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  • Banach space/Related Articles
    {{r|Euclidean space}}
    423 bytes (60 words) - 15:14, 28 July 2009
  • Normed space
    #The Euclidean space <math>\mathbb{R}^n</math> endowed with the Euclidean norm <math>\|x\|=\sqrt
    982 bytes (148 words) - 07:17, 3 December 2007
  • Vector field/Definition
    A vector function on the three-dimensional [[Euclidean space]] <font style="vertical-align: 9.2%"> <math>\scriptstyle \mathbb{E}^3</math
    182 bytes (20 words) - 04:04, 31 October 2008
  • Norm (mathematics)/Definition
    ...vector space that generalises the notion of the distance from a point of a Euclidean space to the origin.
    157 bytes (26 words) - 15:01, 4 January 2009
  • Dot product/Definition
    A type of [[vector]] multiplication in [[Euclidean space]]s which produces a [[scalar]] result.
    131 bytes (17 words) - 13:46, 7 July 2008
  • Topological space/Definition
    A mathematical structure (generalizing some aspects of Euclidean space) defined by a family of open sets.
    141 bytes (19 words) - 17:39, 17 June 2009
  • Rigid motion
    ...'' is a transformation which preserves the geometrical properties of the [[Euclidean space]]. Since Euclidean properties may be defined in terms of [[distance]], the ...of rigid motions are the ''translations'' or displacements. If we regard Euclidean space of ''n'' dimensions as an [[affine space]] built on a [[real number|real]]
    3 KB (392 words) - 14:42, 28 November 2008
  • Geometry
    ...ry)|plane]], a higher dimensional Euclidean space, a sphere or other [[non-Euclidean space]], or more generally, a [[manifold]].
    2 KB (232 words) - 03:09, 8 March 2024
  • Inner product space
    #The [[Euclidean space]] <math>\mathbb{R}^n</math> endowed with the real inner product <math>\lang
    1 KB (204 words) - 14:38, 4 January 2009
  • Heine–Borel theorem/Definition
    In Euclidean space of finite dimension with the usual topology, a subset is compact if and onl
    160 bytes (26 words) - 05:33, 29 December 2008
  • Banach space
    ...dimensional normed space is a Banach space (due to its isomorphism to some Euclidean space).
    2 KB (317 words) - 13:13, 14 July 2008
  • Euclid's Elements/Related Articles
    {{r|Euclidean space}}
    313 bytes (48 words) - 14:58, 7 December 2008
  • Complete metric space
    * The real numbers '''R''', and more generally finite-dimensional [[Euclidean space]]s, with the usual metric are complete.
    3 KB (441 words) - 12:23, 4 January 2009
  • Norm (mathematics)
    ...generalizes to vector spaces the notion of the distance from a point of a Euclidean space to the origin.
    880 bytes (157 words) - 22:28, 20 February 2010
  • Heine–Borel theorem
    A Euclidean space of fixed finite dimension ''n'' also forms a [[metric space]] with the Eucl
    2 KB (381 words) - 08:54, 29 December 2008
  • Hilbert space/Related Articles
    {{r|Euclidean space}}
    955 bytes (150 words) - 15:15, 28 July 2009
  • Compact space
    ...he set is a subset of a finite dimensional [[normed space]], such as the [[Euclidean space]]s, then compactness is equivalent to that set being closed and [[bounded s * The ''[[Heine-Borel theorem]]'': In [[Euclidean space]] with the usual topology, a [[subset]] is compact if and only if it is clo
    4 KB (652 words) - 14:44, 30 December 2008
  • Euclidean space
    A '''Euclidean space''' or, more precisely, a '''Euclidean''' '''''n'''''-'''space''' == Euclidean space ==
    9 KB (1,403 words) - 02:22, 14 October 2013
  • Parallel (geometry)
    ...ce". (This is different, of course, if, more generally (hyper-)planes in [[Euclidean space|higher-dimensional]] spaces are considered.)
