. Until recently, most bathtubs were roughly rectangular in shape but with the advent of acrylic thermoformed baths, more and more shapes are becoming available. Bathtubs are typically white in colour although many other colours can be found. Modern bathtubs encompass an overflow and waste and may or may not have taps mounted on them. They may be built-in or free standing or sometimes sunken. The issues of depth and intended use are what separates a bathtub from a hot tub or other recreational bathing facilities. A bathtub is usually placed in a bathroom either as a stand-alone fixture or in conjunction with a shower. Any historical view of bathtubs should be aware of the 1917 Bathtub hoax.

External links

- [http://www.the-home-improvement-web.com/information/vitreous-enameled-baths.htm Enamelling cast-iron baths]
- [http://www.tjf.or.jp/eng/ge/ge04ofuro.htm The art of Japanese bathing]
- [http://www.driftwood.ie Making wooden baths] Category:Plumbing ja:風呂

Dimension

In common usage, the dimensions (from Latin "measured out") of an object are the parameters or measurements required to define its shape and size, that is, usually, its height, width, and length. In mathematics, the dimensions of a space are the parameters required to describe a particular object in this space. The dimension of a space is the number of these parameters. For example, locating a city on the Earth requires two parameters: longitude and latitude; the corresponding space has therefore two dimensions and its dimension is two. This space is said to be 2-dimensional (for short 2D). Locating an airplane might require a 3D space by addition of the altitude parameter, or even a 6D space, if one adds the three angles required for defining the orientation of the airplane. The airplane is then considered to have six degrees of freedom. Other dimensions can be supplemented like speed, temperature or even colour or price of the plane. Generalisations of the concept are possible and a number of alternative definitions may be introduced. Units are sometimes associated with each dimension, for instance, meters or feet with altitude or dollars with price. In science fiction, a "dimension" can also refer to an alternate universe or plane of existence. This useage is derived from the fact that to get to the alternate universe/plane of existence requires movement in an dimension beyond the normal 3 space + time.

Physical dimensions

The physical dimensions are the parameters required to answer to the question where and when happened or will happen some event; for instance: When did Napoleon die? — On the 5 May 1821 at Saint Helena (15°56′ S 5°42′ W). They play a fundamental role in our perception of the world around us. According to Immanuel Kant, we actually do not perceive them but they form the frame in which we perceive events; they form the a priori background in which events are perceived.

Spatial dimensions

Classical physics theories describe three physical dimensions: from a particular point in space, the basic directions in which we can move are up/down, left/right, and forward/backward. Movement in any other direction can be expressed in terms of just these three. Moving down is the same as moving up a negative amount. Moving diagonally upward and forward is just as the name of the direction implies; i.e., moving in a linear combination of up and forward. In its simplest form: a line describes one dimension, a plane describes two dimensions, and a cube describes three dimensions. (See Cartesian coordinate system)

Time

Time is often referred to as the "fourth dimension". It is, in essence, one way to measure physical change. It is perceived differently from the three spatial dimensions in that there is only one of it, and that movement seems to occur at a fixed rate and in one direction. The equations used by physics to model reality often do not treat time in the same way that humans perceive it. In particular, the equations of classical mechanics are symmetric with respect to time, and equations of quantum mechanics are typically symmetric if both time and other quantities (such as charge and parity) are reversed. In these models, the perception of time flowing in one direction is an artifact of the laws of thermodynamics (we perceive time as flowing in the direction of increasing entropy). The most well-known treatment of time as a dimension is Einstein's theory of general relativity, which treats perceived space and time as parts of a four-dimensional manifold. Some scientists consider that the fact that Einstein's theory allows time to move at different rates indicates a 2nd time dimension exists.

Additional dimensions

Theories such as string theory predict that the space we live in has in fact many more dimensions (frequently 10, 11 or 26), but that the universe measured along these additional dimensions is subatomic in size. As a result, we perceive only the three spatial dimensions that have macroscopic size.

Units

In the physical sciences and in engineering, the dimension of a physical quantity is the expression of the class of physical unit that such a quantity is measured against. The dimension of speed, for example, is length divided by time. In the SI system, the dimension is given by the seven exponents of the fundamental quantities. See Dimensional analysis.

Mathematical dimensions

In mathematics, no definition of dimension adequately captures the concept in all situations where we would like to make use of it. Consequently, mathematicians have devised numerous definitions of dimension for different types of spaces. All, however, are ultimately based on the concept of the dimension of Euclidean n-space Eⁿ. The point E⁰ is 0-dimensional. The line E¹ is 1-dimensional. The plane E² is 2-dimensional. And in general Eⁿ is n-dimensional. A tesseract is an example of a four-dimensional object. In the rest of this section we examine some of the more important mathematical definitions of dimension.

Hamel dimension

For vector spaces, there is a natural concept of dimension, namely the cardinality of a basis. See Hamel dimension for details.

Manifolds

A connected topological manifold is locally homeomorphic to Euclidean n-space, and the number n is called the manifold's dimension. One can show that this yields a uniquely defined dimension for every connected topological manifold. The theory of manifolds, in the field of geometric topology, is characterised by the way dimensions 1 and 2 are relatively elementary, the high-dimensional cases n > 4 are simplified by having extra space in which to 'work'; and the cases n = 3 and 4 are in some senses the most difficult. This state of affairs was highly marked in the various cases of the Poincaré conjecture, where four different proof methods are applied.

Lebesgue covering dimension

For any topological space, the Lebesgue covering dimension is defined to be n if n is the smallest integer for which the following holds: any open cover has a refinement (a second cover where each element is a subset of an element in the first cover) such that no point is included in more than n + 1 elements. For manifolds, this coincides with the dimension mentioned above. If no such n exists, then the dimension is infinite.

Inductive dimension

The inductive dimension of a topological space may refer to the small inductive dimension or the large inductive dimension, and is based on the analogy that n+1-dimensional balls have n dimensional boundaries, permitting an inductive definition based on the dimension of the boundaries of open sets.

