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| [[File:Cell-Shape-Dynamics-From-Waves-to-Migration-pcbi.1002392.s007.ogv|thumb|A migrating, wild-type ''[[Dictyostelium discoideum]]'' cell whose boundary is colored by curvature. Scale bar: 5 µm; duration: 22 seconds.]]
| | == 「傑傑」 == |
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| In [[mathematics]], '''curvature''' refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being ''flat,'' or ''straight'' in the case of a [[line (geometry)|line]], but this is defined in different ways depending on the context. There is a key distinction between '''extrinsic curvature''', which is defined for objects embedded in another space (usually a [[Euclidean space]]) in a way that relates to the [[radius of curvature (mathematics)|radius of curvature]] of circles that touch the object, and ''[[Curvature of Riemannian manifolds|intrinsic curvature]]'', which is defined at each point in a [[Riemannian manifold]]. This article deals primarily with the first concept.
| | ペイかすかな笑顔が暗闇のブランケットから金と鉄の音が広がりに墜落しながら、ゆっくりと、ステップバイステップで前足との間で闇の毛布から間もなくのように、奇妙な黒 [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-9.html カシオ 時計 プロトレック] '色'チェーンで埋め闇のブランケット蛇のようなSenleng光沢を点滅して、チェーンの先端に、アヒルの法執行機関に包まれ全身を拡張します。<br><br>「残りの部分では、すぐに見ることができるようになることが保証される」「カストディアンの時に、古い男は私の魂の家が望んでいることを知っている必要がありますし、誰も逃れることはできない、ほこり！ [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-5.html カシオ 時計] '医学<br><br>シャオヤン顔「色」無関心は、緑の炎が高潮、パームグリップの体から出航し、巨大な謎の重い足が出フラッシュされる、重い足フラット、アヒルの法執行機関を指して、言った： [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-10.html カシオ腕時計 価格] '今日では人生は、私が受け取った！ [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-5.html カシオ 腕時計 スタンダード] '<br><br>「傑傑」<br>シャオヤンの発言のため<br>は、その法執行機関は、ストレート、ショックを受けて、突然、彼の全身黒のチェーンに巻きつけ、すぐに雷のタッチになって黒い霧が流れ、冷たい鴨笑顔です |
| | 相关的主题文章： |
| | <ul> |
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| | <li>[http://www.wfxrmyy.com/plus/feedback.php?aid=221 http://www.wfxrmyy.com/plus/feedback.php?aid=221]</li> |
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| | <li>[http://yjditou.com/home.php?mod=space&uid=16798 http://yjditou.com/home.php?mod=space&uid=16798]</li> |
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| | <li>[http://www.brookfield-tractors.co.uk/cgi-bin/board_single.cgi http://www.brookfield-tractors.co.uk/cgi-bin/board_single.cgi]</li> |
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| | </ul> |
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| The canonical example of extrinsic curvature is that of a [[circle]], which everywhere has curvature equal to the [[Multiplicative inverse|reciprocal]] of its [[radius]]. Smaller circles bend more sharply, and hence have higher curvature. The curvature of a [[smooth curve]] is defined as the curvature of its [[osculating circle]] at each point.
| | == 「宗派間のアセンブリ == |
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| More commonly this is a [[scalar (mathematics)|scalar]] quantity, but one may also define a [[curvature vector]] that takes into account the direction of the bend as well as its sharpness. The curvature of more complex objects (such as [[surface]]s or even curved ''n''-dimensional [[space]]s) is described by more complex objects from [[linear algebra]], such as the general [[Riemann curvature tensor]].
| | 彼らの頭には、それらがやや困難にすると、ビューのアヒルの強点に主張する強い男のカストディアンの日から猶予を得るために、この神秘的な力が、クラウドLANから遠く同等であった カシオ ソーラー 腕時計。<br><br>家の「神秘的な魂、彼らがどこにあるか、少数の人が知っている、私たちのガマ帝国の強さではなく、私が期待したように、それらを適格に連絡十分ではありません、家のこの魂は、さえ執念深い大陸全体を見て、おそらくされている私たちはある場合であっても、強い力としてカウントされ、私たちガマ帝国も、クラウドLANを過ぎて、それが唯一の帝国内で唯一の横暴であり、ましてや本土の恨みの無限のマイルに入れることができるこの大陸北西ゾーンのみ二級部隊としてカウントすることができ、「ハイペリオン東笑顔はすぐに叫んだ： 'この大陸宗派北西ゾーン勤続5年議会で、とても強い帝国にクラウド型LANを見てはいけない、治療はそこに行くしないようにすることができ、傲慢嘲笑は多くを耐えることができます。 カシオ 腕時計 ソーラー 電波 '<br>「宗派間のアセンブリ？ カシオ gps 時計 'ウェンヤン、ヤンシャオを<br> |
| | 相关的主题文章： |
| | <ul> |
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| | <li>?aid=1757538</li> |
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| | <li>http://www.music-space.jp/cgi-bin/epad/msp_epad.cgi</li> |
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| | <li>?mod=space&uid=8987</li> |
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| | </ul> |
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| The remainder of this article discusses, from a mathematical perspective, some geometric examples of curvature: the curvature of a curve embedded in a plane and the curvature of a surface in Euclidean space.
| | == 行う方法長老、' == |
| See the links below for further reading.
