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Coordinates (elementary Mathematics)

Coordinates (elementary mathematics)

This article describes some of the common coordinate systems that appear in elementary mathematics. For advanced topics, please refer to coordinate system. For more background, see Cartesian coordinate system. The coordinates of a point are the components of a tuple of numbers used to represent the location of the point in the plane or space. A coordinate system is a plane or space where the origin and axes are defined so that coordinates can be measured.

Cartesian coordinates

Image:cartesiancoordinates2D.JPG In the two-dimensional Cartesian coordinate system, a point P in the xy-plane is represent by a tuple of two components

(x, y)

.
-

x

is the signed distance from the y-axis to the point P, and
-

y

is the signed distance from the x-axis to the point P. In the three-dimensional Cartesian coordinate system, a point P in the xyz-space is represent by a tuple of three components

(x, y, z)

.
-

x

is the signed distance from the yz-plane to the point P,
-

y

is the signed distance from the xz-plane to the point P, and
-

z

is the signed distance from the xy-plane to the point P. For advanced topics, please refer to Cartesian coordinate system.

Polar coordinates

The polar coordinate systems are coordinate systems in which a point is identified by a distance from some fixed feature in space and one or more subtended angles. They are the most common systems of curvilinear coordinates. The term polar coordinates often refers to circular coordinates (two-dimensional). Other commonly used polar coordinates are cylindrical coordinates and spherical coordinates (both three-dimensional).

Circular coordinates

The circular coordinate system, commonly referred to as the polar coordinate system, is a two-dimensional polar coordinate system, defined by an origin, O, and a semi-infinite line L leading from this point. L is also called the polar axis. In terms of the Cartesian coordinate system, one usually picks O to be the origin (0,0) and L to be the positive x-axis (the right half of the x-axis). Image:CircularCoordinates.png In the circular coordinate system, a point P is represented by a tuple of two components

(r, \theta)

. Using terms of the Cartesian coordinate system,
-

0\leq

(radius) is the distance from the origin to the point P, and
-

0\leq\theta<360^\circ

(azimuth) is the angle between the positive x-axis and the line from the origin to the point P. Possible coordinate transformations from one circular coordinate system to another include:
- change of zero direction
- changing from the angle increasing anticlockwise to increasing clockwise or conversely
- change of scale and combinations. More generally, transformations of the corresponding Cartesian coordinates can be translated into transformations from one circular coordinate system to another by basically transforming to Cartesian coordinates, transforming those, and transforming back to circular coordinates. This is e.g needed for:
- change of origin
- change of scale in one direction A minor change is changing the range

0\leq\theta<360^\circ

to e.g.

-180^\circ<\theta\leq180^\circ

Circular coordinates can be convenient in situations where only the distance, or only the direction to a fixed point matters, rotations about a point, etc. (by taking the special point as the origin). A complex number can be viewed as a point or a position vector on a plane, the so-called complex plane or Argand diagram. Here the circular coordinates are r = |z|, called the absolute value or modulus of z, and φ = arg(z), called the complex argument of z. These coordinates (mod-arg form) are especially convenient for complex multiplication and powers.

Cylindrical coordinates

The cylindrical coordinate system is a three-dimensional polar coordinate system. Image:CylindricalCoordinates.png In the cylindrical coordinate system, a point P is represented by a tuple of three components

(r, \theta, h)

. Using terms of the Cartesian coordinate system,
-

0\leq

(radius) is the distance between the z-axis and the point P,
-

0\leq\theta<360^\circ

(azimuth or longitude) is the angle between the positive x-axis and the line from the origin to the point P projected onto the xy-plane, and
-

h

(height) is the signed distance from xy-plane to the point P. : Note: some sources use

z

for

h

; there is no "right" or "wrong" convention, but it is necessary to be aware of the convention being used. Cylindrical coordinates involve some redundancy;

\theta

loses its significance if

r=0

. Cylindrical coordinates are useful in analyzing systems that are symmetrical about an axis. For example the infinitely long cylinder that has the Cartesian equation

x^2+y^2=c^2

has the very simple equation

r=c

in cylindrical coordinates.

Spherical coordinates

The spherical coordinate system is a three-dimensional polar coordinate system. 500px In the spherical coordinate system, a point

P

is represented by a tuple of three components

(\rho, \phi, \theta)

. Using terms of the Cartesian coordinate system,
-

0\leq\rho

(radius) is the distance between the point

P

and the origin,
-

0\leq\phi\leq 180^\circ

(zenith, colatitude or polar angle) is the angle between the

z

-axis and the line from the origin to the point P, and
-

0\leq\theta<360^\circ

(azimuth or longitude) is the angle between the positive

x

-axis and the line from the origin to the point P projected onto the

xy

-plane. NB: The above convention is the standard used by American mathematicians and American calculus textbooks. However, most physicists, engineers, and non-American mathematicians interchange the symbols

\phi

and

\theta

above, using

\phi

to denote the azimuth and

\theta

the colatitude. One should be very careful to note which convention is being used by a particular author. It should be noted that, regardless of how one labels the coordinates, one argument against the conventional American mathematical definition is the fact that it produces a left-handed coordinate system, rather than the usual convention of a right-handed coordinate system. One argument for it however is that it more closely resembles two dimensional polar notation where

\theta

ranges from 0 to

2\pi

. The latitude

\delta

is the complement of the colatitude

\phi

\delta=90^\circ-\phi

. The latitude is the angle between the

xy

-plane (the equator) and the line from the origin to the point P. Although here indicated with a

\delta

, the latitude is usually also indicated with the symbol

\phi

. The spherical coordinate system also involves some redundancy;

\phi

loses its significance if

\rho=0

, and

\theta

loses its significance if

\rho=0

\phi=0

\phi=180^\circ

. To construct a point from its spherical coordinates: from the origin, go

along the positive

z

-axis, rotate

\phi

about

y

-axis toward the direction of the positive

x

-axis, and rotate

\theta

about the

z

-axis toward the direction of the positive

y

-axis. Spherical coordinates are useful in analyzing systems that are symmetrical about a point; a sphere that has the Cartesian equation

x^2+y^2+z^2=c^2

has the very simple equation

\rho=c

in spherical coordinates. Spherical coordinates are the natural coordinates for physical situations where there is spherical symmetry. In such a situation, one can describe waves using spherical harmonics. Another application is ergonomic design, where

is the arm length of a stationary person and the angles describe the direction of the arm as it reaches out. The concept of spherical coordinates can be extended to higher dimensional spaces and are then referred to as hyperspherical coordinates.

Conversion between coordinate systems

Cartesian and circular

x=r\,\cos\theta \quad

y=r\,\sin\theta \quad

r=\sqrt

\theta
= \arctan\frac + \pi u_0(-x) \, \operatorname y

where u₀ is the Heaviside step function with

u_0(0)=0

and sgn is the signum function. Here the u₀ and sgn functions are being used as "logical" switches which are used as shorthand substitutes for several if ... then statements. Some computer languages include a bivariate arctangent function atan2(y,x) which finds the value for θ in the correct quadrant given x and y.

Cartesian and cylindrical

x=r\,\cos\theta

y=r\,\sin\theta

z=h \quad

r=\sqrt

\theta
=\arctan\frac + \pi u_0(-x) \, \operatorname y

h=z \quad

\begindx\\dy\\dz\end=
\begin
\cos\theta&-r\sin\theta&0\\
\sin\theta&r\cos\theta&0\\
0&0&1
\end
\begindr\\d\theta\\dh\end

\begindr\\d\theta\\dh\end=
\begin
\frac&\frac&0\\
\frac&\frac&0\\
0&0&1
\end
\begindx\\dy\\dz\end

Cartesian and spherical

=\rho \, \sin\phi \, \cos\theta \quad

=\rho \, \sin\phi \, \sin\theta \quad

=\rho \, \cos\phi \quad

=\sqrt

=\arccos\frac

=\arctan\frac

=\arctan\frac + \pi\, u_0(-x)\, \operatorname y

\begindx\\dy\\dz\end=
\begin
\sin\phi\cos\theta&\rho\cos\phi\cos\theta&-\rho\sin\phi\sin\theta\\
\sin\phi\sin\theta&\rho\cos\phi\sin\theta&\rho\sin\phi\cos\theta\\
\cos\phi&-\rho\sin\phi&0
\end
\begind\rho\\d\phi\\d\theta\end

\begind\rho\\d\phi\\d\theta\end=
\begin
\frac&\frac&\frac\\
\frac&\frac&\frac\\
\frac&\frac&0
\end
\begindx\\dy\\dz\end