Hausdorff dimension

For sets which are of a complicated structure, especially fractals, the Hausdorff dimension is useful. The Hausdorff dimension is defined for all metric spaces and, unlike the Hamel dimension, can also attain non-integer real values. The upper and lower box dimensions are a variant of the same idea.

Hilbert spaces

Every Hilbert space admits an orthonormal basis, and any two such bases have the same cardinality. This cardinality is called the dimension of the Hilbert space. This dimension is finite if and only if the space's Hamel dimension is finite, and in this case the two dimensions coincide.

Krull dimension of commutative rings

The Krull dimension of a commutative ring, named after Wolfgang Krull (1899 - 1971), is defined to be the maximal number of strict inclusions in an increasing chain of prime ideals in the ring.

Science fiction

Science fiction texts often mention the concept of dimension, when really referring to parallel universes, alternate universes, or other planes of existence, or concepts that are beyond the reader. The word gives a sense of authority to a film, and inspires imagination and awe in the minds of the reader, that one could travel to "another dimension". This concept is derived from the idea that in order to travel to to parallel/alternate universes/planes of existence one must travel in a spatial direction/dimension besides the standard ones. In effect, the other universes/planes are just a small distance away from our own, but the distance is in a fourth (or higher) spatial dimension, not the standard ones.

Anaglyph

This section should be merged into Anaglyph image. To understand how Anaglyph 3D works, you must understand how Sterioscope 3D works. It's basically blending the two images taken eye width apart. The eye thinks it's seeing one normal image by blending together one image we see with both eyes. You blend two slightly different pictures together and look at it. It looks 3D! Anaglyph is just taking a single framed 3D image and making one eye only see one image... the red sees the blue (because the red of the glasses blends in with the red) and the blue sees the red (the blue of the glasses blends in with the blue). This creates a normal stereo graph image without the need of crossing your eyes the whole time you're looking at the image.

3-D film

This section should be merged into 3-D film. In the 1950's, 3D movies were very popular, but since they didn't have color film then they had an entirely different method. They would take two black and white cameras, line them up eye-with apart and take the film from both at the same time. They took two projectors in the projection booth, both aimed at the screen. The left one with the left film with a blue filter over the lense, and the right with the right film and a red filter over the lense, rather than just having one 3D image blended together.

Modern 3-D films

Spy Kids 3D has developed a new craze of 3D, mostly in amusement parks with motion simulators. Spongebob 3D in Paramount's Great America, California has a Polarization 3D movie along with a motion simulator. Shark Boy and Lava Girl is another modern 3D movie, by the same producer of Spy Kids.

More dimensions

- Dimension of an algebraic variety
- Topological dimension
- Isoperimetric dimension
- Poset dimension
- Pointwise dimension
- Lyapunov dimension
- Kaplan-Yorke dimension
- Exterior dimension
- Hurst exponent
- q-dimension; especially:
- Information dimension (corresponding to q=1)
- Correlation dimension (corresponding to q=2)

Solenoidal vector field

In vector calculus a solenoidal vector field is a vector field v with divergence zero: :

\nabla \cdot \mathbf = 0.\,

This condition is satisfied whenever v has a vector potential, because if :

\mathbf = \nabla \times \mathbf

then :

\nabla \cdot \mathbf = \nabla \cdot (\nabla \times \mathbf) = 0.

The converse also holds: for any solenoidal v there exists a vector potential A such that

\mathbf = \nabla \times \mathbf.

. (Strictly speaking, this holds only subject to certain technical conditions on v, see Helmholtz decomposition.)

Examples

- one of Maxwell's equations states that the magnetic field B is solenoidal;
- the velocity field of an incompressible fluid flow is solenoidal. Category:Vector calculus Category:Fluid dynamics

Euclidean space

In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. The generalization applies Euclid's concept of distance, and the related concepts of length and angle, to a coordinate system in any number of dimensions. It is the "standard" example of a finite-dimensional, real, inner product space. A Euclidean space is a particular metric space that enables the investigation of topological properties such as compactness. An inner product space is a generalization of a Euclidean space. Both inner product spaces and metric spaces are explored within functional analysis. Euclidean space plays a part in the definition of a manifold which embraces the concepts of both Euclidean and non-Euclidean geometry. One mathematical motivation for defining a distance function is the ability to define an open ball around points in the space. This fundamental concept justifies a differential calculus between a Euclidean space and other manifolds. Differential geometry brings such a differential calculus into play, together with a technique of launching a mobile, local Euclidean space, to explore the properties of non-Euclidean manifolds.

Real coordinate space

Let R denote the field of real numbers. For any non-negative integer n, the space of all n-tuples of real numbers forms an n-dimensional vector space over R sometimes called real coordinate space and denoted Rⁿ. An element of Rⁿ is written x = (x₁, x₂, …, x_n) where each x_i is a real number. The vector space operations on Rⁿ are defined by :

\mathbf + \mathbf = (x_1 + y_1, x_2 + y_2, \ldots, x_n + y_n)

a\,\mathbf = (a x_1, a x_2, \ldots, a x_n)

Real coordinate space Rⁿ comes with a standard basis: :

\mathbf_1 = (1, 0, \ldots, 0)

\mathbf_2 = (0, 1, \ldots, 0)

\vdots

\mathbf_n = (0, 0, \ldots, 1)

An arbitrary vector in Rⁿ can then be written in the form :

\mathbf = \sum_^n x_i \mathbf_i

Real coordinate space is the prototypical example of a real n-dimensional vector space. In fact, every real n-dimensional vector space V is isomorphic to Rⁿ. This isomorphism is not canonical however. A choice of isomorphism is equivalent to a choice of basis for V (by looking at the image of the standard basis for Rⁿ in V). The reason for working with arbitrary vector spaces instead of Rⁿ is that it is often preferable to work in a coordinate-free manner (i.e. without choosing a preferred basis).

Euclidean structure

Euclidean space is more than just real coordinate space. In order to do Euclidean geometry one needs to be able to talk about the distance between points and the angles between lines or vectors. The natural way in which to do this is to introduce what is called an inner product or dot product on Rⁿ. This product is defined by :

\mathbf\cdot\mathbf = \sum_^n x_iy_i = x_1y_1+x_2y_2+\cdots+x_ny_n.

