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| {{refimprove|date=April 2013}}[[Image:Real 2-space as square grid.png|thumb|right|256px<!-- do not rescale! -->|The Cartesian product structure of {{math|'''R'''<sup>2</sup>}} on [[Cartesian plane]] of [[ordered pair]]s {{math|(''x'', ''y'')}}. Blue lines denote [[coordinate axes]], horizontal green lines are [[integer]] {{mvar|''y''}}, vertical cyan lines are integer {{mvar|''x''}}, brown-orange lines show [[half-integer]] {{mvar|''x''}} or {{mvar|''y''}}, magenta and its tint show multiples of [[one tenth]] (best seen under magnification)]]
| | == . Isabel Marant Melbourne Australia == |
| In [[mathematics]], '''real coordinate space''' of {{mvar|n}} [[dimension]]s, written '''R'''<sup>{{mvar|n}}</sup> ({{IPAc-en|ɑr|ˈ|ɛ|n|'}} {{respell|ar|EN|'}}) ('''R''' with [[superscript]] ''n'', also written {{math|ℝ<sup>''n''</sup>}} with [[blackboard bold]] R) is a [[coordinate space]] that allows [[tuple|several]] ([[placeholder variable|{{mvar|n}}]]) [[real number|real]] variables to be treated as a single [[variable (mathematics)|variable]]. With various numbers of dimensions (sometimes unspecified), {{math|'''R'''<sup>''n''</sup>}} is used in many areas of pure and applied mathematics, as well as in [[physics]]. It is [[#Vector space|the prototypical real vector space]] and a frequently used [[#Euclidean space|representation of Euclidean {{mvar|n}}-space]]. Due to the latter fact, [[geometry|geometric]] [[conceptual metaphor|metaphors]] are widely used for {{math|'''R'''<sup>''n''</sup>}}, namely a [[plane (geometry)|plane]] for {{math|'''R'''<sup>2</sup>}} and [[three-dimensional space]] for {{math|'''R'''<sup>3</sup>}}.
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| For any [[natural number]] {{mvar|n}}, the [[set (mathematics)|set]] {{math|'''R'''<sup>''n''</sup>}} consists of all {{mvar|n}}-[[tuple]]s of [[real number]]s ({{math|'''R'''}}). It is called (the) "{{mvar|n}}-dimensional real [[space (mathematics)|space]]". As it can be constructed as [[Cartesian product]] of {{mvar|n}} instances of the set {{math|'''R'''}}, it inherits some of its [[mathematical structure|structure]], notably:
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| * {{math|'''R'''<sup>''n''</sup>}} has [[addition]] and [[scalar multiplication]]: see [[#Vector space|below]]
| | <li>[http://jiuye91.com/news/html/?818925.html http://jiuye91.com/news/html/?818925.html]</li> |
| * {{math|'''R'''<sup>''n''</sup>}} is a [[topological space]]: see [[#Topological properties|below]]
| | |
| An element of {{math|'''R'''<sup>''n''</sup>}} is written
| | <li>[http://www.xn--siqr3y30jp5c.cn/forum.php?mod=viewthread&tid=280877 http://www.xn--siqr3y30jp5c.cn/forum.php?mod=viewthread&tid=280877]</li> |
| :<math>\mathbf x = (x_1, x_2, \ldots, x_n)</math>
| | |
| where each {{math|''x''<sub>''i''</sub>}} is a real number.
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| | </ul> |
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| For each {{mvar|n}} there exists only one {{math|'''R'''<sup>''n''</sup>}}, ''the'' real {{mvar|n}}-space.<ref>Unlike many situations in mathematics where certain object is unique [[up to isomorphism]], {{math|'''R'''<sup>''n''</sup>}} in [[uniqueness|unique]] in the strong sense: any its element is described explicitly with its {{mvar|n}} real coordinates.</ref>
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| Purely mathematical uses of {{math|'''R'''<sup>''n''</sup>}} can be roughly classified as follows, although these uses overlap. First, [[linear algebra]] studies its own properties under [[vector addition]] and [[linear transformation]]s and use in as a model of any {{mvar|n}}-[[dimension (mathematics)|dimensional]] real [[vector space]]. Second, it is used in [[mathematical analysis]] to represent the [[domain (function)|domain]] of a [[function (mathematics)|function]] of {{mvar|n}} real variables in a uniform way, as well as a space to which the [[graph of a function|graph]] of a real-valued function of {{math|''n'' − 1}} real variables is a subset. The third use parametrizes [[point (geometry)|geometric points]] with elements of {{math|'''R'''<sup>''n''</sup>}}; it is common in [[analytic geometry|analytic]], [[differential geometry|differential]] and [[algebraic geometry|algebraic]] geometries.