Cylindrical and spherical

=\rho \,\sin\phi

=\theta \quad

=\rho \,\cos\phi

=\sqrt

=\arctan\frac + \pi \, u_0(-r) \, \operatorname h

=\theta \quad

\begindr\\d\theta\\dh\end=
\begin
\sin\phi&\rho\cos\phi&0\\
0&0&1\\
\cos\phi&-\rho\sin\phi&0
\end
\begind\rho\\d\phi\\d\theta\end

\begind\rho\\d\phi\\d\theta\end=
\begin
\frac&0&\frac\\
\frac&0&\frac\\
0&1&0
\end
\begindr\\d\theta\\dh\end

External links

- Frank Wattenberg has made some attractive animations illustrating [http://www.math.montana.edu/frankw/ccp/multiworld/multipleIVP/spherical/body.htm spherical] and [http://www.math.montana.edu/frankw/ccp/multiworld/multipleIVP/cylindrical/body.htm cylindrical] coordinate systems.
- http://www.physics.oregonstate.edu/bridge/papers/spherical.pdf is a description of the different conventions in use for naming components of spherical coordinates, along with a proposal for standardizing this. Category:Coordinate systems Category:Elementary mathematics ko:극좌표 ja:極座標系

Coordinate system

See Cartesian coordinate system or Coordinates (mathematics) for a more elementary introduction to this topic. In mathematics as applied to geometry, physics or engineering, a coordinate system is a system for assigning a tuple of numbers to each point in an n-dimensional space. "Numbers" in many cases means real numbers, but, depending on context, can mean complex numbers or elements of some other field. If the space or manifold is curved, it may not be possible to provide one consistent coordinate system for the entire space. In this case, a set of coordinate systems, called charts, are put together to form an atlas covering the whole space. When the space has some additional algebraic structure, then the co-ordinates will also transform under rings or groups; a particularly famous example in this case are the Lie groups. Although any specific coordinate system is useful for numerical calculations in a given space, the space itself is considered to exist independently of any particular choice of coordinates. By convention the origin of the coordinate system in Cartesian coordinates is the point (0, 0, ..., 0), which may be assigned to any given point of Euclidean space. In physics, a scalar is a physical quantity which assumes a single value which is a "real" quantity independent of the coordinate system. In this sense coordinates are not scalars (although, of course, a scalar field can be defined which for one particular coordinate system corresponds to a particular coordinate). In some coordinate systems some points are associated with multiple tuples of coordinates, e.g. the origin in polar coordinates: r = 0 but θ can be any angle.

Examples

An example of a coordinate system is to describe a point P in the Euclidean space Rⁿ by an n-tuple :P = (r₁, ..., r_n) of real numbers :r₁, ..., r_n. These numbers r₁, ..., r_n are called the coordinates of the point P. If a subset S of a Euclidean space is mapped continuously onto another topological space, this defines coordinates in the image of S. That can be called a parametrization of the image, since it assigns numbers to points. That correspondence is unique only if the mapping is bijective. The system of assigning longitude and latitude to geographical locations is a coordinate system. In this case the parametrization fails to be unique at the north and south poles.

Transformations

A coordinate transformation is a conversion from one system to another, to describe the same space. With every bijection from the space to itself two coordinate transformations can be associated:
- such that the new coordinates of the image of each point are the same as the old coordinates of the original point (the formulas for the mapping are the inverse of those for the coordinate transformation)
- such that the old coordinates of the image of each point are the same as the new coordinates of the original point (the formulas for the mapping are the same as those for the coordinate transformation) For example, in 1D, if the mapping is a translation of 3, the first moves the origin from 0 to 3, so that the coordinate of each point becomes 3 less, while the second moves the origin from 0 to -3, so that the coordinate of each point becomes 3 more. If the bijection is an involution, e.g. a reflection, then the two associated coordinate transformations are the same, e.g., in 1D, x becomes 7-x. Examples of bijections include the invertible affine transformations. Of these, the similarity transformations preserve distance ratios, hence magnitude ratios, and angles, so that e.g. decomposition of a vector into perpendicular components is preserved. In the case that vector quantities are considered in relation to position and displacement, as in vector fields, a similarity transformation of space is normally accompanied by a corresponding linear transformation of the other vector quantities, to preserve angles between e.g. a force and a displacement, hence preserve e.g. dot products up to scaling. The transformation is linear because, as opposed to position, most vector quantities have a natural origin, e.g. zero force. However, velocity translation preserves the laws of motion, because an inertial frame of reference is preserved. (But if there is e.g. air-resistance, a velocity translation will affect tacitly assumed stationarity of air.) In diagrams showing vectors of multiple physical dimensions, e.g. forces and displacements, scaling of one kind of vectors does not affect relevant properties: a force and a displacement having the same length in a diagram has no particular significance.

Singularities

Some choices of coordinate systems may lead to paradoxes, for example, close to a black hole, but can be understood by changing the choice of coordinate system. At an actual mathematical singularity the coordinate system breaks down.

Systems commonly used

Some coordinate systems are the following:
- The Cartesian coordinate system (also called the "rectangular coordinate system"), which, for three-dimensional flat space, uses three numbers representing distances.
- For any finite-dimensional vector space and any basis, the coefficients of the basis vectors can be used as coordinates. Changing the basis is a coordinate transformation, a linear transformation that can be summarized by a matrix, and is computationally the same as a mapping of points to other points keeping the bases the same: e.g. in 2D:
- a clockwise rotation is a mapping of points to other points which changes the coordinates the same as keeping the points in place but rotating the coordinate axes anti-clockwise. The [http://en.wikibooks.org/wiki/Modern_Physics:Math:Vectors rotation of coordinate systems] is covered in depth on wikibooks.
- an expansion by a factor two in the direction of one basis vector is a mapping of points to other points which changes the coordinates the same as keeping the points in place but halving the magnitude of that basis vector (in both cases the corresponding coordinate is doubled).
- a mapping of points to other points which distorts a rectangle to a parallelogram changes the coordinates the same as keeping the points in place but changing the basis vectors from being two sides of that parallelogram to perpendicular ones, two sides of that rectangle.
- Curvilinear coordinates are a generalization of coordinate systems generally; the system is based on the intersection of curves.
- The polar coordinate systems:
- Cylindrical coordinate system represents a point in space by an angle, a distance from the origin and a height.
- Spherical coordinate system represents a point in space with two angles and a distance from the origin.
- Geographic coordinate system
- Generalized coordinates are used in the Lagrangian treatment of mechanics.
- Canonical coordinates are used in the Hamiltonian treatment of mechanics.
- Intrinsic coordinates describe a point upon a curve by the length of the curve to that point and the angle the tangent to that point makes with the x-axis.

Astronomical systems

- Celestial coordinate system
- Horizontal coordinate system
- Equatorial coordinate system - based on Earth rotation
- Ecliptic coordinate system - based on Solar System rotation
- Galactic coordinate system - based on Milky Way rotation
- extragalactic coordinate systems
- supergalactic coordinate system - based on plane of local supercluster of galaxies
- comoving coordinates - valid to particle horizon
- Binary coordinate system

Angle

This article is about angles in geometry. For other articles, see Angle (disambiguation) ---- An Angle (from the Lat. angulus, a corner, a diminutive, of which the primitive form, angus, does not occur in Latin; cognate are the Lat. angere, to compress into a bend or to strangle, and the Greek (angulοs) crooked, curved; both connected with the Aryan or Indo-European root ank-, to bend) is the figure formed by two rays sharing a common endpoint, called the vertex of the angle. Angles provide a means of expressing the difference in slope between two rays meeting at a vertex without the need to explicitly define the slopes of the two rays. Angles are studied in geometry and trigonometry. Euclid defines a plane angle as the inclination to each other, in a plane, of two lines which meet each other, and do not lie straight with respect to each other. According to Proclus an angle must be either a quality or a quantity, or a relationship. The first concept was used by Eudemus, who regarded an angle as a deviation from a straight line; the second by Carpus of Antioch, who regarded it as the interval or space between the intersecting lines; Euclid adopted the third concept, although his definitions of right, acute, and obtuse angles are certainly quantitative.