The dot product of any two vectors x and y gives a real number. This product allows us to define the "length" of a vector x in the following way :

\|\mathbf\| = \sqrt = \sqrt

This length function satisfies the required properties of a norm and is called the Euclidean norm on Rⁿ. The (interior) angle θ between x and y is then given by :

\theta = \cos^\left(\frac\right)

where cos⁻¹ is the arccosine function. Finally, one can use the norm to define a distance function (or metric) on Rⁿ in the following manner :

d(\mathbf, \mathbf) = \|\mathbf - \mathbf\| = \sqrt.

The form of this distance function is based on the Pythagorean theorem, and is called the Euclidean metric. Real coordinate space together with the above Euclidean structure (dot product and the associated norm and metric) is called Euclidean space often denoted by Eⁿ. (Many authors refer to Rⁿ itself as Euclidean space, with the Euclidean structure being understood). The Euclidean structure on Eⁿ gives it the structure of an inner product space (in fact a Hilbert space), a normed vector space, and a metric space.

Euclidean topology

Since Euclidean space is a metric space it is also a topological space with the natural topology induced by the metric. The metric topology on Eⁿ is called the Euclidean topology. A set is open in the Euclidean topology if and only if it contains an open ball around each of its points. The Euclidean topology turns out to be equivalent to the product topology on Rⁿ considered as a product of n copies of the real line R (with its standard topology). An important result on the topology of Rⁿ, that is far from superficial, is Brouwer's invariance of domain. Any subset of Rⁿ (with its subspace topology) which is homeomorphic to another open subset of Rⁿ is itself open. An immediate consequence of this is that R^m is not homeomorphic to Rⁿ if m ≠ n — an intuitively "obvious" result which is nonetheless difficult to prove. Euclidean n-space is the prototypical example of an n-manifold, in fact, a smooth manifold. For n ≠ 4, any differentiable n-manifold that is homeomorphic to Rⁿ is also diffeomorphic to it. The surprising fact that this is not also true for n = 4 was proved by Simon Donaldson in 1982; the counterexamples are called exotic (or fake) 4-spaces. Euclidean space is also known as linear manifold. An m-dimensional linear submanifold of Rⁿ is a Euclidean space of m dimensions embedded in it (as an affine subspace). For example, any straight line in some higher-dimensional Euclidean space is a 1-dimensional linear submanifold of that space.

Unit vector

In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) whose length is 1. A unit vector is often written with a “hat”, thus: î. In Euclidean space, the dot product of two unit vectors is simply the cosine of the angle between them. This follows from the formula for the dot product, since the lengths are both 1. The normalized vector û of a non-zero vector u is the unit vector codirectional with u, i.e., :

\mathbf = \frac.

where ||u|| is the norm (or length) of u. The term normalized vector is sometimes used simply as a synonym for unit vector. As opposed to a general vector, which represents direction and magnitude, a unit vector just represents direction. The components are called direction cosines, because each is the cosine of the angle between the vector and one coordinate axis. The elements of a basis are often chosen to be unit vectors. In the 3-Dimensional Cartesian coordinate system, these are usually i, j, and k—unit vectors along the x, y, and z axes, respectively: :

\mathbf = \begin1\\0\\0\end, \,\, \mathbf = \begin0\\1\\0\end, \,\,  \mathbf = \begin0\\0\\1\end.

These are not always written with a hat; but it can generally be assumed that i, j, and k are unit vectors in most contexts. Very often these unit vectors are written as e₁, e₂ and e₃ respectively. Other coordinate systems, such as polar coordinates or spherical coordinates use different unit vectors; notations vary. Category:Linear algebra th:เวกเตอร์หนึ่งหน่วย

Differentiable

In mathematics, the derivative is one of the two central concepts of calculus. (The other is the integral; the two are related via the fundamental theorem of calculus.) The simplest type of derivative is the derivative of a real-valued function of a single real variable. It has several interpretations:
- The derivative gives the slope of a tangent to the graph of the function at a point. In this way, derivatives can be used to determine many geometrical properties of the graph, such as concavity or convexity.
- The derivative provides a mathematical formulation of rate of change; it measures the rate at which the function's value changes as the function's argument changes. This derivative is the kind usually encountered in a first course on calculus, and historically was the first to be discovered. However, there are also many generalizations of the derivative. The remainder of this article discusses only the simplest case (real-valued functions of real numbers).

Differentiation and differentiability

In physical terms, differentiation expresses the rate at which a quantity, y, changes with respect to the change in another quantity, x, on which it has a functional relationship. Using the symbol Δ to refer to change in a quantity, this rate is defined as a limit of difference quotients :

\frac

as Δx approaches 0. In Leibniz's notation for derivatives, the derivative of y with respect to x is written :

\frac

suggesting the ratio of two infinitesimal quantities. The above expression is pronounced in various ways such as "dy by dx" or "dy over dx". The form "dy dx" is also used conversationally, although it may be confused with the notation for element of area. Modern mathematicians do not bother with "dependent quantities", but simply state that differentiation is a mathematical operation on functions. The precise definition of this operation (which therefore need not deal with infinitesimal quantities) is given as: :

\lim_\frac.

A function is differentiable at a point x if its derivative exists at that point; a function is differentiable on an interval if it is differentiable at every x within the interval. If a function is not continuous at x, then there is no tangent line and the function is therefore not differentiable at x; however, even if a function is continuous at x, it may not be differentiable there. In other words, differentiability implies continuity, but not vice versa. One famous example of a function that is continuous everywhere but differentiable nowhere is the Weierstrass function. The derivative of a differentiable function can itself be differentiable. The derivative of a derivative is called a second derivative. Similarly, the derivative of a second derivative is a third derivative, and so on.