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| | | <li>[http://www.510mg.com/news/html/?74827.html http://www.510mg.com/news/html/?74827.html]</li> |
| {{math|'''R'''<sup>''n''</sup>}}, together with supplemental structures on it, is also extensively used in [[mathematical physics]], [[dynamical systems theory]], [[mathematical statistics]] and [[probability theory]].
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| | | <li>[http://ymhc.yyrr8.com/forum.php?mod=viewthread&tid=12664 http://ymhc.yyrr8.com/forum.php?mod=viewthread&tid=12664]</li> |
| In [[applied mathematics]], [[numerical analysis]], and so on, [[array data structure|arrays]], [[sequence]]s, and other collections of [[number]]s in applications can be seen as the use of {{math|'''R'''<sup>''n''</sup>}} too.
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| | | <li>[http://ying25.86.web1268.com/forum.php?mod=viewthread&tid=153712&fromuid=118637 http://ying25.86.web1268.com/forum.php?mod=viewthread&tid=153712&fromuid=118637]</li> |
| == The domain of a function of several variables ==
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| {{main|Multivariable calculus|Real multivariable function}}
| | <li>[http://uyouyl.com/forum.php?mod=viewthread&tid=3106754&fromuid=179936 http://uyouyl.com/forum.php?mod=viewthread&tid=3106754&fromuid=179936]</li> |
| Any function {{math|''f''(''x''<sub>1</sub>, ''x''<sub>2</sub>, … , ''x''<sub>''n''</sub>)}} of {{mvar|n}} real variables can be considered as a function on {{math|'''R'''<sup>''n''</sup>}} (with this [[domain of a function|domain]]). The use of the real {{mvar|n}}-space, instead of several variables considered separately, can simplify a notation and suggest reasonable definitions. Consider, for {{math|1=''n'' = 2}}, a [[function composition]] of the following form:
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| :<math> F(t) = f(g_1(t),g_2(t)),</math>
| | </ul> |
| where functions {{math|''g''<sub>1</sub>}} and {{math|''g''<sub>2</sub>}} are [[continuous function|continuous]]. If
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| :{{math|∀''x''<sub>1</sub> ∈ '''R''' : ''f''(''x''<sub>1</sub>, ·<!-- not a multiplication sign! -->)}} is continuous (by {{math|''x''<sub>2</sub>}}) | |
| :{{math|∀''x''<sub>2</sub> ∈ '''R''' : ''f''(·<!-- not a multiplication sign! -->, ''x''<sub>2</sub>)}} is continuous (by {{math|''x''<sub>1</sub>}})
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| then {{mvar|F}} is not necessarily continuous. It is a stronger condition: the continuity of {{mvar|f}} in the natural {{math|'''R'''<sup>2</sup>}} topology ([[#Topological properties|discussed below]]), also called ''multivariable continuity'', which is sufficient for continuity of the composition {{mvar|F}}.
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| {{expand section|date=April 2013}}
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| == Vector space ==
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| <!-- The initial (16 April, 2013) revision of this section used a content from the [[coordinate space]] article. See the history there for attribution. -->
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| {{math|'''R'''<sup>''n''</sup>}} forms an {{mvar|n}}-dimensional [[vector space]] over the [[field (mathematics)|field]] of real numbers.
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| The operations on {{math|'''R'''<sup>''n''</sup>}} are defined by
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| :<math>\mathbf x + \mathbf y = (x_1 + y_1, x_2 + y_2, \ldots, x_n + y_n)</math>
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| :<math>\alpha \mathbf x = (\alpha x_1, \alpha x_2, \ldots, \alpha x_n).</math>
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| The [[additive identity|zero vector]] is given by
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| :<math>\mathbf 0 = (0, 0, \ldots, 0)</math>
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| and the [[additive inverse]] of the vector {{math|'''x'''}} is given by
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| :<math>-\mathbf x = (-x_1, -x_2, \ldots, -x_n).</math>
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| This structure is important because any {{mvar|n}}-dimensional real vector space becomes isomorphic to {{math|'''R'''<sup>''n''</sup>}} after choosing a [[basis (linear algebra)|basis]].
