Three-dimensional space (mathematics)

نوشته شده در موضوع خرید اینترنتی در ۳۱ مرداد ۱۳۹۴

Geometry

History

Geometers

Three-dimensional space (also: tri-dimensional space) is a geometric three-parameter indication of a earthy star (without deliberation time) in that all famous matter exists. These 3 measure can be labeled by a mixed of 3 selected from a terms length, width, height, depth, and breadth. Any 3 directions can be chosen, supposing that they do not all distortion in a same plane.

In production and mathematics, a method of n numbers can be accepted as a plcae in n-dimensional space. When n = 3, a set of all such locations is called three-dimensional Euclidean space. It is ordinarily represented by a pitch . This space is customarily one instance of a good accumulation of spaces in 3 measure called 3-manifolds.

In geometry[edit]

Coordinate systems[edit]

In mathematics, analytic geometry (also called Cartesian geometry) describes any indicate in three-dimensional space by means of 3 coordinates. Three coordinate axes are given, any perpendicular to a other dual during a origin, a indicate during that they cross. They are customarily labeled x, y, and z. Relative to these axes, a position of any indicate in three-dimensional space is given by an systematic triple of genuine numbers, any series giving a stretch of that indicate from a start totalled along a given axis, that is equal to a stretch of that indicate from a craft dynamic by a other dual axes.

Other renouned methods of describing a plcae of a indicate in three-dimensional space embody cylindrical coordinates and round coordinates, yet there is an gigantic series of probable methods. See Euclidean space.

Below are images of a above-mentioned systems.

Polytopes[edit]

In 3 dimensions, there are 9 unchanging polytopes: a 5 convex Platonic solids and a 4 nonconvex Kepler-Poinsot polyhedra.

Regular polytopes in 3 dimensions

Class
Platonic solids
Kepler-Poinsot polyhedra
Symmetry
Td
Oh
Ih
Coxeter group
A3, [3,3]
B3, [4,3]
H3, [5,3]
Order
۲۴
۴۸
۱۲۰
Regular
polyhedron


{۳,۳}

{۴,۳}

{۳,۴}

{۵,۳}

{۳,۵}

{۵/۲,۵}

{۵,۵/۲}

{۵/۲,۳}

{۳,۵/۲}

Sphere[edit]

A globe in 3-space (also called a 2-sphere since a aspect is 2-dimensional) consists of a set of all points in 3-space during a bound stretch r from a executive indicate P. The volume enclosed by this aspect is:

Another form of sphere, though carrying a three-dimensional aspect is a 3-sphere: points inner to a start of a euclidean space during stretch one. If any position is , afterwards impersonate a indicate in a 3-sphere.

Orthogonality[edit]

In a informed 3-dimensional space that we live in, there are 3 pairs of principal directions: north/south (latitude), east/west (longitude) and up/down (altitude). These pairs of directions are jointly orthogonal: They are during right angles to any other. Movement along one pivot does not change a coordinate value of a other dual axes. In mathematical terms, they distortion on 3 coordinate axes, customarily labelled x, y, and z. The z-buffer in mechanism graphics refers to this z-axis, representing abyss in a 2-dimensional imagery displayed on a mechanism screen.

In linear algebra[edit]

Another mathematical approach of observation three-dimensional space is found in linear algebra, where a suspicion of autonomy is crucial. Space has 3 measure since a length of a box is eccentric of a extent or breadth. In a technical denunciation of linear algebra, space is three-dimensional since any indicate in space can be described by a linear mixed of 3 eccentric vectors.

Dot product, angle, and length[edit]

The dot product of dual vectors A = [A1, A2,A3] and B = [B1, B2,B3] is tangible as:[1]

A matrix can be graphic as an arrow. Its bulk is a length, and a instruction is a instruction a arrow points. The bulk of a matrix A is denoted by . In this viewpoint, a dot product of dual Euclidean vectors A and B is tangible by[2]

where θ is a angle between A and B.

The dot product of a matrix A by itself is

which gives

the regulation for a Euclidean length of a vector.

Cross product[edit]

The cranky product or vector product is a binary operation on dual vectors in three-dimensional space and is denoted by a pitch ×. The cranky product a × b of a vectors a and b is a matrix that is perpendicular to both and therefore normal to a craft containing them. It has many applications in mathematics, physics, and engineering.