Newton's difference quotient

The derivative of a function f at x is geometrically the slope of the tangent line to the graph of f at x. Without the concept which we are about to define, it is impossible to directly find the slope of the tangent line to a given function, because we only know one point on the tangent line, namely (x, f(x)). Instead, we will approximate the tangent line with multiple secant lines that have progressively shorter distances between the two intersecting points. When we take the limit of the slopes of the nearby secant lines in this progression, we will get the slope of the tangent line. The derivative is then defined by taking the limit of the slope of secant lines as they approach the tangent line. tangent tangent To find the slopes of the nearby secant lines, choose a small number h. h represents a small change in x, and it can be either positive or negative. The slope of the line through the points (x,f(x)) and (x+h,f(x+h)) is :

.

This expression is Newton's difference quotient. The derivative of f at x is the limit of the value of the difference quotient as the secant lines get closer and closer to being a tangent line: :

f'(x)=\lim_.

difference quotient If the derivative of f exists at every point x in the domain, we can define the derivative of f to be the function whose value at a point x is the derivative of f at x. Since immediately substituting 0 for h results in division by zero, calculating the derivative directly can be unintuitive. One technique is to simplify the numerator so that the h in the denominator can be cancelled. This happens easily for polynomials; see calculus with polynomials. For almost all functions however, the result is a mess. Fortunately, many guidelines exist.

Notations for differentiation

Lagrange's notation

The simplest notation for differentiation that is in current use is due to Joseph Louis Lagrange and uses the prime mark:

Leibniz's notation

The other common notation is Leibniz's notation for differentiation which is named after Leibniz. For the function whose value at x is the derivative of f at x, we write: : $\frac.$ We can write the derivative of f at the point a in two different ways: : $\frac\left.\right|_ = \left(\frac\right)(a).$ If the output of f(x) is another variable, for example, if y=f(x), we can write the derivative as: : $\frac.$ Higher derivatives are expressed as : $\frac$ or $\frac$ for the n-th derivative of f(x) or y respectively. Historically, this came from the fact that, for example, the 3rd derivative is: : $\frac$ which we can loosely write as: : $\left(\frac\right)^3 \left(f(x)\right) =
\frac \left(f(x)\right).$ Dropping brackets gives the notation above. Leibniz's notation allows one to specify the variable for differentiation (in the denominator). This is especially relevant for partial differentiation. It also makes the chain rule easy to remember, because the "du" terms appear symbolically to cancel: : $\frac = \frac \cdot \frac.$ (In the popular formulation of calculus in terms of limits, the "du" terms cannot literally cancel, because on their own they are undefined; they are only defined when used together to express a derivative. In nonstandard analysis, however, they can be viewed as infinitesimal numbers that cancel.)
Newton's notation
Newton's notation for differentiation (also called the dot notation for differentiation) requires placing a dot over the function name: : $\dot = \frac = x'(t)$ : $\ddot = x$ (t) and so on. Newton's notation is mainly used in mechanics, normally for time derivatives such as velocity and acceleration, and in ODE theory. It is usually only used for first and second derivatives.

Euler's notation

Euler's notation uses a differential operator, denoted as D, which is prefixed to the function with the variable as a subscript of the operator: This notation can also be abbreviated when taking derivatives of expressions that contain a single variable. The subscript to the operator is dropped and is assumed to be the only variable present in the expression. In the following examples, u represents any expression of a single variable: Euler's notation is useful for stating and solving linear differential equations.

Critical points

Points on the graph of a function where the derivative is undefined or equals zero are called critical points or sometimes stationary points (in the case where the derivative equals zero). If the second derivative is positive at a critical point, that point is a local minimum; if negative, it is a local maximum; if zero, it may or may not be a local minimum or local maximum. Taking derivatives and solving for critical points is often a simple way to find local minima or maxima, which can be useful in optimization. In fact, local minima and maxima can only occur at critical points. This is related to the extreme value theorem.

Physics

Arguably the most important application of calculus to physics is the concept of the "time derivative"—the rate of change over time—which is required for the precise definition of several important concepts. In particular, the time derivatives of an object's position are significant in Newtonian physics:
- Velocity (instantaneous velocity; the concept of average velocity predates calculus) is the derivative (with respect to time) of an object's position.
- Acceleration is the derivative (with respect to time) of an object's velocity.
- Jerk is the derivative (with respect to time) of an object's acceleration. For example, if an object's position