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| == Curvature of plane curves ==
| | 「7日」 [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-0.html casio 腕時計]<br>古代チンヤンが正面を見て、叫び笑顔<br>は、古南海古いの多くを持っているように見えた一瞬ためらった、今言った： [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-12.html 腕時計 メンズ casio] '？。行う方法長老、'<br><br>区南海は頭、声かすれ道路を横に振った「わからない」 [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-7.html カシオ 腕時計 gps]。<br>余談<br>、「医学 [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-13.html カシオ アナログ 腕時計] '古い顔'色 'は、また、いくつかの疲れ徐ギャングゆう話し、わずかな「スイング」紫の影の研究の前のスペースは、後者の「露」表面を点滅さ明らかにする、彼は首を横に振って言った直面している： '。いいえ火災悪魔空間は、空間が、私はその方向を知覚することはできませんシールの一枚を持っていない」<br><br>は、聞いたような「色」として '医療'老人の顔、、だけでなく、暗い。<br><br>「彼の事故が、その後、ドラゴンプリントが消えた場合シャオヤンは、問題ないはずです [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-14.html casio 腕時計 phys]。「手のうちパープルの研究は、その手のひらに、淡い金色のドラゴンプリント「色」で、彼女Pianguoの塔は、古代の南に見える |
| [[Cauchy]] defined the centre of curvature ''C'' as the intersection point of two [[infinitesimal|infinitely close]] normals to the curve, the radius of curvature as the distance from the point to ''C'', and the curvature itself as the inverse of the radius of curvature.<ref>*{{citation | | 相关的主题文章： |
| | last1 = Borovik | first1 = Alexandre
| | <ul> |
| | author1-link = Alexandre Borovik
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| | last2 = Katz | first2 = Mikhail G.
| | <li>[http://www.zxxxcx.com/plus/feedback.php?aid=250 http://www.zxxxcx.com/plus/feedback.php?aid=250]</li> |
| | author2-link = Mikhail Katz
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| | arxiv = 1108.2885
| | <li>[http://bbsc.ziyu.net/bbs/hit10000/index.cgi http://bbsc.ziyu.net/bbs/hit10000/index.cgi]</li> |
| | doi = 10.1007/s10699-011-9235-x
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| | issue =
| | <li>[http://www.52813.cc/thread-16818-1-1.html http://www.52813.cc/thread-16818-1-1.html]</li> |
| | journal = [[Foundations of Science]]
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| | pages =
| | </ul> |
| | title = Who gave you the Cauchy--Weierstrass tale? The dual history of rigorous calculus
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| | volume =
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| | year = 2011}}</ref> | |
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| Let ''C'' be a [[plane curve]] (the precise technical assumptions are given below). The curvature of ''C'' at a point is a measure of how sensitive its [[tangent line]] is to moving the point to other nearby points. There are a number of equivalent ways that this idea can be made precise.
| | == ジャングルのいくつかを経て、道に沿って、山へのNalanは甘い == |
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| [[Image:Osculating circle.svg|float|right|250px]] | | 愛が起こっている、とにかく変更することはできませんので、言及していない。 [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-5.html カシオの時計] '<br><br>白い歯が唇をかじるので、シャオヤンを見て見て、苦いNalan甘い口が突破し、それは苦い醸造、それ自体であるように、今まで食べて、それは本当に心臓に難しいです [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-12.html 時計 カシオ]。<br><br>控えめな笑顔、Nalan甘い [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-12.html 腕時計 メンズ casio] '私は、その年あなたがそれらの事を忘れたくない、ちょうどその私Nalan甘い、確かにやや近視眼的なので、今日だけでなく、トラブルを求めて下さい。その後を言いたい」離れて山のスキミングになって、「私と一緒に来て。 [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-5.html 時計 casio] '<br><br>影の前にかなりのダンスを見て、シャオヤンの目は少しルマンちらつき、一瞬の後、火の後ろの翼の振動がすぐに続く [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-5.html カシオ 腕時計 スタンダード]。<br>ジャングルのいくつかを経て、道に沿って、山へのNalanは甘い<br>従う、険しい崖の最後で停止しました。<br><br>「先生が崖にあり、あなたが行く |
| One way is geometrical. It is natural to define the curvature of a [[straight line]] to be identically zero. The curvature of a circle of radius ''R'' should be large if ''R'' is small and small if ''R'' is large. Thus the curvature of a circle is defined to be the reciprocal of the radius:
| | 相关的主题文章： |
| | <ul> |
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| | <li>[http://221.2.159.212:90/jpk12/yuyan/plus/view.php?aid=32786 http://221.2.159.212:90/jpk12/yuyan/plus/view.php?aid=32786]</li> |
| | |
| | <li>[http://www.shiboyjy.com/plus/feedback.php?aid=149 http://www.shiboyjy.com/plus/feedback.php?aid=149]</li> |
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| | <li>[http://www.zushu.org/plus/view.php?aid=40187 http://www.zushu.org/plus/view.php?aid=40187]</li> |
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| | </ul> |
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| : <math> \kappa = \frac{1}{R}.</math>
| | == 「何ですか == |
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| Given any curve ''C'' and a point ''P'' on it, there is a unique circle or line which most closely approximates the curve near ''P'', the [[osculating circle]] at ''P''. The curvature of ''C'' at ''P'' is then defined to be the curvature of that circle or line. The radius of curvature is defined as the reciprocal of the curvature.