    5 KB (852 words) - 05:23, 17 April 2010
  • Bounded set
    The [[Heine–Borel theorem]] states that a subset of the [[Euclidean space]] '''R'''<sup>''n''</sup> is [[compact space|compact]] if and only if it is
    1 KB (188 words) - 05:37, 29 December 2008
  • Heine–Borel theorem/Related Articles
    {{r|Euclidean space}}
    923 bytes (145 words) - 10:03, 29 December 2008
  • Metric space
    ...aking, an abstract mathematical structure that generalizes the notion of a Euclidean space <math>\mathbb{R}^n</math> which has been equipped with the Euclidean distan ..., and indeed what motivated the general definition of such a space, is the Euclidean space <math>\mathbb{R}^n</math> endowed with the Euclidean distance <math>d_E\,</
    6 KB (1,068 words) - 07:30, 4 January 2009
  • Cartesian coordinates
    ...(black) lines, labeled ''X'', ''Y'', ''Z'', are shown in three-dimensional Euclidean space. They intersect in a point ''O'', again called the origin. The lines are th
    4 KB (679 words) - 03:09, 8 March 2024
  • Great circle/Related Articles
    {{r|Euclidean space}}
    506 bytes (65 words) - 16:59, 11 January 2010
  • Mercator projection/Related Articles
    {{r|Euclidean space}}
    536 bytes (68 words) - 18:26, 11 January 2010
  • Basis (linear algebra)
    ...e provides the space with an [[invertible]] [[linear transformation]] to [[Euclidean space]], given by taking the [[coordinate (mathematics)/advanced|coordinates]] of
    3 KB (464 words) - 19:45, 1 December 2008
  • Line (Euclidean geometry)
    ...clidean space, while solid geometry is the theory of the three-dimensional Euclidean space. ...nts and lines are situated in the plane (assumed to be a two-dimensional [[Euclidean space]]).''
    10 KB (1,620 words) - 03:09, 8 March 2024
  • Line (Euclidean geometry)/Citable Version
    ...clidean space, while solid geometry is the theory of the three-dimensional Euclidean space. ...nts and lines are situated in the plane (assumed to be a two-dimensional [[Euclidean space]]).''
    10 KB (1,620 words) - 03:09, 8 March 2024
  • Measure theory
    ...a suitable notion of "size" or "measure" to certain classes of sets in the Euclidean space <math>\scriptstyle \mathbb{R}^n</math> would be crucial to developing a mor
    4 KB (604 words) - 05:50, 12 May 2008
  • Tangent space
    ...as an n-dimensional [[hyperplane]] at that point. However, this ambient [[Euclidean space]] is unnecessary to the definition of the tangent space.
    4 KB (676 words) - 00:52, 15 November 2007
  • Affine space
    ==Euclidean space== ...h a positive-definite inner product is defined. Then ''A'' is called a ''Euclidean space''. The distance between two point ''P'' and ''Q''
    15 KB (2,366 words) - 09:09, 4 April 2010
  • Neighbourhood (topology)
    ...s or real analysis) is <math> \mathbb R^d </math>, the ''d''-dimensional [[Euclidean space]]:
    7 KB (1,205 words) - 09:52, 8 September 2013
  • Neighbourhood (topology)/Citable Version
    ...s or real analysis) is <math> \mathbb R^d </math>, the ''d''-dimensional [[Euclidean space]]:
    7 KB (1,205 words) - 09:51, 8 September 2013
  • Hutchinson operator
    ...the case when the ''f''<sub>''i''</sub> are [[contraction mapping]]s on a Euclidean space ''X'' = '''R'''<sup>d</sup>. He showed that such a system of functions has
    2 KB (327 words) - 15:52, 27 October 2008
  • Space (mathematics)
    ...of a non-Euclidean geometry is a clever choice of some objects existing in Euclidean space and some relations between these objects that satisfy all axioms (therefore ...pace among other spaces. The modern approach defines the three-dimensional Euclidean space more algebraically, via linear spaces and quadratic forms, namely, as an af
    28 KB (4,311 words) - 08:36, 14 October 2010
  • Plane (geometry)/Citable Version
    ...and planes are situated in the space (assumed to be a three-dimensional [[Euclidean space]]), and by lines we mean [[Line (Euclidean geometry)|straight lines]].'' The definitions given above assume implicitly that the 3-dimensional Euclidean space is already defined, together with (at least one of) such notions as distanc