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| ===Matrix notation===
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| {{main|Matrix (mathematics)}}
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| <!-- The initial (16 April, 2013) revision of this subsection used a content from the [[coordinate space]] article. See the history there for attribution. -->
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| In standard [[matrix (mathematics)|matrix]] notation, each element of {{math|'''R'''<sup>''n''</sup>}} is typically written as a [[column vector]]
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| :<math>\mathbf x = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}</math>
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| and sometimes as a [[row vector]]:
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| :<math>\mathbf x = \begin{bmatrix} x_1 & x_2 & \dots & x_n \end{bmatrix}.</math>
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| The coordinate space {{math|'''R'''<sup>''n''</sup>}} may then be interpreted as the space of all {{math|''n'' × 1}} [[column vector]]s, or all {{math|1 × ''n''}} [[row vector]]s with the ordinary matrix operations of addition and [[scalar multiplication]].
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| [[Linear transformation]]s from {{math|'''R'''<sup>''n''</sup>}} to {{math|'''R'''<sup>''m''</sup>}} may then be written as {{math|''m'' × ''n''}} matrices which act on the elements of {{math|'''R'''<sup>''n''</sup>}} via [[left and right (algebra)|left]] multiplication (when the elements of {{math|'''R'''<sup>''n''</sup>}} are column vectors) and on elements of {{math|'''R'''<sup>''m''</sup>}} via right multiplication (when they are row vectors). The formula for left multiplication, a special case of [[matrix multiplication]], is:
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| :<math>(A{\mathbf x})_k = \sum\limits_{l=1}^n A_{kl} x_l</math>
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| {{anchor|continuity of linear maps}}Any linear transformation is a [[continuous function]] (see [[#Topological properties|below]]). Also, a matrix define an [[open map]] from {{math|'''R'''<sup>''n''</sup>}} to {{math|'''R'''<sup>''m''</sup>}} if and only if the [[rank (matrix theory)|rank of the matrix]] equals to {{mvar|m}}.
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| ===Standard basis===
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| {{main|Standard basis}}
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| <!-- The initial (16 April, 2013) revision of this subsection used a content from the [[coordinate space]] article. See the history there for attribution. -->
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| The coordinate space {{math|'''R'''<sup>''n''</sup>}} comes with a standard basis:
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| :<math>
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| \begin{align}
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| \mathbf e_1 & = (1, 0, \ldots, 0) \\
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| \mathbf e_2 & = (0, 1, \ldots, 0) \\
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| & {}\ \vdots \\
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| \mathbf e_n & = (0, 0, \ldots, 1)
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| \end{align}
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| </math>
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| To see that this is a basis, note that an arbitrary vector in {{math|'''R'''<sup>''n''</sup>}} can be written uniquely in the form
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| :<math>\mathbf x = \sum_{i=1}^n x_i \mathbf{e}_i.</math>
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| == Geometric properties and uses ==
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| === Orientation ===
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| The fact that [[real numbers]], unlike many other [[field (mathematics)|fields]], constitute an [[ordered field]] yields an [[orientation (vector space)|orientation structure]] on {{math|'''R'''<sup>''n''</sup>}}. Any [[rank (matrix theory)|full-rank]] linear map of {{math|'''R'''<sup>''n''</sup>}} to itself either preserves or reverses orientation of the space depending on the [[sign (mathematics)|sign]] of the [[determinant]] of its matrix. If one [[permutation|permutes]] coordinates (or, in other words, elements of the basis), the resulting orientation will depend on the [[parity of a permutation|parity of the permutation]].
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| [[Diffeomorphism]]s of {{math|'''R'''<sup>''n''</sup>}} or [[domain (mathematical analysis)|domains in it]], by their virtue to avoid zero [[Jacobian matrix and determinant|Jacobian]], are also classified to orientation-preserving and orientation-reversing. It has important consequences for the theory of [[differential form]]s, whose applications include [[electrodynamics]].
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| Another manifestation of this structure is that the [[point reflection]] in {{math|'''R'''<sup>''n''</sup>}} has different properties depending on [[even and odd numbers|evenness of {{mvar|n}}]]. For even {{mvar|n}} it preserves orientation, while for odd {{mvar|n}} it is reversed (see also [[improper rotation]]).