Units of measure for angles

In order to measure an angle, a circle centered at the vertex is drawn. Since the circumference of a circle is always directly proportional to the length of its radius, the measure of the angle is independent of the size of the circle. Note that angles are dimensionless, since they are defined as the ratio of lengths.
- The radian measure of the angle is the length of the arc cut out by the angle, divided by the circle's radius. The SI system of units uses radians as the (derived) unit for angles. Because of the relationship to arc length, radians are a special unit. Sines and cosines whose argument is in radians have particular analytic properties, just as do exponential functions in the base e. (As we've discovered, this is no coincidence).
- The degree measure of the angle is the length of the arc, divided by the circumference of the circle, and multiplied by 360. The symbol for degrees is a small superscript circle, as in 360°. 2π radians is equal to 360° (a full circle), so one radian is about 57° and one degree is π/180 radians. Degrees are further broken down into minutes of arc and seconds of arc, which are 1/60th and 1/3600th of a degree, respectively. Minutes of arc are commonly encountered in discussions of external ballistics, as a minute of arc covers almost exactly 1 inch at 100 yards (1 m at 1200 m). A rifle capable of shooting "1 MOA", one minute of arc, can place all shots within 1 inch at 100 yards, 2 inches at 200 yards, etc. Minutes of arc were also used in navigation, and a nautical mile is roughly defined as one minute of arc of the earth's surface.
- The grad, also called grade, gradian or gon, is an angular measure where the arc is divided by the circumference, and multiplied by 400. It is used mostly in triangulation.
- The point is used in navigation, and is defined as 1/32 of a circle, or exactly 11.25°.
- The full circle or full turns represents the number or fraction of complete full turns. For example, π/2 radians = 90° = 1/4 full circle

Conventions on measurement

A convention universally adopted in mathematical writing is that angles given a sign are positive angles if measured counterclockwise, and negative angles if measured clockwise, from a given line. If no line is specified, it can be assumed to be the x-axis in the Cartesian plane. In navigation, bearings are measured from north, increasing clockwise, so a bearing of 45 is north-east. Negative bearings are not used in navigation, so north-west is 315. In mathematics radians are assumed unless specified otherwise because this removes the arbitrariness of the number 360 in the degree system and because the trigonometric functions can be developed into particularly simple Taylor series if their arguments are specified in radians.

Types of angles

An angle of π/2 radians or 90°, one-quarter of the full circle is called a right angle. Two line segments, rays, or lines (or any combination) which form a right angle are said to be either perpendicular or orthogonal:
- Angles smaller than a right angle are called acute angles
- Angles larger than a right angle are called obtuse angles.
- Angles equal to two right angles are called straight angles.
- Angles larger than two right angles are called reflex angles.
- The difference between an acute angle and a right angle is termed the complement of the angle
- The difference between an angle and two right angles is termed the supplement of the angle.

Some facts

In Euclidean geometry, the inner angles of a triangle add up to π radians or 180°; the inner angles of a quadrilateral add up to 2π radians or 360°. In general, the inner angles of a simple polygon with n sides add up to (n − 2) × π radians or (n − 2) × 180°. If two straight lines intersect, four angles are formed. Each one has an equal measure to the angle across from it; these congruent angles are called vertical angles. If a straight transversal line intersects two parallel lines, corresponding (alternate) angles at the two points of intersection are equal; adjacent angles are supplementary, that is they add to π radians or 180°.

A formal definition

A Euclidean angle is completely determined by the corresponding right triangle. In particular, if

\theta

is a Euclidean angle, it is true that :

\cos \theta = \frac

and :

\sin \theta = \frac

for two numbers

x

and

y

. So an angle can be legitimately given by two numbers

x

and

y

, or by a ratio

\frac

. What's more, to any such ratio there corresponds exactly one angle, since :

\frac = \frac = \frac

(i.e., changing the ratio will necessarily change the sin and cos, which in the geometric range

0 < \theta < 2\pi

are one-to-one - one sin or cos corresponds to one

\theta

Angles in different contexts

In the Euclidean plane, the angle θ between two vectors u and v is related to their dot product and their lengths by the formula :

\mathbf \cdot \mathbf = \cos(\theta)\ \|\mathbf\|\ \|\mathbf\|.

This allows one to define angles in any real inner product space, replacing the Euclidean dot product · by the Hilbert space inner product <·,·>. The angle between a line and a curve (mixed angle) or between two intersecting curves (curvilinear angle) is defined to be the angle between the tangents at the point of intersection. Various names (now rarely, if ever, used) have been given to particular cases:—amphicyrtic (Gr. ἀμφί, on both sides, κυρτόσ, convex) or cissoidal (Gr. κισσόσ, ivy), biconvex; xystroidal or sistroidal (Gr. ξυστρίσ, a tool for scraping), concavo-convex; amphicoelic (Gr. κοίλη, a hollow) or angulus lunularis, biconcave. Two intersecting planes form an angle, called their dihedral angle. It is defined as the angle between two lines normal to the planes. Also a plane and an intersecting line form an angle. This angle is equal to π/2 radians minus the angle between the intersecting line and the line that goes through the point of intersection and is perpendicular to the plane.

Angles in Riemannian geometry

In Riemannian geometry, the metric tensor is used to define the angle between two tangents. Where U and V are tangent vectors and g_ij are the components of the metric tensor G, :

\cos \theta = \frac
.

Angles in astronomy

In astronomy, one can measure the angular separation of two stars by imagining two lines through the Earth, each one intersecting one of the stars. Then the angle between those lines can be measured; this is the angular separation between the two stars. Astronomers also measure the apparent size of objects. For example, the full moon has an angular measurement of approximately 0.5°, when viewed from Earth. One could say, "The Moon subtends an angle of half a degree." The small-angle formula can be used to convert such an angular measurement into a distance/size ratio.

Angles in maritime navigation

The modern format of angle used to indicate longitude or latitude is hemisphere degree minute.decimal, where there are 60 minutes in a degree, for instance N 51 23.438 or E 090 58.928. The obsolete (but still commonly used) format of angle used to indicate longitude or latitude is hemisphere degree minute' second", where there are 60 minutes in a degree and 60 seconds in a minute, for instance N 51 23′26″ or E 090 58′57″

External links

- [http://www.unitconversion.org/unit_converter/angle.html Online Angle Converter - convert between various units of angle, such as degree, radian, grad, gon, minute, second, sign, mil, and so on]
- [http://www.unitconversion.org/unit_converter/angle-v.html Interactive Angle Conversion Table - convert selected unit to all other units of angle]
- [http://www.cut-the-knot.org/triangle/ABisector.shtml Angle Bisectors] at cut-the-knot
- [http://www.cut-the-knot.org/Curriculum/Geometry/PerpBiInQuadri.shtml Angle Bisectors and Perpendiculars in a Quadrilateral] at cut-the-knot
- [http://www.cut-the-knot.org/Curriculum/Geometry/CyQuadri.shtml Angle Bisectors in a Quadrilateral] at cut-the-knot
- [http://www.cut-the-knot.org/triangle/TriangleFromBisectors.shtml Constructing a triangle from its angle bisectors] at cut-the-knot
- [http://calc.skyrocket.de/en/ Online Unit Converter - Conversion of many different units] Category:Elementary geometry Category:Trigonometry
- ko:각도 ja:角度 simple:Angle

Curvilinear coordinates

Curvilinear coordinates are a coordinate system based on some transformation of the standard coordinate system. We need the same number of coordinates. If we consider the 2D case, then instead of Cartesian coordinates x and y we use e.g. p and q; the level curves of p and q in the xy-plane, as well as those of x and y in the pq-plane are in general curved. Required is that the transformation is locally invertible at each point. This means that we can convert a point given in one coordinate system to its curvilinear coordinates and back. Depending on the application, a curvilinear coordinate system may be simpler than the Cartesian coordinate system. This also has consequences that we can express many of the concepts in vector calculus which are given in Cartesian or spherical coordinates or any other arbitrary coordinate system, also in curvilinear coordinates.

Terminology

In R³, for example, if we have some transformation :

H(x_1', x_2', x_3')=(x_1, x_2, x_3)=\mathbf

giving curvilinear coordinates x₁′, x₂′,x₃′, for x₁, x₂, x₃, if this transformation is locally invertible everywhere, the Jacobian determinant :

\partial(x_1, x_2, x_3) \over \partial(x_1', x_2', x_3')

is nonzero, and for this to happen, the vectors :

must form a basis for R³. From these basis vectors, we define scale factors :

h_=h_i=\left|\right|

and thus arrive at the unit basis vectors for the curvilinear coordinates to be :

\mathbf_=1/h_i

Note that the coordinate system we choose need not be orthogonal, but for the purposes of this article, they are treated as being so. The system is orthogonal iff :

\cdot  = \delta_

where δ_ij is the Kronecker delta.

Example

If we consider polar coordinates for R², note that :

(r \mathrm\theta, r \mathrm \theta) = (x, y)\,\!