p(t) = -16t^2 + 16t + 32

; then, the object's velocity is

\dot p(t) = p'(t) = -32t + 16

; the object's acceleration is

\ddot p(t) = p

(t) = -32; and the object's jerk is $p(t) = 0.$ If the velocity of a car is given, as a function of time, then, the derivative of said function with respect to time describes the acceleration of said car, as a function of time.
Algebraic manipulation
Messy limit calculations can be avoided, in certain cases, because of differentiation rules which allow one to find derivatives via algebraic manipulation; rather than by direct application of Newton's difference quotient. One should not infer that the definition of derivatives, in terms of limits, is unnecessary. Rather, that definition is the means of proving the following "powerful differentiation rules"; these rules are derived from the difference quotient.
- Constant rule: The derivative of any constant is zero.
- Constant multiple rule: If c is some real number; then, the derivative of $cf(x)$ equals c multiplied by the derivative of f(x) (a consequence of linearity below).
- Linearity: $(af + bg)' = af' + bg'$ for all functions f and g and all real numbers a and b.
- Power rule: If $f(x) = x^r$ , for some real number r; $f'(x) = rx^$ .
- Product rule: $(fg)' = f'g + fg'$ for all functions f and g.
- Quotient rule: $(f/g)' = (f'g - fg')/(g^2)$ unless g is zero.
- Chain rule: If $f(x) = h(g(x))$ , then $f'(x) = h'(g(x)) g'(x)$ .
- Inverse function: If $g(x) = f^(x)$ , and $f(x)$ is injective, then $g'(x) = 1/f'(f^(x))$ .
- Derivative of one variable with respect to another when both are functions of a third variable: Let $x = f(t)$ and $y = g(t)$ . Now $d y/d x = (d y/d t)/(d x/d t)$ . This is the chain rule in the Leibniz notation.
- Implicit differentiation: If $f(x,y) = 0$ is an implicit function, we have: dy/dx = - (∂f / ∂x) / (∂f / ∂y). In addition, the derivatives of some common functions are useful to know. See the table of derivatives. As an example, the derivative of : $f(x) = 2x^4 + \sin (x^2) - \ln (x)\;e^x + 7$ is : $f'(x) = 8x^3 + 2x\cos (x^2) - \frac\;e^x - \ln (x)\;e^x.$
Using derivatives to graph functions
Derivatives are a useful tool for examining the graphs of functions. In particular, the points in the interior of the domain of a real-valued function which take that function to local extrema will all have a first derivative of zero. However, not all critical points are local extrema; for example, f(x)=x³ has a critical point at x=0, but it has neither a maximum nor a minimum there. The first derivative test and the second derivative test provide ways to determine if the critical points are maxima, minima or neither. In the case of multidimensional domains, the function will have a partial derivative of zero with respect to each dimension at local extrema. In this case, the Second Derivative Test can still be used to characterize critical points, by considering the eigenvalues of the Hessian matrix of second partial derivatives of the function at the critical point. If all of the eigenvalues are positive, then the point is a local minimum; if all are negative, it is a local maximum. If there are some positive and some negative eigenvalues, then the critical point is a saddle point, and if none of these cases hold then the test is inconclusive (e.g., eigenvalues of 0 and 3). Once the local extrema have been found, it is usually rather easy to get a rough idea of the general graph of the function, since (in the single-dimensional domain case) it will be uniformly increasing or decreasing except at critical points, and hence (assuming it is continuous) will have values in between its values at the critical points on either side.
Generalizations
Where a function depends on more than one variable, the concept of a partial derivative is used. Partial derivatives can be thought of informally as taking the derivative of the function with all but one variable held temporarily constant near a point. Partial derivatives are represented as ∂/∂x (where ∂ is a rounded 'd' known as the 'partial derivative symbol'). Some people pronounce the partial derivative symbol as 'der' rather than the 'dee' used for the standard derivative symbol, 'd'. The concept of derivative can be extended to more general settings. The common thread is that the derivative at a point serves as a linear approximation of the function at that point. Perhaps the most natural situation is that of functions between differentiable manifolds; the derivative at a certain point then becomes a linear transformation between the corresponding tangent spaces and the derivative function becomes a map between the tangent bundles. In order to differentiate all continuous functions and much more, one defines the concept of distribution. For complex functions of a complex variable differentiability is a much stronger condition than that the real and imaginary part of the function are differentiable with respect to the real and imaginary part of the argument. For example, the function f(x + iy) = x + 2iy satisfies the latter, but not the first. See also the article on holomorphic functions.
See also

- Derivative (examples)
- Derivative (generalizations)
- Partial derivative
- Total derivative
- Table of derivatives
- Smooth function
- Differintegral
- Automatic differentiation
External links

- [http://wims.unice.fr/wims/wims.cgi?module=tool/analysis/function.en WIMS Function Calculator] makes online calculation of derivatives; this software enables also interactive exercises.
References

- Spivak, Michael; Calculus (3rd edition, 1994) Publish or Perish Press. ISBN 0914098896. Explains why all this works.
- Thompson, Silvanus Phillips, Calculus made easy : being a very-simplest introduction to those beautiful methods of reckoning which are generally called by the terrifying names of the differential calculus and the integral calculus New York : St. Martin's Press, 1998 ISBN 0312185480. Introduced by Martin Gardner. "What one fool can do, another can."
- Larson, Ron; Hostetler, Robert P.; and Edwards, Bruce H. (2003). Calculus of a Single Variable: Early Transcendental Functions (3rd edition). Houghton Mifflin Company. ISBN 061822307X.
- Anton, Howard (1980). Calculus with analytical geometry.. New York:John Wiley and Sons. ISBN 0-471-03248-4.
- ko:미분 ja:微分 simple:Derivative th:อนุพันธ์

Vector field

Definition

Given a subset S in Rⁿ a vector field is represented by a vector-valued function

V: S \to \mathbf^n

in standard Euclidean coordinates (x₁, ..., x_n). If there is another coordinate system y, then

V_y := \frac V

V(p) = 0

(fV)(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.

Notes

Examples

Gradient field

V = \nabla f.

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

\int_\gamma \langle V(x), \mathrmx \rangle = \int_\gamma \langle \nabla f(x), \mathrmx \rangle = f((\gamma)(1)) - f((\gamma)(0))

Central field

A C^∞-vector field over Rⁿ \ is called a central field if :

V(T(p)) = T(V(p)) \qquad (T \in \mathrm(n, \mathbf))

Curve integral

\int_\gamma \langle V(x), \mathrmx \rangle = \int_0^1 \langle V(\gamma(t)), \gamma'(t)\;\mathrmt \rangle

Flow curves

\gamma'(t) = V(\gamma(t))

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

\gamma_x(0) = x

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

Difference between scalar and vector field

Example 1

s_:(r, \theta) \mapsto 1, \quad v_:(r, \theta) \mapsto (1, 0).

s_:(x, y) \mapsto 1, \quad v_:(x, y) \mapsto (\cos \theta, \sin \theta) = \left(\frac, \frac\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

s_:\xi \mapsto 1, \quad v_:\xi \mapsto 1.

s_:x \mapsto 1, \quad v_:x \mapsto 2,

which are different.

Example 3

External links

Orthogonal matrix

In matrix theory, a real orthogonal matrix is a square matrix Q whose transpose is its inverse: :

Q^T Q = Q Q^T = I\,\! .

Overview

The definition of orthogonal matrix is more restrictive than that of orthonormal matrix, which requires only Q^TQ = I and may have fewer columns than rows. An orthogonal matrix is also the real specialization of a unitary matrix, and thus always a normal matrix. Although we consider only real matrices here, the definition can be used for matrices with entries from any field. However, orthogonal matrices arise naturally from inner products, and for matrices of complex numbers that leads instead to the unitary requirement. To see the inner product connection, consider a vector v in an n-dimensional real inner product space. Written with respect to an orthonormal basis, the squared length of v is v^Tv. If a linear transformation, in matrix form Qv, preserves vector lengths, then :

^T = (Q)^T(Q) = ^T Q^T Q  .