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| === Affine space ===
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| {{details|Affine space}}
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| <!-- The initial (16 April, 2013) revision of this section used some pieces of content from the [[Euclidean space]] article (see the history there for attribution) -->
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| {{math|'''R'''<sup>''n''</sup>}} understood as an affine space is the same space, where {{math|'''R'''<sup>''n''</sup>}} as a vector space [[group action|acts]] by [[translation (geometry)|translations]]. Conversely, a vector has to be understood as a "[[displacement (vector)|difference]] between two points", usually illustrated by a directed [[line segment]] connecting two points. The distinction says that there is no [[canonical form|canonical]] choice of where the [[origin (mathematics)|origin]] should go in an affine {{mvar|n}}-space, because it can be translated anywhere.
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| === Convexity ===
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| [[File:2D-simplex.svg|thumb|The ''n''-simplex (see [[#Polytopes in Rn|below]]) is the standard convex set, that maps to every polytope, and is the intersection of the standard {{math|(''n'' + 1)}} affine hyperplane (standard affine space) and the standard {{math|(''n'' + 1)}} orthant (standard cone).]]
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| {{details|Convex analysis}}
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| In a real vector space, such as {{math|'''R'''<sup>''n''</sup>}}, one can define a convex [[cone (linear algebra)|cone]], which contains all ''non-negative'' linear combinations of its vectors. Corresponding concept in an affine space is a [[convex set]], which allows only [[convex combination]]s (non-negative linear combinations that sum to 1).
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| In the language of [[universal algebra]], a vector space is an algebra over the universal vector space {{math|'''R'''<sup>∞</sup>}} of finite sequences of coefficients, corresponding to finite sums of vectors, while an affine space is an algebra over the universal affine hyperplane in this space (of finite sequences summing to 1), a cone is an algebra over the universal [[orthant]] (of finite sequences of nonnegative numbers), and a convex set is an algebra over the universal [[simplex]] (of finite sequences of nonnegative numbers summing to 1). This geometrizes the axioms in terms of "sums with (possible) restrictions on the coordinates".
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| Another concept from convex analysis is a [[convex function]] from {{math|'''R'''<sup>''n''</sup>}} to real numbers, which is defined through an [[inequality (mathematics)|inequality]] between its value on a convex combination of [[point (geometry)|points]] and sum of values in those points with the same coefficients.
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| === Euclidean space ===
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| {{main|Euclidean space|Cartesian coordinate system}}
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| The [[dot product]]
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| :<math>\mathbf{x}\cdot\mathbf{y} = \sum_{i=1}^n x_iy_i = x_1y_1+x_2y_2+\cdots+x_ny_n</math>
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| defines [[normed vector space|the norm]] {{math|1={{!}} '''x''' {{!}} = }}{{sqrt|'''x'''⋅'''x'''}} on the vector space {{math|'''R'''<sup>''n''</sup>}}. If every vector has its [[Euclidean norm]], then for any pair of points the distance
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| :<math>d(\mathbf{x}, \mathbf{y}) = \|\mathbf{x} - \mathbf{y}\| = \sqrt{\sum_{i=1}^n (x_i - y_i)^2}</math>
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| is defined, providing a [[metric space]] structure on {{math|'''R'''<sup>''n''</sup>}} in addition to its affine structure.
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| Like for vector space structure, the dot product and Euclidean distance usually are assumed existing on {{math|'''R'''<sup>''n''</sup>}} without special explanations. Though, the real {{mvar|n}}-space and a Euclidean {{mvar|n}}-space are distinct objects, strictly speaking. Any Euclidean {{mvar|n}}-space has a [[coordinate system]] where dot product and Euclidean distance have the form shown above, called [[Renatus Cartesius|''Cartesian'']]. But there are ''many'' Cartesian coordinate systems on a Euclidean space.