(r, θ) are the curvilinear coordinates, and the Jacobian determinant of the transformation (r,θ) → (r cos θ, r sin θ) is r. The basis vectors are b_r = (cos θ, sin θ), b_θ = (-r sin θ, r cos θ), with unit basis vectors e_r = (cos θ, sin θ), e_θ = (- sin θ, cos θ) with scale factors h_r = 1 and h_θ= r.

Line, surface, and volume integrals

Since we use curvilinear coordinates to aid in the calculation in vector calculus, there are adjustments we need to make in the calculation of line, surface and volume integrals.

Line integrals

Normally in the calculation of line integrals we are interested in calculating :

\int_C f \,ds = \int_a^b f(\mathbf(t))\left|\right|\; dt

where x(t) parametrizes C in Cartesian coordinates. In curvilinear coordinates, the term :

\left|\right| = \left|\right|

by the chain rule. But from the definition of the curvilinear coordinates, :

= h_i \mathbf_

and thus :

\left|\right| = \sqrt

and we can proceed normally.

Surface integrals

Likewise, if we are interested in a surface integral, the relevant calculation, with the parametrisation of the surface in Cartesian coordinates is: :

\int_S f \,ds = \iint_T f(\mathbf(s, t)) \left|\times \right| ds dt

Again, in curvilinear coordinates, the term :

\left|\times \right| = \left| \times \right|

and we make use of the definition of curvilinear coordinates again to yield :

= \sum  h_ \mathbf_

and :

= \sum  h_ \mathbf_

where the cross product, in terms of curvilinear coordinates, will be: :

\begin 
\mathbf_                    & \mathbf_                    & \mathbf_ \\
 && \\
h_1  & h_2  & h_3  \\
&& \\
h_1  & h_2  & h_3  \end

Div, curl, grad

In curvilinear coordinates, we can express the divergence, curl and gradient of a function or vector field as follows: :

\nabla f = \sum  \mathbf_

\nabla\times F =  \begin h_1 \partial_1 \\ \vdots \\ h_n \partial_n \end\times F

\nabla\cdot F = \sum 1/h_i

Category:Coordinate systems

Circular coordinates

Coordinates (mathematics)#Cartesian and circular

Cylindrical coordinates

Coordinates (mathematics)#Cylindrical coordinates

RADIUS

RADIUS (Remote Authentication Dial In User Service) is an AAA (authentication, authorization and accounting) protocol for applications such as network access or IP mobility. It is intended to work in both local and roaming situations. When you connect to an ISP using a modem, DSL, cable or wireless connection, you must enter your username and password. This information is passed to a Network Access Server (NAS) device over the Point-to-Point Protocol (PPP), then to a RADIUS server over the RADIUS protocol. The RADIUS server checks that the information is correct using authentication schemes like PAP, CHAP or EAP. If accepted, the server will then authorize access to the ISP system and select an IP address, L2TP parameters, etc. The RADIUS server will also be notified when the session starts and stops, so that the user can be billed accordingly; or the data can be used for statistical purposes. RADIUS was originally developed by Livingston Enterprises for their PortMaster series of Network Access Servers, but later (1997) published as RFC 2058 and RFC 2059 (current versions are RFC 2865 and RFC 2866). Now, several commercial and open-source RADIUS servers exist. Features can vary, but most can look up the users in text files, LDAP servers, various databases, etc. Accounting tickets can be written to text files, various databases, forwarded to external servers, etc. SNMP is often used for remote monitoring. RADIUS proxy servers are used for centralized administration and can rewrite RADIUS packets on the fly (for security reasons, or to convert between vendor dialects). RADIUS is extensible; most vendors of RADIUS hardware and software implement their own dialects. The DIAMETER protocol is the planned replacement for RADIUS, but is still backwards compatible.

Standards

The RADIUS protocol is currently defined in:
- RFC 2865 Remote Authentication Dial In User Service (RADIUS)
- RFC 2866 RADIUS Accounting Other relevant RFCs are:
- RFC 2548 Microsoft Vendor-specific RADIUS Attributes
- RFC 2607 Proxy Chaining and Policy Implementation in Roaming
- RFC 2618 RADIUS Authentication Client MIB
- RFC 2619 RADIUS Authentication Server MIB
- RFC 2620 RADIUS Accounting Client MIB
- RFC 2621 RADIUS Accounting Server MIB
- RFC 2809 Implementation of L2TP Compulsory Tunneling via RADIUS
- RFC 2867 RADIUS Accounting Modifications for Tunnel Protocol Support
- RFC 2868 RADIUS Attributes for Tunnel Protocol Support
- RFC 2869 RADIUS Extensions
- RFC 2882 Network Access Servers Requirements: Extended RADIUS Practices
- RFC 3162 RADIUS and IPv6
- RFC 3576 Dynamic Authorization Extensions to RADIUS

External links

- [http://www.untruth.org/~josh/security/radius/radius-auth.html An Analysis of the RADIUS Authentication Protocol]
- [http://www.freeradius.org/rfc/attributes.html List of RADIUS attributes] Category:Authentication methods Category:Internet protocols Category:Internet standards ja:RADIUS

Complex number

In mathematics, a complex number is an expression of the form a + bi, where a and b are real numbers, and i stands for the square root of minus one (−1), which cannot be represented by any real number. For example, :3 + 2i is a complex number, where 3 is called the real part and 2 the imaginary part. Since a complex number a + bi is uniquely specified by an ordered pair (a, b) of real numbers, the complex numbers are in one-to-one correspondence with points on a plane, called the complex plane. The set of all complex numbers is usually denoted by C, or in blackboard bold by

\mathbb

. It includes the real numbers because every real number can be regarded as complex: a = a + 0i. Complex numbers are added, subtracted, and multiplied by formally applying the associative, commutative and distributive laws of algebra, together with the equation i ² = −1: :(a + bi) + (c + di) = (a+c) + (b+d)i :(a + bi) − (c + di) = (a−c) + (b−d)i :(a + bi)(c + di) = ac + bci + adi + bd i ² = (ac−bd) + (bc+ad)i Division of complex numbers can also be defined (see below). Thus the set of complex numbers forms a field which, in contrast to the real numbers, is algebraically closed. In mathematics, the adjective "complex" means that the field of complex numbers is the underlying number field considered, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra.

Definition

The complex number field

Formally, the complex numbers can be defined as ordered pairs of real numbers (a, b) together with the operations:
-

( a , b ) + ( c , d ) = ( a + c , b + d ) \,

( a , b ) \cdot ( c , d ) = ( ac - bd , bc + ad ). \,

So defined, the complex numbers form a field, the complex number field, denoted by C. We identify the real number a with the complex number (a, 0), and in this way the field of real numbers R becomes a subfield of C. The imaginary unit i is the complex number (0, 1). In C, we have:
- additive identity ("zero"): (0, 0)
- multiplicative identity ("one"): (1, 0)
- additive inverse of (a,b): (−a, −b)
- multiplicative inverse (reciprocal) of non-zero (a, b):

\left(,\right).

C can also be defined as the topological closure of the algebraic numbers or as the algebraic closure of R, both of which are described below.

The complex plane

image:complex.png

A complex number can be viewed as a point or a position vector on a two-dimensional Cartesian coordinate system called the complex plane or Argand diagram (named after Jean-Robert Argand). The Cartesian coordinates of the complex number are the real part x and the imaginary part y, while the circular coordinates are r = |z|, called the absolute value or modulus, and φ = arg(z), called the complex argument of z (mod-arg form). Together with Euler's formula we have :

z = x + iy = r (\cos \phi + i\sin \phi ) = r e^. \,

Additionally the notation r cis φ is sometimes used. Note that the complex argument is unique modulo 2π, that is, if any two values of the complex argument exactly differ by an integer multiple of 2π, they are considered equivalent. By simple trigonometric identities, we see that :

r_1 e^ \cdot r_2 e^ 
= r_1 r_2 e^ \,

and that :

\frac

= \frac e^. \,

Now the addition of two complex numbers is just the vector addition of two vectors, and the multiplication with a fixed complex number can be seen as a simultaneous rotation and stretching. Multiplication with i corresponds to a counter clockwise rotation by 90 degrees (

\pi/2

radians). The geometric content of the equation i² = −1 is that a sequence of two 90 degree rotations results in a 180 degree (

\pi

radians) rotation. Even the fact (−1) · (−1) = +1 from arithmetic can be understood geometrically as the combination of two 180 degree turns.