Thus finite-dimensional linear isometries—rotations, reflections, and their combinations—produce orthogonal matrices. The converse is also true: orthogonal matrices imply orthogonal transformations. However, linear algebra includes orthogonal transformations between spaces which may be neither finite-dimensional nor of the same dimension, and these have no orthogonal matrix equivalent. Orthogonal matrices are important for a number of reasons, both theoretical and practical. The n×n orthogonal matrices form a group, the orthogonal group denoted by O(n), which—with its subgroups—is widely used in mathematics and the physical sciences. For example, the point group of a molecule is a subgroup of O(3). Because floating point versions of orthogonal matrices have advantageous properties, they are key to many algorithms in numerical linear algebra, such as QR decomposition. As another example, with appropriate normalization the discrete cosine transform (used in MP3 compression) is represented by an orthogonal matrix.

Examples

Below are a few examples of small orthogonal matrices and possible interpretations.
-

\begin
1 & 0 \\
0 & 1 \\
\end

(identity transformation)
-

\begin
0.96 & -0.28 \\
0.28 & \;\;\,0.96 \\
\end

(rotation by 16.26°)
-

\begin
1 & 0 \\
0 & -1 \\
\end

(reflection across the x-axis)
-

\begin
0 & -0.80 & -0.60 \\
0.80 & -0.36 & \;\;\,0.48 \\
0.60 & \;\;\,0.48 & -0.64
\end

(rotoinversion with axis (0,−3/5,4/5), angle 90°)
-

\begin
0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0 \\
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0
\end

(permutation of axes)

Elementary constructions

Lower dimensions

The simplest orthogonal matrices are the 1×1 matrices [1] and [−1] which we can interpret as the identity and a reflection of the real line across the origin. The 2×2 matrices have the form :

\begin
p & t\\
q & u
\end,

which orthogonality demands satisfy the three equations : In consideration of the first equation, without loss of generality let p = cos θ, q = sin θ; then either t = −q, u = p or t = q, u = −p. We can interpret the first case as a rotation by θ (where θ = 0 is the identity), and the second as a reflection across a line at an angle of θ/2. :

\begin
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta \\
\end

(rotation),

\begin
\cos \theta & \sin \theta \\
\sin \theta & -\cos \theta \\
\end

(reflection) The reflection at 45° exchanges x and y; it is a permutation matrix, with a single 1 in each column and row (and otherwise 0): :

\begin
0 & 1\\
1 & 0
\end.

The identity is also a permutation matrix. A reflection is its own inverse, which implies that a reflection matrix is symmetric (equal to its transpose) as well as orthogonal. The product of two rotation matrices is a rotation matrix, and the product of two reflection matrices is also a rotation matrix.

Higher dimensions

Regardless of the dimension, it is always possible to classify orthogonal matrices as purely rotational or not, but for 3×3 matrices and larger the non-rotational matrices can be more complicated than reflections. For example, :

\begin
-1 & 0 & 0\\
0 & -1 & 0\\
0 & 0 & -1
\end

and

\begin
0 & -1 & 0\\
1 & 0 & 0\\
0 & 0 & -1
\end

represent an inversion through the origin and a rotoinversion about the z axis. Rotations also become more complicated; they can no longer be completely characterized by an angle, and may affect more than one planar subspace. While it is common to describe a 3×3 rotation matrix in terms of an axis and angle, the existence of an axis is an accidental property of this dimension that applies in no other. However, we have elementary building blocks for permutations, reflections, and rotations that apply in general.

Primitives

The most elementary permutation is a transposition, obtained from the identity matrix by exchanging two rows. Any n×n permutation matrix can be constructed as a product of at most n−1 transpositions. A Householder reflection is constructed from a non-null vector v as :

Q = I - 2  .

Here the numerator is a symmetric matrix while the denominator is a number, the squared magnitude of v. This is a reflection in the hyperplane perpendicular to v (negating any vector component parallel to v). If v is a unit vector, then Q = I−2vv^T suffices. A Householder reflection is typically used to simultaneously zero the lower part of a column. Any orthogonal matrix of size n×n can be constructed as a product of at most n such reflections. A Givens rotation acts on a two-dimensional (planar) subspace spanned by two coordinate axes, rotating by a chosen angle. It is typically used to zero a single subdiagonal entry. Any rotation matrix of size n×n can be constructed as a product of at most n(n−1)/2 such rotations. In the case of 3x3 matrices, three such rotations suffice; and by fixing the sequence we can thus describe all 3×3 rotation matrices (though not uniquely) in terms of the three angles used, often called Euler angles. A Jacobi rotation has the same form as a Givens rotation, but is used as a similarity transformation chosen to zero both off-diagonal entries of a 2×2 symmetric submatrix.

Properties

Matrix properties

A real square matrix is orthogonal if and only if its columns form an orthonormal basis of the Euclidean space Rⁿ with the ordinary Euclidean dot product, which is the case if and only if its rows form an orthonormal basis of Rⁿ. It might be tempting to suppose a matrix with orthogonal (not orthonormal) columns would be called an orthogonal matrix, but such matrices have no special interest and no special name; they only satisfy M^TM = D, with D a diagonal matrix. The determinant of any orthogonal matrix is +1 or −1. This follows from basic facts about determinants, as follows: :

1=\det(I)=\det(Q^TQ)=\det(Q^T)\det(Q)=(\det(Q))^2\,\! .

The converse is not true; having a determinant of +1 is no guarantee of orthogonality, even with orthogonal columns, as shown by the following counterexample. :

\begin
2 & 0 \\
0 & \frac
\end

With permutation matrices the determinant matches the signature, being +1 or −1 as the parity of the permutation is even or odd, for the determinant is an alternating function of the rows. Stronger than the determinant restriction is the fact that an orthogonal matrix can always be diagonalized over the complex numbers to exhibit a full set of eigenvalues, all of which must have (complex) absolute value 1.