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| Conversely, the formula for Euclidean metric above defines the ''standard'' Euclidean structure on {{math|'''R'''<sup>''n''</sup>}}, but it is not the only possible one. Actually, any [[positive-definite quadratic form]] {{mvar|q}} defines its own "distance" {{sqrt|''q''('''x''' − '''y''')}}, but it is not very different from the Euclidean one in the sense that
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| :<math>\exist C_1 > 0,\ \exist C_2 > 0,\ \forall \mathbf{x}, \mathbf{y} \in \mathbb{R}^n:
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| C_1 d(\mathbf{x}, \mathbf{y}) \le \sqrt{q(\mathbf{x} - \mathbf{y})} \le
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| C_2 d(\mathbf{x}, \mathbf{y}). </math>
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| Such change of the metric preserves some its properties, for example the property to be a [[complete metric space]].
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| This also implies that any full-rank linear transformation of {{math|'''R'''<sup>''n''</sup>}}, or its [[affine transformation]], does not magnify distances more than by some fixed {{math|C<sub>2</sub>}}, and does not make distances smaller than {{math|1 ∕ C<sub>1</sub>}} times, a fixed finite number times smaller.
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| Aforementioned equivalence of metric functions remains in place if {{sqrt|''q''('''x''' − '''y''')}} is replaced with {{math|''M''('''x''' − '''y''')}}, where {{mvar|M}} is any convex positive [[homogeneous function]] of degree 1, i.e. a [[normed vector space|vector norm]] (see [[Minkowski distance]] for useful examples). Because of this fact that any "natural" metric on {{math|'''R'''<sup>''n''</sup>}} is not especially different from Euclidean, {{math|'''R'''<sup>''n''</sup>}} is not always distinguished from a Euclidean {{mvar|n}}-space even in professional mathematical works.
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| === In algebraic and differential geometry ===
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| Although the definition of a [[manifold]] does not require that its model space should be {{math|'''R'''<sup>''n''</sup>}}, this choice is the most common, and almost exclusive one in [[differential geometry]].
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| On the other hand, [[Whitney embedding theorem]]s state that any real [[differentiable manifold|differentiable {{mvar|m}}-dimensional manifold]] can be [[embedding|embedded]] into {{math|'''R'''<sup>2''m''</sup>}}.
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| {{expand section|date=April 2013}}
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| === Other appearances ===
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| Other structures considered on {{math|'''R'''<sup>''n''</sup>}} include the one of a [[pseudo-Euclidean space]], [[symplectic structure]] (even {{mvar|n}}), and [[contact structure]] (odd {{mvar|n}}). All these structures, although can be defined in a coordinate-free manner, admit standard (and reasonably simple) forms in coordinates.
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| {{math|'''R'''<sup>''n''</sup>}} is also a real vector subspace of [[complex coordinate space|{{math|'''C'''<sup>''n''</sup>}}]] which is invariant to [[complex conjugation]]; see also [[complexification]].
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| === Polytopes in R<sup>''n''</sup> ===
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| {{see also|Linear programming|Convex polytope}}
| |
| There are three families of [[polytope]]s which have simple representations in {{math|'''R'''<sup>''n''</sup>}} spaces, for any {{mvar|n}}, and can be used to visualize any affine coordinate system in a real {{mvar|n}}-space. Vertices of [[hypercube]] have coordinates {{math|(''x''<sub>1</sub>, ''x''<sub>2</sub>, … , ''x''<sub>''n''</sub>)}} where each {{mvar|x<sub>k</sub>}} is either 0 or 1. Actually, any two numbers can be chosen instead of 0 and 1, for example {{num|−1}} and 1. An {{mvar|n}}-hypercube can be thought of as the Cartesian product of {{mvar|n}} identical [[interval (mathematics)|intervals]] (such as the [[unit interval]] {{closed-closed|0,1}}) on the real line. As an {{mvar|n}}-dimensional subset it can be described with a [[system of inequalities|system of {{math|2''n''}} inequalities]]:
| |
| {| align=left
| |
| |<math>\displaystyle\begin{matrix}
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| 0 \le x_1 \le 1 \\
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| \vdots \\
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| 0 \le x_n \le 1
| |
| \end{matrix}
| |
| </math> (for {{closed-closed|0,1}})
| |
| |<math>\displaystyle\begin{matrix}
| |
| |x_1| \le 1 \\
| |
| \vdots \\
| |
| |x_n| \le 1
| |
| \end{matrix}
| |
| </math> (for {{closed-closed|−1,1}})
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| |}<br clear="left"/>
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| Each vertex of the [[cross-polytope]] has, for some {{mvar|k}}, the {{mvar|x<sub>k</sub>}} coordinate equal to [[±1]] and all other coordinates equal to 0 (such that it is the {{mvar|k}}th [[#Standard basis|standard basis vector]] up to [[sign (mathematics)|sign]]). This is a [[dual polytope]] of hypercube. As an {{mvar|n}}-dimensional subset it can be described with a single inequality which uses the [[absolute value]] operation:
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| :<math>\sum\limits_{k=1}^n |x_k| \le 1\,,</math>
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| but this can be expressed with a system of [[power of two|{{math|2<sup>''n''</sup>}}]] linear inequalities as well.