Absolute value, conjugation and distance

The absolute value (or modulus or magnitude) of a complex number z = r e^iφ is defined as |z| = r. Algebraically, if z = a + ib, then

| z | = \sqrt.

One can check readily that the absolute value has three important properties: :

| z | = 0 \,

iff

z = 0 \,

| z + w | \leq | z | + | w | \,

| z w | = | z | \; | w | \,

for all complex numbers z and w. It then follows, for example, that

| 1 | = 1

and

|z/w|=|z|/|w|

. By defining the distance function d(z, w) = |z − w| we turn the complex numbers into a metric space and we can therefore talk about limits and continuity. The addition, subtraction, multiplication and division of complex numbers are then continuous operations. Unless anything else is said, this is always the metric being used on the complex numbers. The complex conjugate of the complex number z = a + ib is defined to be a - ib, written as

\bar

z^ - \,

. As seen in the figure,

\bar

is the "reflection" of z about the real axis. The following can be checked: :

\overline = \bar + \bar

\overline = \bar\bar

\overline = \bar/\bar

\bar=z

\bar=z

iff z is real :

|z|=|\bar|

|z|^2 = z\bar

z^ = \bar|z|^

if z is non-zero. The latter formula is the method of choice to compute the inverse of a complex number if it is given in rectangular coordinates. That conjugation commutes with all the algebraic operations (and many functions; e.g.

\sin\bar z=\overline

) is rooted in the ambiguity in choice of i (−1 has two square roots); note, however, that conjugation is not differentiable (see holomorphic).

Complex number division

Given a complex number (a + ib) which is to be divided by another complex number (c + id) whose magnitude is non-zero, there are two ways to do this; in either case it is the same as multiplying the first by the multiplicative inverse of the second. The first way has already been implied: to convert both complex numbers into exponential form, from which their quotient is easy to derive. The second way is to express the division as a fraction, then to multiply both numerator and denominator by the complex conjugate of the denominator. This causes the denominator to simplify into a real number: :

=  =

:::

= \left(\right) + i\left(  \right).

Matrix representation of complex numbers

While usually not useful, alternative representations of complex fields can give some insight into their nature. One particularly elegant representation interprets every complex number as 2×2 matrix with real entries which stretches and rotates the points of the plane. Every such matrix has the form :

\begin
  a &   -b  \\
  b & \;\; a  
\end

with real numbers a and b. The sum and product of two such matrices is again of this form. Every non-zero such matrix is invertible, and its inverse is again of this form. Therefore, the matrices of this form are a field. In fact, this is exactly the field of complex numbers. Every such matrix can be written as :

\begin
  a &     -b  \\
  b & \;\; a  
\end
=
a \begin
  1 & \;\; 0  \\
  0 & \;\; 1 
\end
+
b \begin
  0 &     -1  \\
  1 & \;\; 0 
\end

which suggests that we should identify the real number 1 with the matrix :

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

and the imaginary unit i with :

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

a counter-clockwise rotation by 90 degrees. Note that the square of this latter matrix is indeed equal to −1. The absolute value of a complex number expressed as a matrix is equal to the square root of the determinant of that matrix. If the matrix is viewed as a transformation of a plane, then the transformation rotates points through an angle equal to the argument of the complex number and scales by a factor equal to the complex number's absolute value. The conjugate of the complex number z corresponds to the transformation which rotates through the same angle as z but in the opposite direction, and scales in the same manner as z; this can be described by the transpose of the matrix corresponding to z. If the matrix elements are themselves complex numbers, then the resulting algebra is that of the quaternions. In this way, the matrix representation can be seen as a way of expressing the Cayley-Dickson construction of algebras.

Geometric interpretation of the operations on complex numbers

Cayley-Dickson construction Choose a point in the plane which will be the origin,

0

. Given two points A and B in the plane, their sum is the point X in the plane such that the triangles with vertices 0, A, B and X, B, A are similar. similar Choose in addition a point in the plane different from zero, which will be the unity, 1. Given two points A and B in the plane, their product is the point X in the plane such that the triangles with vertices 0, 1, A, and 0, B, X are similar. similar Given a point A in the plane, its complex conjugate is a point X in the plane such that the triangles with vertices 0, 1, A and 0, 1, X are mirror image of each other.

Some properties

Real vector space

C is a two-dimensional real vector space. Unlike the reals, complex numbers cannot be ordered in any way that is compatible with its arithmetic operations: C cannot be turned into an ordered field. R-linear maps C → C have the general form :

f(z)=az+b\overline

with complex coefficients a and b. Only the first term is C-linear; also only the first term is holomorphic; the second term is real-differentiable, but does not satisfy the Cauchy-Riemann equations. The function :

f(z)=az\,

corresponds to rotations combined with scaling, while the function :

f(z)=b\overline

corresponds to reflections combined with scaling.

Solutions of polynomial equations

A root of the polynomial p is a complex number z such that p(z) = 0. A most striking result is that all polynomials of degree n with real or complex coefficients have exactly n complex roots (counting multiple roots according to their multiplicity). This is known as the fundamental theorem of algebra, and shows that the complex numbers are an algebraically closed field. Indeed, the complex number field is the algebraic closure of the real number field, and Cauchy constructed complex numbers in this way. It can be identified as the quotient ring of the polynomial ring R[X] by the ideal generated by the polynomial X² + 1: :

\mathbb = \mathbb[ X ] / ( X^2 + 1). \,

This is indeed a field because X² + 1 is irreducible, hence generating a maximal ideal, in R[X]. The image of X in this quotient ring becomes the imaginary unit i.

Algebraic characterization

The field C is (up to field isomorphism) characterized by the following three facts:
- its characteristic is 0
- its transcendence degree over the prime field is the cardinality of the continuum
- it is algebraically closed Consequently, C contains many proper subfields which are isomorphic to C. Another consequence of this characterization is that the Galois group of C over the rational numbers is enormous, with cardinality equal to that of the power set of the continuum.

Characterization as a topological field

As noted above, the algebraic characterization of C fails to capture some of its most important properties. These properties, which underpin the foundations of complex analysis, arise from the topology of C. The following properties characterize C as a topological field:
- C is a field.
- C contains a subset P of nonzero elements satisfying:
- P is closed under addition, multiplication and taking inverses.
- If x and y are distinct elements of P, then either x-y or y-x is in P
- If S is any nonempty subset of P, then S+P=x+P for some x in C.
- C has a nontrivial involutive automorphism x->x
- , fixing P and such that xx
- is in P for any nonzero x in C. Given these properties, one can then define a topology on C by taking the sets
-

B(x,p) = \

as a base, where x ranges over C, and p ranges over P. To see that these properties characterize C as a topological field, one notes that P ∪ ∪ -P is an ordered Dedekind-complete field and thus can be identified with the real numbers R by a unique field isomorphism. The last property is easily seen to imply that the Galois group over the real numbers is of order two, completing the characterization. Pontryagin has shown that the only connected locally compact topological fields are R and C. This gives another characterization of C as a topological field, since C can be distinguished from R by noting the nonzero complex numbers are connected whereas the nonzero real numbers are not.

Complex analysis

The study of functions of a complex variable is known as complex analysis and has enormous practical use in applied mathematics as well as in other branches of mathematics. Often, the most natural proofs for statements in real analysis or even number theory employ techniques from complex analysis (see prime number theorem for an example). Unlike real functions which are commonly represented as two dimensional graphs, complex functions have four dimensional graphs and may usefully be illustrated by color coding a three dimensional graph to suggest four dimensions, or by animating the complex function's dynamic transformation of the complex plane.

Applications

Control theory

In control theory, systems are often transformed from the time domain to the frequency domain using the Laplace transform. The system's poles and zeros are then analyzed in the complex plane. The root locus, Nyquist plot, and Nichols plot techniques all make use of the complex plane. In the root locus method, it is especially important whether the poles and zeros are in the left or right half planes, i.e. have real part greater than or less than zero. If a system has poles that are
- in the right half plane, it will be unstable,
- all in the left half plane, it will be stable,
- on the imaginary axis, it will be marginally stable. If a system has zeros in the right half plane, it is a nonminimum phase system.

