atlas

Dictionary: at·las¹ (ăt'ləs) pronunciation

n., pl., -las·es.

A book or bound collection of maps, sometimes with supplementary illustrations and graphic analyses.
A volume of tables, charts, or plates that systematically illustrates a particular subject: an anatomical atlas.
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.

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.
Anatomy. The top or first cervical vertebra of the neck, which supports the skull.

[From ATLAS.]

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US Military Dictionary: Atlas

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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.

Columbia Encyclopedia: atlas

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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.

Geography: atlas

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

Veterinary Dictionary: atlas

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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.

Word Tutor: atlas

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IN BRIEF: A collection of maps in book form.

If you travel, take a road atlas along.

Wikipedia: Atlas (topology)

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

In mathematics, particularly topology, an atlas describes how a manifold is equipped with a differential structure. Each piece is given by a chart (also known as coordinate chart or local coordinate system).

Before giving the formal definition of an atlas, we recall that a chart on a manifold M is defined to be a homeomorphism $φ$ from an open subset U of M to an open subset V of $\mathbb{R}^n$ . If $(U_{\alpha}, \varphi_{\alpha})$ and $(U_{\beta}, \varphi_{\beta})$ are two charts on M such that $U_{\alpha} \cap U_{\beta}$ is non-empty, then define the transition map

$\varphi_{\alpha,\beta} : \varphi_{\alpha}(U_{\alpha} \cap U_{\beta}) \to \varphi_{\beta}(U_{\alpha} \cap U_{\beta})$ , $\varphi_{\alpha,\beta} = \varphi_{\beta} \circ \varphi_{\alpha}^{-1}.$

Note that since $\varphi_{\alpha}$ and $\varphi_{\beta}$ are both homeomorphisms, the transition maps are also homeomorphisms. So, the transition maps are already endowed with a kind of compatibility in the sense that changing from the coordinate system on one chart to the coordinate system on another chart is continuous.

Then an atlas on a manifold M is a collection $\mathcal{A} = \{(U_{\alpha}, \varphi_{\alpha})\}$ of charts on M whose domains cover M.

Now, we say that two overlapping charts $(U_{\alpha}, \varphi_{\alpha})$ and $(U_{\beta}, \varphi_{\beta})$ are smoothly compatible if the transition map between them is infinitely differentiable as a map from Euclidean space to itself.

Having defined these notions, a smooth atlas on M is an atlas where we make the additional requirement that, for any two overlapping charts on M, the transition maps between them are smoothly compatible.

Two atlases $\mathcal{A}$ and $\mathcal{B}$ on M are smoothly compatible if all charts in $\mathcal{A}$ which overlap charts in $\mathcal{B}$ are smoothly compatible. If this is the case then $\mathcal{A} \cup \mathcal{B}$ is also a smooth atlas on M. This gives a natural equivalence relation, from which we can consider an equivalence class of smoothly compatible atlases, which we call the maximal atlas. A manifold M together with a maximal atlas is said to have a smooth structure. There are, in higher dimensions, examples of topological manifolds with multiple different smooth structures. One of the first examples was John Milnor's discovery of an exotic sphere, a 7-manifold which is homeomorphic to the 7-sphere but not diffeomorphic.

In general, doing computations with the maximal atlas of a manifold is unwieldy and we need only choose one particular smooth atlas to work with. Maximal atlases are needed for the unambiguous definition of smooth maps from one manifold to another.

The differentiability requirements on the transition functions can be weakened, so that we only require the transition maps to be k-times continuously differentiable; or strengthened, so that we require the transition maps to real-analytic. Accordingly, this gives a $C k$ or analytic structure on the manifold rather than a smooth one. Similarly, we can define a complex manifold by requiring the transition maps to be holomorphic.

References

Lee, John M. (2006). Introduction to Smooth Manifolds. Springer-Verlag. ISBN 978-0-387-95448-6.

Sepanski, Mark R. (2007). Compact Lie Groups. Springer-Verlag. ISBN 978-0-387-30263-8.

External links

Atlas by Rowland, Todd

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Translations: Atlas

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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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		Dictionary. The American Heritage® Dictionary of the English Language, Fourth Edition Copyright © 2007, 2000 by Houghton Mifflin Company. Updated in 2009. Published by Houghton Mifflin Company. All rights reserved. Read more
		US Military Dictionary. The Oxford Essential Dictionary of the U.S. Military. Copyright © 2001, 2002 by Oxford University Press, Inc. All rights reserved. Read more
		Columbia Encyclopedia. The Columbia Electronic Encyclopedia, Sixth Edition Copyright © 2003, Columbia University Press. Licensed from Columbia University Press. All rights reserved. www.cc.columbia.edu/cu/cup/. Read more
		Geography. The New Dictionary of Cultural Literacy, Third Edition Edited by E.D. Hirsch, Jr., Joseph F. Kett, and James Trefil. Copyright © 2002 by Houghton Mifflin Company. Published by Houghton Mifflin. All rights reserved. Read more
		Veterinary Dictionary. Saunders Comprehensive Veterinary Dictionary 3rd Edition. Copyright © 2007 by D.C. Blood, V.P. Studdert and C.C. Gay, Elsevier. All rights reserved. Read more
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