Group properties

The inverse of every orthogonal matrix is again orthogonal, as is the matrix product of two orthogonal matrices. In fact, the set of all n×n orthogonal matrices satisfies all the axioms of a group. It is a compact Lie group of dimension n(n − 1)/2, called the orthogonal group and denoted by O(n). The orthogonal matrices whose determinant is +1 form a path-connected normal subgroup of O(n) of index 2, the special orthogonal group SO(n) of rotations. The quotient group O(n)/SO(n) is isomorphic to O(1), with the projection map choosing [+1] or [−1] according to the determinant. Orthogonal matrices with determinant −1 do not include the identity, and so do not form a subgroup but only a coset; it is also (separately) connected. Thus each orthogonal group falls into two pieces; and because the projection map splits, O(n) is a semidirect product of SO(n) by O(1). In practical terms, a comparable statement is that any orthogonal matrix can be produced by taking a rotation matrix and possibly negating one of its columns, as we saw with 2×2 matrices. If n is odd, then the semidirect product is in fact a direct product, and any orthogonal matrix can be produced by taking a rotation matrix and possibly negating all of its columns. Now consider (n+1)×(n+1) orthogonal matrices with bottom right entry equal to 1. The remainder of the last column (and last row) must be zeros, and the product of any two such matrices has the same form. The rest of the matrix is an n×n orthogonal matrix; thus O(n) is a subgroup of O(n+1) (and of all higher groups). :

\begin
  &      & & 0\\
  & O(n) & & \vdots\\
  &      & & 0\\
0 & \cdots & 0 & 1
\end

Since an elementary reflection in the form of a Householder matrix can reduce any orthogonal matrix to this constrained form, a series of such reflections can bring any orthogonal matrix to the identity; thus an orthogonal group is a reflection group. The last column can be fixed to any unit vector, and each choice gives a different copy of O(n) in O(n+1); in this way O(n+1) is a bundle over the unit sphere Sⁿ with fiber O(n). Similarly, SO(n) is a subgroup of SO(n+1); and any special orthogonal matrix can be generated by Givens plane rotations using an analogous procedure. The bundle structure persists: SO(n) ↪ SO(n+1) → Sⁿ. A single rotation can produce a zero in the first row of the last column, and series of n−1 rotations will zero all but the last row of the last column of an n×n rotation matrix. Since the planes are fixed, each rotation has only one degree of freedom, its angle. By induction, SO(n) therefore has :

(n-1) + (n-2) + \cdots + 1 = n(n-1)/2

degrees of freedom, and so does O(n). Permutation matrices are simpler still; they form, not a Lie group, but only a finite group, the order n! symmetric group S_n. By the same kind of argument, S_n is a subgroup of S_n+1. The even permutations produce the subgroup of permutation matrices of determinant +1, the order n!/2 alternating group.

Canonical form

More broadly, the effect of any orthogonal matrix separates into independent actions on orthogonal two-dimensional subspaces. That is, if Q is special orthogonal then one can always find an orthogonal matrix P, a (rotational) change of basis, that brings Q into block diagonal form: :

P^QP = \begin
R_1 & & \\ & \ddots & \\ & & R_k
\end

(n even),

P^QP = \begin
R_1 & & & \\ & \ddots & & \\ & & R_k & \\ & & & 1
\end

(n odd). where the matrices R₁,...,R_k are 2×2 rotation matrices, and with the remaining entries zero. Exceptionally, a rotation block may be diagonal, ±I. Thus, negating one column if necessary, and noting that a 2×2 reflection diagonalizes to a +1 and −1, any orthogonal matrix can be brought to the form :

P^QP = \begin
\beginR_1 & & \\ & \ddots & \\ & & R_k\end & 0 \\
0 & \begin\pm 1 & & \\ & \ddots & \\ & & \pm 1\end \\
\end

, The matrices R₁,…,R_k give conjugate pairs of eigenvalues lying on the unit circle in the complex plane; so this decomposition confirms that all eigenvalues have absolute value 1. If n is odd, there is at least one real eigenvalue, +1 or −1; for a 3×3 rotation, the eigenvector associated with +1 is the rotation axis.

Lie algebra

Suppose the entries of Q are differentiable functions of t, and that t = 0 gives Q = I. Differentiating the orthogonality condition :

Q^T Q = I \,\!

yields :

\dot^T Q + Q^T \dot = 0

Evaluation at t = 0 (Q = I) then implies :

\dot^T = -\dot .

In Lie group terms, this means that the Lie algebra of an orthogonal matrix group consists of skew-symmetric matrices. Going the other direction, the matrix exponential of any skew-symmetric matrix is an orthogonal matrix (in fact, special orthogonal). For example, the three-dimensional object physics calls angular velocity is a differential rotation, thus a vector in the Lie algebra

\mathfrak(3)

tangent to SO(3). Given ω = (xθ,yθ,zθ), with v = (x,y,z) a unit vector, the correct skew-symmetric matrix form of ω is :

\Omega = \begin
0 & -z\theta & y\theta \\
z\theta & 0 & -x\theta \\
-y\theta & x\theta & 0
\end .

The exponential of this is the orthogonal matrix for rotation around axis v by angle θ; setting c = cos θ/2, s = sin θ/2, :

\exp(\Omega) = 
\begin
1  -  2s^2  +  2x^2 s^2  &  2xy s^2  -  2z sc  &  2xz s^2  +  2y sc\\
2xy s^2  +  2z sc  &  1  -  2s^2  +  2y^2 s^2  &  2yz s^2  -  2x sc\\
2xz s^2  -  2y sc  &  2yz s^2  +  2x sc  &  1  -  2s^2  +  2z^2 s^2   
\end
.