Signal analysis

Complex numbers are used in signal analysis and other fields as a convenient description for periodically varying signals. The absolute value |z| is interpreted as the amplitude and the argument arg(z) as the phase of a sine wave of given frequency. If Fourier analysis is employed to write a given real-valued signal as a sum of periodic functions, these periodic functions are often written as the real part of complex valued functions of the form :

f ( t ) = z e^ \,

where ω represents the angular frequency and the complex number z encodes the phase and amplitude as explained above. In electrical engineering, the Fourier transform is used to analyze varying voltages and currents. The treatment of resistors, capacitors, and inductors can then be unified by introducing imaginary, frequency-dependent resistances for the latter two and combining all three in a single complex number called the impedance. (Electrical engineers and some physicists use the letter j for the imaginary unit since i is typically reserved for varying currents and may come into conflict with i.) This use is also extended into digital signal processing and digital image processing, which utilize digital versions of Fourier analysis (and Wavelet analysis) to transmit, compress, restore, and otherwise process digital audio signals, still images, and video signals.

Improper integrals

In applied fields, the use of complex analysis is often used to compute certain real-valued improper integrals, by means of complex-valued functions. Several methods exist to do this, see methods of contour integration.

Quantum mechanics

The complex number field is also of utmost importance in quantum mechanics since the underlying theory is built on (infinite dimensional) Hilbert spaces over C.

Relativity

In special and general relativity, some formulas for the metric on spacetime become simpler if one takes the time variable to be imaginary.

Applied mathematics

In differential equations, it is common to first find all complex roots r of the characteristic equation of a linear differential equation and then attempt to solve the system in terms of base functions of the form f(t) = e^rt.

Fluid dynamics

In fluid dynamics, complex functions are used to describe potential flow in 2d.

Fractals

Certain fractals are plotted in the complex plane e.g. Mandelbrot set and Julia set.

History

The earliest fleeting reference to square roots of negative numbers occurred in the work of the Greek mathematician and inventor Heron of Alexandria in the 1st century AD, when he considered the volume of an impossible frustum of a pyramid. They became more prominent when in the 16th century closed formulas for the roots of third and fourth degree polynomials were discovered by Italian mathematicians (see Niccolo Fontana Tartaglia, Gerolamo Cardano). It was soon realized that these formulas, even if one was only interested in real solutions, sometimes required the manipulation of square roots of negative numbers. For example, Tartaglia's cubic formula gives the following solution to the equation

x^3-x=0

: :

\frac\left(\sqrt^+\frac\right).

At first glance this looks like nonsense. However formal calculations with complex numbers show that the equation

z^3=i

has solutions −i,

\frac+\fraci

and

\frac+\fraci

. Substituting these in turn for

\sqrt^

into the cubic formula and simplifying, one gets 0, 1 and -1 as the solutions of

x^3-x=0.

This was doubly unsettling since not even negative numbers were considered to be on firm ground at the time. The term "imaginary" for these quantities was coined by René Descartes in the 17th century and was meant to be derogatory (see imaginary number for a discussion of the "reality" of complex numbers). A further source of confusion was that the equation

\sqrt^2=\sqrt\sqrt=-1

seemed to be capriciously inconsistent with the algebraic identity

\sqrt\sqrt=\sqrt

, which is valid for positive real numbers a and b, and which was also used in complex number calculations with one of a, b positive and the other negative. The incorrect use of this identity (and the related identity

\frac=\sqrt

) in the case when both a and b are negative even bedeviled Euler. This difficulty eventually led to the convention of using the special symbol i in place of

\sqrt

to guard against this mistake. The 18th century saw the labors of Abraham de Moivre and Leonhard Euler. To De Moivre is due (1730) the well-known formula which bears his name, de Moivre's formula: :

(\cos \theta + i\sin \theta)^ = \cos n \theta + i\sin n \theta \,

and to Euler (1748) Euler's formula of complex analysis: :

\cos \theta + i\sin \theta = e ^. \,

The existence of complex numbers was not completely accepted until the geometrical interpretation (see below) had been described by Caspar Wessel in 1799; it was rediscovered several years later and popularized by Carl Friedrich Gauss, and as a result the theory of complex numbers received a notable expansion. The idea of the graphic representation of complex numbers had appeared, however, as early as 1685, in Wallis's De Algebra tractatus. Wessel's memoir appeared in the Proceedings of the Copenhagen Academy for 1799, and is exceedingly clear and complete, even in comparison with modern works. He also considers the sphere, and gives a quaternion theory from which he develops a complete spherical trigonometry. In 1804 the Abbé Buée independently came upon the same idea which Wallis had suggested, that

\pm\sqrt

should represent a unit line, and its negative, perpendicular to the real axis. Buée's paper was not published until 1806, in which year Jean-Robert Argand also issued a pamphlet on the same subject. It is to Argand's essay that the scientific foundation for the graphic representation of complex numbers is now generally referred. Nevertheless, in 1831 Gauss found the theory quite unknown, and in 1832 published his chief memoir on the subject, thus bringing it prominently before the mathematical world. Mention should also be made of an excellent little treatise by Mourey (1828), in which the foundations for the theory of directional numbers are scientifically laid. The general acceptance of the theory is not a little due to the labors of Augustin Louis Cauchy and Niels Henrik Abel, and especially the latter, who was the first to boldly use complex numbers with a success that is well known. The common terms used in the theory are chiefly due to the founders. Argand called

\cos \phi + i  -  \sin \phi

the direction factor, and

r = \sqrt

the modulus; Cauchy (1828) called

\cos \phi + i  - \sin \phi

the reduced form (l'expression réduite); Gauss used i for

\sqrt

, introduced the term complex number for

a+bi

, and called

a^2+b^2

the norm. The expression direction coefficient, often used for

\cos \phi + i - 
\sin \phi

, is due to Hankel (1867), and absolute value, for modulus, is due to Weierstrass. Following Cauchy and Gauss have come a number of contributors of high rank, of whom the following may be especially mentioned: Kummer (1844), Leopold Kronecker (1845), Scheffler (1845, 1851, 1880), Bellavitis (1835, 1852), Peacock (1845), and De Morgan (1849). Möbius must also be mentioned for his numerous memoirs on the geometric applications of complex numbers, and Dirichlet for the expansion of the theory to include primes, congruences, reciprocity, etc., as in the case of real numbers. A complex ring or field is a set of complex numbers which is closed under addition, subtraction, and multiplication. Gauss studied complex numbers of the form

a + bi

, where a and b are integral, or rational (and i is the root of

x^2 + 1 = 0

). His student, Ferdinand Eisenstein, studied the type

a + b\omega

, where

\omega

is a complex root of

x^3 - 1 = 0

. Other such classes (called cyclotomic fields) of complex numbers are derived from the roots of unity

x^k - 1 = 0

for higher values of

k

. This generalization is largely due to Kummer, who also invented ideal numbers, which were expressed as geometrical entities by Felix Klein in 1893. The general theory of fields was created by Évariste Galois, who studied the fields generated by the roots of any polynomial equation :

\ F(x) = 0

The late writers (from 1884) on the general theory include Weierstrass, Schwarz, Richard Dedekind, Otto Hölder, Berloty, Henri Poincaré, Eduard Study, and Alexander MacFarlane. The formally correct definition using pairs of real numbers was given in the 19th century.

External links

- Complex numbers at Wikibooks
- [http://mathforum.org/johnandbetty/ John and Betty's Journey Through Complex Numbers]
- [http://mathworld.wolfram.com/ComplexNumber.html Complex Number from MathWorld]
- [http://www.sosmath.com/complex/complex.html SOS Math - Complex Variables]
- [http://www.binarythings.com/hidigit/ Windows calculator that supports complex numbers] Category:Complex numbers Category:Elementary mathematics ko:복소수 ja:複素数 nb:Komplekst tall th:จำนวนเชิงซ้อน

Vector (spatial)

:This article discusses vectors that have a particular relation to the spatial coordinates. For a generalization, see vector space. In physics and in vector calculus, a spatial vector is a concept characterized by a magnitude, which is a scalar, and a direction (which can be defined in a 3-dimensional space by the Euler angles). Although it is often described by a number of "components", each of which is dependent upon the particular coordinate system being used, a vector is an object with properties which do not depend on the coordinate system used to describe it. A common example of a vector is force — it has a magnitude and an orientation in three dimensions (or however many spatial dimensions one has), and multiple forces sum according to the parallelogram law. A spatial vector can be formally defined by its relationship to the spatial coordinate system under rotations. Alternatively, it can be defined in a coordinate-free fashion via a tangent space of a three-dimensional manifold in the language of differential geometry. These definitions are discussed in more detail below. A spatial vector is a special case of a tensor and is also analogous to a four-vector in relativity (and is sometimes therefore called a three-vector in reference to the three spatial dimensions, although this term also has another meaning for p-vectors of differential geometry). Vectors are the building blocks of vector fields and vector calculus. The word vector is also now used for more general concepts (see also vector and generalizations below), but this article describes the original spatial meaning except where otherwise noted.