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| | |
| The third polytope with simply enumerable coordinates is the [[standard simplex]], whose vertices are {{mvar|n}} standard basis vectors and [[origin (mathematics)|the origin]] {{math|(0, 0, … , 0)}}. As an {{mvar|n}}-dimensional subset it is described with a system of {{math|''n'' + 1}} linear inequalities:
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| : <math>\begin{matrix}
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| 0 \le x_1 \\
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| \vdots \\
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| 0 \le x_n \\
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| \sum\limits_{k=1}^n x_k \le 1
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| \end{matrix}
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| </math>
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| Replacement of all "≤" with "<" gives interiors of these polytopes.
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| | |
| == Topological properties ==
| |
| The [[topology (structure)|topological structure]] of {{math|'''R'''<sup>''n''</sup>}} (called '''standard topology''', '''Euclidean topology''', or '''usual topology''') can be obtained not only [[#Definition and uses|from Cartesian product]]. It is also identical to the [[natural topology]] induced by [[#Euclidean space|Euclidean metric discussed above]]: a set is [[open set|open]] in the Euclidean topology [[if and only if]] it contains an [[open ball]] around each of its points. Also, {{math|'''R'''<sup>''n''</sup>}} is a [[linear topological space]] (see [[#continuity of linear maps|continuity of linear maps]] above), and there is only one possible (non-trivial) topology compatible with its linear structure. As there are many open linear maps from {{math|'''R'''<sup>''n''</sup>}} to itself which are not [[isometry|isometries]], there can be many Euclidean structures on {{math|'''R'''<sup>''n''</sup>}} which correspond to the same topology. Actually, it does not depend much even on the linear structure: there are many non-linear [[diffeomorphism]]s (and other homeomorphisms) of {{math|'''R'''<sup>''n''</sup>}} onto itself, or its parts such as a Euclidean open ball or [[#Polytopes in Rn|the interior of a hypercube]]).
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| | |
| {{math|'''R'''<sup>''n''</sup>}} has the [[topological dimension]] {{mvar|n}}.
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| <!-- The initial (16 April, 2013) revision of this section used a content from the [[Euclidean space]] article. See the history there for attribution. -->
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| An important result on the topology of {{math|'''R'''<sup>''n''</sup>}}, that is far from superficial, is [[L. E. J. Brouwer|Brouwer]]'s [[invariance of domain]]. Any subset of {{math|'''R'''<sup>''n''</sup>}} (with its [[subspace topology]]) that is [[homeomorphic]] to another open subset of {{math|'''R'''<sup>''n''</sup>}} is itself open. An immediate consequence of this is that {{math|'''R'''<sup>''m''</sup>}} is not [[homeomorphism|homeomorphic]] to {{math|'''R'''<sup>''n''</sup>}} if {{math|''m'' ≠ ''n''}} – an intuitively "obvious" result which is nonetheless difficult to prove.
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| | |
| Despite the difference in topological dimension, and contrary to a naïve perception, it is possible to map a lesser-dimensional real space continuously and [[surjective function|surjectively]] onto {{math|'''R'''<sup>''n''</sup>}}. A continuous (although not smooth) [[space-filling curve]] (an image of {{math|'''R'''<sup>1</sup>}}) is possible.