The space and product form an algebra over a field, that is conjunction commutative nor associative, though is a Lie algebra with a cranky product being a Lie bracket.

One can in n measure take a product of n − ۱ vectors to furnish a matrix perpendicular to all of them. But if a product is singular to non-trivial binary products with matrix results, it exists customarily in 3 and 7 dimensions.[3]

In calculus[edit]

Gradient, dissimilarity and curl[edit]

In a rectilinear coordinate system, a slope is given by

The dissimilarity of a invariably differentiable matrix margin F = U i + V j + W k is equal to a scalar-valued function:

Expanded in Cartesian coordinates (see Del in cylindrical and round coordinates for round and cylindrical coordinate representations), a twist ∇ × F is, for F stoical of [Fx, Fy, Fz]:

where i, j, and k are a section vectors for a x-, y-, and z-axes, respectively. This expands as follows:[4]

Line integrals, aspect integrals, and volume integrals[edit]

For some scalar margin f : URnR, a line constituent along a piecewise well-spoken bend CU is tangible as

where r: [a, b] → C is an capricious bijective parametrization of a bend C such that r(a) and r(b) give a endpoints of C and .

For a matrix margin F : URnRn, a line constituent along a piecewise well-spoken bend CU, in a instruction of r, is tangible as

where · is a dot product and r: [a, b] → C is a bijective parametrization of a bend C such that r(a) and r(b) give a endpoints of C.

A aspect constituent is a generalization of mixed integrals to formation over surfaces. It can be suspicion of as a double constituent analog of a line integral. To find an pithy regulation for a aspect integral, we need to parameterize a aspect of interest, S, by deliberation a complement of curvilinear coordinates on S, like a embodiment and longitude on a sphere. Let such a parameterization be x(s, t), where (s, t) varies in some segment T in a plane. Then, a aspect constituent is given by

where a countenance between bars on a right-hand side is a bulk of a cranky product of a prejudiced derivatives of x(s, t), and is famous as a aspect element. Given a matrix margin v on S, that is a duty that assigns to any x in S a matrix v(x), a aspect constituent can be tangible component-wise according to a clarification of a aspect constituent of a scalar field; a outcome is a vector.

A volume constituent refers to an constituent over a 3-dimensional domain.

It can also meant a triple constituent within a segment D in R3 of a duty and is customarily created as:

Fundamental postulate of line integrals[edit]

The elemental postulate of line integrals, says that a line constituent by a slope margin can be evaluated by evaluating a strange scalar margin during a endpoints of a curve.

Let . Then

Stokes’ theorem[edit]

Stokes’ postulate relates a aspect constituent of a twist of a matrix margin F over a aspect Σ in Euclidean three-space to a line constituent of a matrix margin over a range ∂Σ:

Divergence theorem[edit]

Suppose V is a subset of (in a box of n = 3, V represents a volume in 3D space) that is compress and has a piecewise well-spoken range S (also indicated with V = S). If F is a invariably differentiable matrix margin tangible on a area of V, afterwards a dissimilarity postulate says:[5]

The left side is a volume constituent over a volume V, a right side is a aspect constituent over a range of a volume V. The sealed plural V is utterly generally a range of V oriented by outward-pointing normals, and n is a external indicating section normal margin of a range V. (dS might be used as a shorthand for ndS.)

In topology[edit]

Three-dimensional space has a series of topological properties that heed it from spaces of other dimension numbers. For example, during slightest 3 measure are compulsory to tie a tangle in a square of string.[6]

With a space , a topologists locally indication all other 3-manifolds.

See also[edit]

  • ۳D scholarship and technology
  • ۳-manifolds
  • Dimensional analysis
  • Distance from a indicate to a plane
  • Skew lines#Distance
  • Space
  • Three-dimensional graph
  • Two-dimensional space

References[edit]

External links[edit]

Wikiquote has quotations associated to: Three-dimensional space (mathematics)

Wikimedia Commons has media associated to 3D.

Dimensional spaces

Other dimensions

Polytopes and shapes

Dimensions by number

Article source: http://en.wikipedia.org/wiki/Three-dimensional_space

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