Definitions

Informally, a vector is a quantity characterized by a magnitude (in mathematics a number, in physics a number times a unit) and a direction, often represented graphically by an arrow. Examples are "moving north at 90 km/h" or "pulling towards the center of Earth with a force of 70 newtons". The notion of having a "magnitude" and "direction" is formalized by saying that the vector has components that transform like the coordinates under rotations. That is, if the coordinate system undergoes a rotation described by a rotation matrix R, so that a coordinate vector x is transformed to x' = Rx, then any other vector v is similarly transformed via v' = Rv. This ensures the invariance of the operations dot product, Euclidean norm, cross product, gradient, divergence, curl, and scalar triple product, and trivially for vector addition and subtraction, and scalar multiplication. The terms scalar and vector as used here include pseudoscalars and pseudovectors or axial vectors (see also below). Accordingly, let, for example, each of two vectors be expressed as three space coordinates, and apply the formula for the cross product, resulting in three coordinates, which represent a third vector. If we rewrite the two vectors in rotated coordinates, and apply the formula for the cross product again, then the result is the original cross product in terms of rotated coordinates. Also, let, for example, a vector field be expressed as three space coordinate functions of three variables, and apply the formula for the curl based on these functions, resulting in three additional functions, which represent a second vector field. If we rewrite the original vector field in terms of rotated position coordinates and correspondingly rotated coordinates for the vector function values, and apply the formula for the curl based on these functions, then the result is the rewritten version of the original curl: also in terms of rotated position coordinates and correspondingly rotated coordinates for the vector function values. The same applies for dot product, gradient, divergence, vector addition and scalar multiplication. For these, also reflection in a plane can be applied. The scalars involved should not be transformed (e.g. in the case of a rotation by 180°, the scalar should not be multiplied by -1). Thus even in 1D we have to distinguish scalars and vectors: 2 × 3 = 6 can be interpreted as a scalar multiplication or a dot product, but not as a product of two vectors. Similarly differentiation in 1D can be interpreted as a gradient or a divergence: one of the two functions is scalar and one a vector, and the argument is a vector, ensuring invariance under inversion of the vectors without changing the scalars. Since rotation of the three Cartesian coordinate axes changes the formulas the same as an inverse rotation of the field itself, we can also conclude:
- if the same rotation is applied to two vectors, then the cross product is correspondingly rotated, but the dot product remains the same
- rotation of a scalar field results in a correspondingly rotated vector field for the gradient
- rotation of a vector field results in a correspondingly rotated scalar field for the divergence and a correspondingly rotated vector field for the curl where rotation of a scalar field involves only rotation of the position vectors, while rotation of a vector field involves also a corresponding rotation of the vector field values. Note that the concept of corresponding rotations applies even if different coordinate systems are used for field values and position vectors, so that e.g. for one we multiply by an orthogonal matrix and for the other we add an angle to an angle coordinate. In order to use the usual formulas, e.g. to compute mechanical work, the x-axis of forces should be in the same direction as the x-axis of position, etc. When, as described above, coordinate rotations of position are accompanied by corresponding coordinate rotations of forces, this property is preserved. On the other hand, the origin of forces is simply at the zero force (no force), while the origin of position can be chosen as desired. For example, work depends on displacement, which is the difference of positions and therefore does not depend on the origin. Position and function value of a vector field are often, but not necessarily, expressed in similar coordinate systems. For example gravitational field strength due to a particular point mass may be

g_r = -9.8 (r/r_0)^ m/s^2

, with both the function value and the position vector in spherical coordinates. For the position vector the origin is chosen here at the center of the point mass; for the field strength the origin is simply at "zero field strength" anyway. How the other two coordinates are chosen does not matter in this case, because the field does not depend on them, and the field has no components in their directions. More generally, a vector is a tensor of contravariant rank one. In differential geometry, the term vector usually refers to quantities that are closely related to tangent spaces of a differentiable manifold (assumed to be three-dimensional and equipped with a positive definite Riemannian metric). (A four-vector is a related concept when dealing with a 4 dimensional spacetime manifold in relativity.) Examples of vectors include displacement, velocity, electric field, momentum, force, and acceleration. Vectors can be contrasted with scalar quantities such as distance, speed, energy, time, temperature, charge, power, work, and mass, which have magnitude, but no direction (they are invariant under coordinate rotations). The magnitude of any vector is a scalar. A related concept is that of a pseudovector (or axial vector). This is a quantity that transforms like a vector under proper rotations, but gains an additional sign flip under improper rotations. Examples of pseudovectors include magnetic field, torque, and angular momentum. (This distinction between vectors and pseudovectors is often ignored, but it becomes important in studying symmetry properties.) To distinguish from pseudo/axial vectors, an ordinary vector is sometimes called a polar vector. See also parity (physics). Sometimes, one speaks informally of bound or fixed vectors, which are vectors additionally characterized by a "base point". Most often, this term is used for position vectors (relative to an origin point). More generally, however, the physical interpretation of a particular vector can be parameterized by any number of quantities.

Examples in one dimension

A force may be "15N to the right", with coordinate 15N if the basis vector is to the right, and −15N if the basis vector is to the left. The magnitude of the vector is 15N in both cases. A displacement may be "4m to the right", with coordinate 4m if the basis vector is to the right, and −4m if the basis vector is to the left. The magnitude of the vector is 4m in both cases. The work done by the force in the case of this displacement is 60J in both cases. The force and displacement are vectors, the magnitudes are scalars, and the coordinates are neither.

Generalizations

In mathematics, a vector is any element of a vector space over some field. The spatial vectors of this article are a very special case of this general definition (they are not simply any element of R^d in d dimensions), which includes a variety of mathematical objects (algebras, the set of all functions from a given domain to a given linear range, and linear transformations). Note that under this definition, a tensor is a special vector!

Representation of a vector

Symbols standing for vectors are usually printed in boldface as a; this is also the convention adopted in this encyclopedia. Other conventions include

\vec

or a, especially in handwriting. Alternately, some use a tilde (~) placed under the vector. The length or magnitude or norm of the vector a is denoted by |a|. Vectors are usually shown in graphs or other diagrams as arrows, as illustrated below: Image:vecab.png Here the point A is called the tail, base, start, or origin; point B is called the head, tip, endpoint, or destination. The length of the arrow represents the vector's magnitude, while the direction in which the arrow points represents the vector's direction. If a vector is itself spatial, the length of the arrow depends on a dimensionless scale. If it represents e.g. a force, the "scale" is of physical dimension length/force. Thus there is typically consistency in scale among quantities of the same dimension, but otherwise scale ratios may vary; for example, if "1 newton" and "5 m" are both represented with an arrow of 2cm, the scales are 1:250 and 1m:50N respectively. Equal length of vectors of different dimension has no particular significance unless there is some proportionality constant inherent in the system that the diagram represents. Also length of a unit vector (of dimension length, not length/force, etc.) has no coordinate-system-invariant significance. In the figure above, the arrow can also be written as

\overrightarrow

or AB In order to calculate with vectors, the graphical representation is too cumbersome. Vectors in a n-dimensional Euclidean space can be represented as a linear combination of n mutually perpendicular unit vectors. In this article, we will consider R³ as an example. In R³, we usually denote the unit vectors parallel to the x-, y- and z-axes by i, j and k respectively. Any vector a in R³ can be written as a = a₁i + a₂j + a₃k with real numbers a₁, a₂ and a₃ which are uniquely determined by a. Sometimes a is then also written as a 3-by-1 or 1-by-3 matrix: :

= \begin
 a_1\\
 a_2\\
 a_3\\
\end

= \begin
 a_1 & a_2 & a_3 \\
\end

even though this notation suppresses the dependence of the coordinates a₁, a₂ and a₃ on the specific choice of coordinate system i, j and k.

Length of a vector

The length of the vector a = a₁i + a₂j + a₃k can be computed with the Euclidian norm :

\left\|\mathbf\right\|=\sqrt

which is a consequence of the Pythagorean theorem.

Vector equality

Two vectors are said to be equal if they have the same magnitude and direction. However if we are talking about bound vector, then two bound vectors are equal if they have the same base point and end point. For example, the vector i + 2j + 3k with base point (1,0,0) and the vector i+2j+3k with base point (0,1,0) are different bound vectors, but the same (unbounded) vector.