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| | |
| == Examples ==
| |
| {| align=right style="margin: 2ex 0 2ex 2em"
| |
| | align=center |[[Image:Real 0-space.svg|52px]]
| |
| |-
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| | style="font-size:80%" |[[empty matrix|Empty]] column vector,<br/>the only element of {{math|'''R'''<sup>0</sup>}}
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| |}
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| {| align=left style="margin: 0 2em 1ex 0"
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| |[[Image:Real 1-space, orthoplex.svg|52px]]
| |
| |-
| |
| |{{math|'''R'''<sup>1</sup>}}
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| |}
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| | |
| === ''n'' ≤ 1 ===
| |
| Cases of {{math|0 ≤ ''n'' ≤ 1}} do not offer anything new: {{math|'''R'''<sup>1</sup>}} is the [[real line]], whereas {{math|'''R'''<sup>0</sup>}} (the space of empty vectors) is a [[singleton (mathematics)|singleton]], understood as [[zero vector space]]. Though, it is useful to include these "trivial" cases to theories which are appropriate for different {{mvar|n}}.
| |
| | |
| === ''n'' = 2 ===
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| [[Image:Real 2-space, orthoplex.svg|thumb|right|264px|Both hypercube and cross-polytope in {{math|'''R'''<sup>2</sup>}} are [[square]]s, but coordinates of vertices are arranged differently]]
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| {{details|Two-dimensional space}}
| |
| {{details|Cartesian plane}}
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| {{see also|SL2(R)}}
| |
| {{expand section|date=April 2013}}
| |
| | |
| === ''n'' = 3 ===
| |
| [[Image:Duality_Hexa-Okta_SVG.svg|thumb|left|[[Cube]] (the hypercube) and [[octahedron]] (the cross-polytope) of {{math|'''R'''<sup>3</sup>}}. Coordinates are not shown]]
| |
| {{details|Three-dimensional space}}
| |
| {{expand section|date=April 2013}}
| |
| <br clear="left"/>
| |
| | |
| === ''n'' = 4 ===
| |
| [[Image:4-cube 3D.png|thumb|right]]
| |
| {{details|Four-dimensional space}}
| |
| {{math|'''R'''<sup>4</sup>}} can be imagined using the fact that {{num|16}} points {{math|(''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>, ''x''<sub>4</sub>)}}, where each {{mvar|x<sub>k</sub>}} is either 0 or 1, are vertices of a [[tesseract]] (pictured), the 4-hypercube (see [[#Polytopes in Rn|above]]).
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| | |
| The first major use of {{math|'''R'''<sup>4</sup>}} is a [[spacetime]] model: three spacial coordinates plus one [[time|temporal]]. This is usually associated with [[theory of relativity]], although four dimensions were continuously used for such models since [[Galileo Galilei|Galilei]]. The choice of theory leads to different structure, though: in [[Galilean relativity]] the {{mvar|t}} coordinate is privileged, but in Einsteinian relativity it is not. Special relativity is set in [[Minkowski space]]. General relativity uses curved spaces, which may be thought of as {{math|'''R'''<sup>4</sup>}} with a [[metric tensor (general relativity)|curved metric]] for most practical purposes, though. Any of these structures does not provide a (positive) [[metric (mathematics)|metric]] on {{math|'''R'''<sup>4</sup>}}.
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| | |
| Euclidean {{math|'''R'''<sup>4</sup>}} also attracts the attention of mathematicians, for example due to its relation to [[quaternion]]s, a 4-dimensional [[algebra over a field|real algebra]] themselves. See [[rotations in 4-dimensional Euclidean space]] for some information.
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| | |
| In differential geometry, {{math|1=''n'' = 4}} is the only case where {{math|'''R'''<sup>''n''</sup>}} admits a non-standard [[differential structure]]: see [[exotic R4|exotic R<sup>4</sup>]].
| |
| | |
| == Generalizations ==
| |
| <!-- something about infinite dimension, please… -->
| |
| {{expand section|date=April 2013}}
| |
| | |
| == See also ==
| |
| * [[Exponential object]], for theoretical explanation of the superscript notation
| |
| * [[Real projective space]]
| |
| | |
| == Footnotes ==
| |
| {{reflist}}
| |
| | |
| ==References==
| |
| *{{cite book | author=Kelley, John L. | title=General Topology | publisher=Springer-Verlag | year=1975 | isbn= 0-387-90125-6 }}
| |
| *{{cite book | author=Munkres, James | title=Topology | publisher=Prentice-Hall | year=1999 | isbn= 0-13-181629-2 }}
| |
| | |
| [[Category:Real numbers|ⁿ]]
| |
| [[Category:Topological vector spaces]]
| |
| [[Category:Analytic geometry]]
| |
| [[Category:Multivariable calculus]]
| |
| [[Category:Mathematical analysis]]
| |
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