Numerical linear algebra

Benefits

Numerical analysis takes advantage of many of the properties of orthogonal matrices for numerical linear algebra, and they arise naturally. For example, it is often desirable to compute an orthonormal basis for a space, or an orthogonal change of bases; both take the form of orthogonal matrices. Having determinant ±1 and all eigenvalues of magnitude 1 is of great benefit for numeric stability. One implication is that the condition number is 1 (which is the minimum), so errors are not magnified when multiplying with an orthogonal matrix. Many algorithms use orthogonal matrices like Householder reflections and Givens rotations for this reason. It is also helpful that, not only is an orthogonal matrix invertible, but its inverse is available essentially free, by exchanging indices. Permutations are essential to the success of many algorithms, including the workhorse Gaussian elimination with partial pivoting (where permutations do the pivoting). However, they rarely appear explicitly as matrices; their special form allows more efficient representation, such as a list of n indices. Likewise, algorithms using Householder and Givens matrices typically use specialized methods of multiplication and storage. For example, a Givens rotation affects only two rows of a matrix it multiplies, changing a full multiplication of order n³ to a much more efficient order n. When uses of these reflections and rotations introduce zeros in a matrix, the space vacated is enough to store sufficient data to reproduce the transform, and to do so robustly. (Following Stewart (1976), we do not store a rotation angle, which is both expensive and badly behaved.)

Decompositions

A number of important matrix decompositions (Golub & Van Loan, 1996) involve orthogonal matrices, including especially: :;QR decomposition: M = QR, Q orthogonal, R upper triangular :;Singular value decomposition: M = UΣV^T, U and V orthogonal, Σ non-negative diagonal :;Spectral decomposition: S = QΛQ^T, S symmetric, Q orthogonal, Λ diagonal. :;Polar decomposition: M = QS, Q orthogonal, S symmetric non-negative definite

Examples

Consider an overdetermined system of linear equations, as might occur with repeated measurements of a physical phenomenon to compensate for experimental errors. Write Ax = b, where A is m×n, m > n. A QR decomposition reduces A to upper triangular R. For example, if A is 5×3 then R has the form :

R = \begin
\star & \star & \star \\
0 & \star & \star \\
0 & 0 & \star \\
0 & 0 & 0 \\
0 & 0 & 0
\end

The linear least squares problem is to find the x that minimizes ‖Ax−b‖, which is equivalent to projecting b to the subspace spanned by the columns of A. (Think of a pigeon flying over a parking lot at noon; its shadow hits the nearest point on the ground.) Assuming the columns of A (and hence R) are independent, the projection solution is found from A^TAx = A^Tb. Now A^TA is square (n×n) and invertible, and also equal to R^TR. But the lower rows of zeros in R are superfluous in the product, which is thus already in lower-triangular upper-triangular factored form, as in Gaussian elimination (Cholesky decomposition). Here orthogonality is important not only for reducing A^TA = (R^TQ^T)QR to R^TR, but also for allowing solution without magnifying numerical problems. In the case of a linear system which is underdetermined, or an otherwise non-invertible matrix, singular value decomposition (SVD) is equally useful. With A factored as UΣV^T, a satisfactory solution uses the Moore-Penrose pseudoinverse, VΣ⁺U^T, where Σ⁺ merely replaces each non-zero diagonal entry with its reciprocal. Set x to VΣ⁺U^Tb. The case of a square invertible matrix also holds interest. Suppose, for example, that A is a 3×3 rotation matrix which has been computed as the composition of numerous twists and turns. Floating point does not match the mathematical ideal of real numbers, so A has gradually lost its true orthogonality. A Gram-Schmidt process could orthogonalize the columns, but it is not the most reliable, nor the most efficient, nor the most invariant method. The polar decomposition factors a matrix into a pair, one of which is the unique closest orthogonal matrix to the given matrix, or one of the closest if the given matrix is singular. (Closeness can be measured by any matrix norm invariant under an orthogonal change of basis, such as the spectral norm or the Frobenius norm.) For a near-orthogonal matrix, rapid convergence to the orthogonal factor can be achieved by a "Newton's method" approach due to Higham (1986) (1990), repeatedly averaging the matrix with its inverse transpose. Dubrulle (1994) has published an accelerated method with a convenient convergence test. For example, consider a (very!) non-orthogonal matrix for which the simple averaging algorithm takes seven steps :

\begin3 & 1\\7 & 5\end
\rightarrow
\begin1.8125 & 0.0625\\3.4375 & 2.6875\end
\rightarrow \cdots \rightarrow
\begin0.8 & -0.6\\0.6 & 0.8\end

and which acceleration trims to two steps (with γ = 0.353553, 0.565685). :

\begin3 & 1\\7 & 5\end
\rightarrow
\begin1.41421 & -1.06066\\1.06066 & 1.41421\end
\rightarrow
\begin0.8 & -0.6\\0.6 & 0.8\end

Gram-Schmidt yields an inferior solution, shown by a Frobenius distance of 8.28659 instead of the minimum 8.12404. :

\begin3 & 1\\7 & 5\end
\rightarrow
\begin0.393919 & -0.919145\\0.919145 & 0.393919\end

Randomization

Some numerical applications, such as Monte Carlo methods and exploration of high-dimensional data spaces, require generation of uniformly distributed random orthogonal matrices. In this context, "uniform" is defined in terms of Haar measure, which essentially requires that the distribution not change if multiplied by any freely chosen orthogonal matrix. It does not work to fill a matrix with independent uniformly distributed random entries and then orthogonalize it. It does work to fill it with independent normally distributed random entries, then use QR decomposition. Stewart (1980) replaced this with a more efficient idea that Diaconis & Shahshahani (1987) later generalized as the "subgroup algorithm" (in which form it works just as well for permutations and rotations). To generate an (n+1)×(n+1) orthogonal matrix, take an n×n one and a uniformly distributed unit vector of dimension n+1. Construct a Householder reflection from the vector, then apply it to the smaller matrix (embedded in the larger size with a 1 in the bottom corner).

Spin and Pin

A subtle technical problem afflicts some uses of orthogonal matrices. Not only are the group components with determinant +1 and −1 not connected to each other, even the +1 component, SO(n), is not simply connected (except for SO(1), which is trivial). Thus it is sometimes advantageous, or even necessary, to work with a covering group of SO(n), the spin group, Spin(n). Likewise, O(n) has covering groups, the pin groups, Pin(n). For n > 2, Spin(n) is simply connected, and thus the universal coveri

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