Vector addition and subtraction

Let a=a₁i + a₂j + a₃k and b=b₁i + b₂j + b₃k. The sum of a and b is: :

\mathbf+\mathbf
=(a_1+b_1)\mathbf
+(a_2+b_2)\mathbf
+(a_3+b_3)\mathbf

The addition may be represented graphically by placing the start of the arrow b at the tip of the arrow a, and then drawing an arrow from the start of a to the tip of b. The new arrow drawn represents the vector a + b, as illustrated below: Pythagorean theorem This addition method is sometimes called the parallelogram rule because a and b form the sides of a parallelogram and a + b is one of the diagonals. If a and b are bound vectors, then the addition is only defined if a and b have the same base point, which will then also be the base point of a + b. One can check geometrically that a + b = b + a and (a + b) + c = a + (b + c). The difference of a and b is: :

\mathbf-\mathbf
=(a_1-b_1)\mathbf
+(a_2-b_2)\mathbf
+(a_3-b_3)\mathbf

Subtraction of two vectors can be geometrically defined as follows: to subtract b from a, place the ends of a and b at the same point, and then draw an arrow from the tip of b to the tip of a. That arrow represents the vector a − b, as illustrated below: parallelogram If a and b are bound vectors, then the subtraction is only defined if they share the same base point which will then also become the base point of their difference. This operation deserves the name "subtraction" because (a − b) + b = a. In physics, vectors of different physical dimension may occur in the same diagram. However, adding or subtracting them (graphically or otherwise) is meaningless.

Scalar multiplication

A vector may also be multiplied by a real number r. In mathematics numbers are often called scalars to distinguish them from vectors, and this operation is therefore called scalar multiplication. The resulting vector is: :

r\mathbf=(ra_1)\mathbf
+(ra_2)\mathbf
+(ra_3)\mathbf

The length of ra is |r||a|. If the scalar is negative, it also changes the direction of the vector by 180^o. Two examples (r = -1 and r = 2) are given below: real number Here it is important to check that the scalar multiplication is compatible with vector addition in the following sense: r(a + b) = ra + rb for all vectors a and b and all scalars r. One can also show that a - b = a + (-1)b. The set of all geometrical vectors, together with the operations of vector addition and scalar multiplication, satisfies all the axioms of a vector space. Similarly, the set of all bound vectors with a common base point forms a vector space. This is where the term "vector space" originated. In physics, scalars also have a unit. The scale of acceleration in the diagram is e.g. 2 m/s² : cm, and that of force 5 N : cm. Thus a scale ratio of 2.5 kg : 1 is used for mass. Similarly, if displacement has a scale of 1:1000 and velocity of 0.2 cm : 1 m/s, or equivalently, 2 ms : 1, a scale ratio of 0.5 : s is used for time.

Unit vector

Main article: Unit vector A unit vector is any vector with a length of one. If you have a vector of arbitrary length, you can use it to create a unit vector. This is known as normalizing a vector. Unit vector To normalize a vector a = [a₁, a₂, a₃], scale the vector by the inverse of its length ||a||. That is: :

\mathbf=\frac=\frac\mathbf+\frac\mathbf+\frac\mathbf

Dot product

Main article: Dot product The dot product of two vectors a and b (sometimes called inner product, or, since its result is a scalar, the scalar product) is denoted by a·b and is defined as: :

\mathbf\cdot\mathbf
=\left\|\mathbf\right\|\left\|\mathbf\right\|\cos(\theta)

where ||a|| and ||b|| denote the norm (or length) of a and b, and θ is the measure of the angle between a and b (see trigonometric function for an explanation of cosine). Geometrically, this means that a and b are drawn with a common start point and then the length of a is multiplied with the length of that component of b that points in the same direction as a. This operation is often useful in physics; for instance, work is the dot product of force and displacement.

Cross product

The cross product (also vector product or outer product) differs from the dot product primarily in that the result of a cross product of two vectors is a vector. While everything that was said above can be generalized in a straightforward manner to more than three dimensions, the cross product is only meaningful in three dimensions (although a related product exists in seven dimensions - see below). The cross product, denoted a×b, is a vector perpendicular to both a and b and is defined as: :

\mathbf\times\mathbf
=\left\|\mathbf\right\|\left\|\mathbf\right\|\sin(\theta)\,\mathbf

where θ is the measure of the angle between a and b, and n is a unit vector perpendicular to both a and b. The problem with this definition is that there are two unit vectors perpendicular to both b and a. Which vector is the correct one depends upon the orientation of the vector space, i.e. on the handedness of the coordinate system. The coordinate system i, j, k is called right handed, if the three vectors are situated like the thumb, index finger and middle finger (pointing straight up from your palm) of your right hand. Graphically the cross product can be represented by this figure Image:crossproduct.png In such a system, a×b is defined so that a, b and a×b also becomes a right handed system. If i, j, k is left-handed, then a, b and a×b is defined to be left-handed. Because the cross product depends on the choice of coordinate systems, its result is referred to as a pseudovector. Fortunately, in nature cross products tend to come in pairs, so that the "handedness" of the coordinate system is undone by a second cross product. The length of a×b can be interpreted as the area of the parallelogram having a and b as sides.

Scalar triple product

The scalar triple product (also called the box product or mixed triple product) isn't really a new operator, but a way of applying the other two multiplication operators to three vectors. The scalar triple product is denoted by (a b c) and defined as: :

(\mathbf\ \mathbf\ \mathbf)
=\mathbf\cdot(\mathbf\times\mathbf)

It has three primary uses. First, the absolute value of the box product is the volume of the parallelepiped which has edges that are defined by the three vectors. Second, the scalar triple product is zero if and only if the three vectors are linearly dependent, which can be easily proved by considering that in order for the three vectors to not make a volume, they must all lie in the same plane. Third, the box product is positive if and only if the three vectors a, b and c are oriented like the coordinate system i, j and k. In coordinates, if the three vectors are thought of as rows, the scalar triple product is simply the determinant of the 3-by-3 matrix having the three vectors as rows. The scalar triple product is linear in all three entries and anti-symmetric in the following sense: : Technically, the scalar triple product is not a scalar, it is a pseudoscalar: under a coordinate inversion (x goes to −x), it flips sign.

Vectors as directional derivatives

A vector may also be defined as a directional derivative: consider a function

f(x^\alpha)

and a curve

x^\alpha (\sigma)

. Then the directional derivative of

f

is a scalar defined as

\frac = \frac\frac.

where the index

\alpha

is summed over the appropriate number of dimensions (e.g. from 1 to 3 in 3-dimensional Euclidian space, from 0 to 3 in 4-dimensional spacetime, etc.). Then consider a vector tangent to

x^\alpha (\sigma)

t^\alpha = \frac.

We can rewrite the directional derivative in differential form (without a given function

f

) as

\frac = t^\alpha\frac.

Therefore any directional derivative can be identified with a corresponding vector, and any vector can be identified with a corresponding directional derivative. We can therefore define a vector precisely: :

\mathbf \equiv a^\alpha \frac.

External links

- [http://wwwppd.nrl.navy.mil/nrlformulary/vector_identities.pdf Online vector identities] (PDF)
- Vectors at Wikibooks Category:Abstract algebra Category:Vector calculus Category:Linear algebra Category:Introductory physics ko:벡터 ja:ベクトル (数学)

Mod-arg form

\mathbb

Coordinates (elementary mathematics)

Cartesian coordinates

Polar coordinates

Circular coordinates

Cylindrical coordinates

Spherical coordinates

Conversion between coordinate systems

Cartesian and circular

Cartesian and cylindrical

Cartesian and spherical

Cylindrical and spherical

See also

Spherical coordinates

External links

Coordinate system

Examples

Transformations

Singularities

Systems commonly used

Astronomical systems

See also

Angle

Units of measure for angles

Conventions on measurement

Types of angles

Some facts

A formal definition

Angles in different contexts

Angles in Riemannian geometry

Angles in astronomy

Angles in maritime navigation

See also

External links

Curvilinear coordinates

Terminology

Example

Line, surface, and volume integrals

Line integrals

Surface integrals

Div, curl, grad

Circular coordinates

Cylindrical coordinates

RADIUS

Standards

See also

External links

Complex number

Definition

The complex number field

The complex plane

Absolute value, conjugation and distance

Complex number division

Matrix representation of complex numbers

Geometric interpretation of the operations on complex numbers

Some properties

Real vector space

Solutions of polynomial equations

Algebraic characterization

Characterization as a topological field

Complex analysis

Applications

Control theory

Signal analysis

Improper integrals

Quantum mechanics

Relativity

Applied mathematics

Fluid dynamics

Fractals

History

See also

Further reading

External links

Vector (spatial)

Definitions

Examples in one dimension

Generalizations

Representation of a vector

Length of a vector

Vector equality