atlas

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(ăt'ləs) pronunciation
n., pl., -las·es.
  1. A book or bound collection of maps, sometimes with supplementary illustrations and graphic analyses.
  2. A volume of tables, charts, or plates that systematically illustrates a particular subject: an anatomical atlas.
  3. A large size of drawing paper, measuring 26 × 33 or 26 × 34 inches.

[After Atlas, probably from depictions of him holding the world on his shoulders that appeared on the frontispieces of early works of this kind.]


at·las2 (ăt'ləs) pronunciation
n., pl., -es.
  1. pl., at·lan·tes (ăt-lăn'tēz). Architecture. A standing or kneeling figure of a man used as a supporting column, as for an entablature or balcony.
  2. Anatomy. The top or first cervical vertebra of the neck, which supports the skull.

[From ATLAS.]


The first intercontinental ballistic missile built by the United States, tested in 1958 and deployed in 1959. See also Minuteman.

See the Introduction, Abbreviations and Pronunciation for further details.

atlas, in geography, collection of maps or charts. It usually includes data on various features of a country, e.g., its topography, natural resources, climate, and population, as well as its agriculture and main industries. In astronomy, a star atlas is a collection of maps or photographs covering much or all of the celestial sphere and showing the locations of stars and other objects. Although the first known atlas was compiled by the Greek geographer Ptolemy in the 2d cent. A.D., its modern form was introduced in 1570 with the publication of Theatrum orbis terrarum by the Flemish geographer Abraham Ortelius. In 1595 his close friend Gerardus Mercator published Atlas sive cosmographicae. Its frontispiece was a figure of the titan Atlas holding a globe on his shoulders. The name Atlas subsequently came to be applied to volumes of maps and information in this format.


A bound collection of maps. Atlases are named after the Greek god Atlas.

The first cervical vertebra, the uppermost segment of the backbone which supports the skull, characterized by the absence of a body and a wide vertebral canal.

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Atlas (topology)

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For other uses of "atlas", see Atlas (disambiguation).

In mathematics, particularly topology, one describes a manifold using an atlas. An atlas consists of individual charts that, roughly speaking, describe individual regions of the manifold. If the manifold is the surface of the Earth, then an atlas has its more common meaning. In general, the notion of atlas underlies the formal definition of a manifold.

Contents

Charts

The definition of an atlas depends on the notion of a chart. A chart for a topological space M is a homeomorphism \varphi from an open subset U of M to an open subset of Euclidean space. The chart is traditionally recorded as the ordered pair  (U, \varphi).

Formal definition of atlas

An atlas for a topological space M is a collection  \{(U_{\alpha}, \varphi_{\alpha})\} of charts on M such that  \bigcup U_{\alpha} = M. If the range of each chart is the n-dimensional Euclidean space, then M is said to be an n-dimensional manifold.

Transition maps

A transition map provides a way of comparing two charts of an atlas. To make this comparison, we consider the composition of one chart with the inverse of the other. This composition is not well-defined unless we restrict both charts to the intersection of their domains of definition. (For example, if we have a chart of Europe and a chart of Russia, then we can compare these two charts on their overlap, namely the European part of Russia.)

To be more precise, suppose that (U_{\alpha}, \varphi_{\alpha}) and (U_{\beta}, \varphi_{\beta}) are two charts for a manifold M such that U_{\alpha} \cap U_{\beta} is non-empty. The transition map  \tau_{\alpha,\beta} is the map defined on the intersection U_{\alpha} \cap U_{\beta} by

\tau_{\alpha,\beta} = \varphi_{\beta} \circ \varphi_{\alpha}^{-1}.

Note that since \varphi_{\alpha} and \varphi_{\beta} are both homeomorphisms, the transition map  \tau_{\alpha, \beta} is also a homeomorphism.

More structure

One often desires more structure on a manifold than simply the topological structure. For example, if one would like an unambiguous notion of differentiation of functions on a manifold, then it is necessary to construct an atlas whose transition functions are differentiable. Then one can unambiguously define the notion of tangent vectors and then directional derivatives.

If each transition function is a smooth map, then the atlas is called a smooth atlas. Alternatively, one could require that the transition maps have only k continuous derivatives in which case the atlas is said to be  C^k .

Very generally, if each transition function belongs to a pseudo-group  {\mathcal G} of homeomorphisms of Euclidean space, then the atlas is called a  {\mathcal G}-atlas.

References

External links


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Dansk (Danish)
1.
n. - atlas, kortbog

2.
n. - atlashvirvel, øverste halshvirvel

n. - Atlas

Nederlands (Dutch)
atlas, bovenste halswervel, steunpilaar

Français (French)
1.
n. - (Géog) atlas, (Anat) atlas, atlas (un format), (Archit) atlante, télamon

2.
n. - (Mythol) Atlas, (Monts) de l'Atlas

Deutsch (German)
1.
n. - Atlas

2.
n. - Atlasseide

n. - Atlas

Ελληνική (Greek)
n. - 'Ατλας, (γεωγραφικός) άτλαντας, άτλας, (ανατ.) άτλας, επιστροφέας

Italiano (Italian)
atlante

Português (Portuguese)
n. - atlas (m)

Русский (Russian)
атлас, титан, могучая опора, карта

Español (Spanish)
1.
n. - atlas

2.
n. - atlante

Svenska (Swedish)
n. - atlas, kartbok, atlas (med.)

中文(简体)(Chinese (Simplified))
1. 地图集, 图解集

2. 巨神阿特拉斯, 身负重担的人

中文(繁體)(Chinese (Traditional))
1.
n. - 地圖集, 圖解集

2.
n. - 巨神阿特拉斯, 身負重擔的人

한국어 (Korean)
1.
n. - 지도책, 대역, 미국의 우주 개발용 로켓

2.
n. - 아틀라스(신들을 배반한 죄로 하늘을 짊어지게 된 신)

日本語 (Japanese)
n. - 地図帳, 図解, アトラス, 図表集, アトラス判

العربيه (Arabic)
‏(الاسم) أطلس, مصور جغرافي , مجموعه خرائط جغرافيه مجلده, الفهقه : فقهرة العنق الأولى‏

עברית (Hebrew)
n. - ‮אטלס, מפון‬
n. - ‮חוליית הצוואר‬
n. - ‮אטלס‬


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