    16 KB (2,609 words) - 03:09, 8 March 2024
  • Plane (geometry)
    ...and planes are situated in the space (assumed to be a three-dimensional [[Euclidean space]]), and by lines we mean [[Line (Euclidean geometry)|straight lines]].'' The definitions given above assume implicitly that the 3-dimensional Euclidean space is already defined, together with (at least one of) such notions as distanc
    16 KB (2,638 words) - 03:10, 8 March 2024
  • Manifold (geometry)
    5 KB (805 words) - 17:01, 28 November 2008
  • Square circle
    A '''square circle''' would be a shape or object in [[Euclidean space]] that is ''both'' [[square]] ''and'' [[circle|circular]]. [[Philosophy|Ph
    2 KB (267 words) - 23:37, 25 October 2013
  • Cauchy-Schwarz inequality
    If ''V'' is the [[Euclidean space]] '''R'''<sup>''n''</sup>, whose inner product is defined by
    4 KB (629 words) - 16:46, 17 December 2008
  • Nonlinear programming
    If the objective function ''f'' is linear and the constrained [[Euclidean space|space]] is a [[polytope]], the problem is a [[linear programming]] problem,
    3 KB (405 words) - 18:02, 1 January 2008
  • Vector field
    ...tivistic physics, the space in question is usually the three-dimensional [[Euclidean space]] <math>\scriptstyle \mathbb{E}^3</math>&mdash;the infinite (Newtonian) wor
    5 KB (928 words) - 04:59, 9 December 2009
  • Polar coordinates
    In modern [[vector]] language one identifies the plane with a real [[Euclidean space]] <math>\scriptstyle \mathbb{R}^2</math> that has a [[Cartesian coordinates
    3 KB (451 words) - 23:53, 20 February 2010
  • Cauchy-Riemann equations
    .../var></sup>≡ℝ<sup><var>2n</var></sup>, identifying the [[point]]s of the [[euclidean space]]s on the [[Complex field|complex]] and [[real field]]s as follows
    6 KB (874 words) - 03:45, 7 October 2013
  • Bernhard Riemann
    ...tances along curves in the space. He argued that the space need not be a [[Euclidean space]] and that it could have any finite dimension. Riemann's work on different
    5 KB (751 words) - 11:37, 25 March 2022
  • Frame of reference (physics)
    ...atical definition:… the reference frame is... the set of all points in the Euclidean space with the rigid body motion of the observer. The frame, denoted <math>\mathf According to Hawking and Ellis: "A manifold is a space locally similar to Euclidean space in that it can be covered by coordinate patches. This structure allows diff
    29 KB (4,366 words) - 09:10, 26 March 2011
  • Measure (mathematics)
    Not all subsets of [[Euclidean space]] are [[Lebesgue measurable]]; examples of such sets include the [[Vitali s
    14 KB (2,350 words) - 17:37, 10 November 2007
  • Magnetic induction
    ...physics, the space on which '''B''' is defined is the three-dimensional [[Euclidean space]] <math>\scriptstyle \mathbb{E}^3</math>&mdash;the infinite world that we l
    9 KB (1,330 words) - 16:37, 31 March 2011
  • Magnetic field
    ...non-relativistic physics, the space in question is the three-dimensional [[Euclidean space]] <math> \mathbb{E}^3</math>&mdash;the infinite world that we live in.
    9 KB (1,428 words) - 08:19, 1 April 2011
  • Vector product
    ...''B''' &times; '''A''' of two vectors '''A''' and '''B''' in 3-dimensional Euclidean space <math>\scriptstyle \mathbb{R}^3</math>. The vector product is again a 3-d
    12 KB (1,961 words) - 13:04, 11 June 2009
  • Theory (mathematics)
    "A Euclidean space consists, by definition, of three sets, whose elements are called points, l
    34 KB (5,174 words) - 21:32, 25 October 2013
  • Molecular Hamiltonian
    ...ractions]], which are inversely proportional to the [[Distance#Distance in Euclidean space|distance]] ''r''<sub>ij</sub>
    31 KB (4,757 words) - 02:20, 27 October 2013
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