| | ボックスを開き、突然硬い体が、すぐに急に暗いとなり、迅速になって、石門面「色」を見た [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-12.html カシオ 時計 電波]。<br><br>「誰かが来ている！ '<br><br>「何ですか？」聞いた、均等に驚い小さな医療セント、すぐに首を横に振ったと言った： [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-13.html 時計 カシオ] '！が出来ないが、ここでは両方がそれを知っている」<br><br>「私はまだ多くの人の数に、それは間違っていない！ [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-2.html casio 腕時計 デジタル] 'シャオヤンの顔「色」醜い小さな医療セントを見つめ、Hanmangの目が光った。<br><br>は「私はあなたがそれである疑いがあると呼ばれる？ [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-0.html カシオ 腕時計 バンド] '<br><br>は小さな医療セントが突然怒っQiaolian、シャオヤンの顔を見て： '私はあなたを扱う場合には、あなたの早死に数回！'<br>怒っていないようだ小型医療セント詐欺を見て<br>、シャオヤンはすぐに、眉をひそめ振り向く、手が絶えず伸ばし鍵穴の鍵は、あなたが状況で緊張を感じることができるが、常に「プラグ」になりません [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-10.html カシオ 腕時計 スタンダード]。<br><br>「くそ！ '怒って呪われ、シャオヤン |
| | | 相关的主题文章： |
| Another way to understand the curvature is physical. Suppose that a particle moves along the curve with unit speed. Taking the time ''s'' as the parameter for ''C'', this provides a natural parametrization for the curve. The unit tangent vector '''T''' (which is also the velocity vector, since the particle is moving with unit speed) also depends on time. The curvature is then the magnitude of the rate of change of '''T'''. Symbolically,
| | <ul> |
| :<math>\kappa = \left\|\frac{d\mathbf{T}}{ds}\right\|.</math> | | |
| | | <li>[http://bbs.zhuanglizhuangwai.com/bbs/forum.php?mod=viewthread&tid=23633&fromuid=5858 http://bbs.zhuanglizhuangwai.com/bbs/forum.php?mod=viewthread&tid=23633&fromuid=5858]</li> |
| [[Image:FrenetTN.svg|thumb|right|350px|The '''T''' and '''N''' vectors at two points on a plane curve, a translated version of the second frame (dotted), and the change in '''T''': δ'''T'''. δs is the distance between the points. In the limit <math>\tfrac{d\mathbf{T}}{ds}</math> will be in the direction '''N''' and the curvature describes the speed of rotation of the frame.]]
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| This is the magnitude of the acceleration of the particle and the vector <math>d\mathbf{T} / ds</math> is the acceleration vector. Geometrically, the curvature <math>\kappa</math> measures how fast the unit tangent vector to the curve rotates. If a curve keeps close to the same direction, the unit tangent vector changes very little and the curvature is small; where the curve undergoes a tight turn, the curvature is large.
| | <li>[http://ktm-rc8.de/cgi-bin/guestbook-rc8/guestbook.cgi http://ktm-rc8.de/cgi-bin/guestbook-rc8/guestbook.cgi]</li> |
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| These two approaches to the curvature are related geometrically by the following observation. In the first definition, the curvature of a circle is equal to the ratio of the angle of an arc to its length. Likewise, the curvature of a plane curve at any point is the limiting ratio of ''dθ'', an infinitesimal angle (in radians) between tangents to that curve at the ends of an infinitesimal segment of the curve, to the length of that segment ''ds'', i.e., ''dθ/ds''. If the tangents at the ends of the segment are represented by [[unit vectors]], it is easy to show that in this limit, the magnitude of the difference vector is equal to ''dθ'', which leads to the given expression in the second definition of curvature.
| | <li>[http://www.sghsalumni.com/cgi-bin/sghs/guestbook/guestbook.cgi http://www.sghsalumni.com/cgi-bin/sghs/guestbook/guestbook.cgi]</li> |
| | | |
| ===Precise definition===
| | </ul> |
| Suppose that ''C'' is a twice [[continuously differentiable]] [[immersion (mathematics)|immersed]] [[plane curve]], which here means that there exists [[parametric representation]] of ''C'' by a pair of functions {{nowrap|γ(''t'') {{=}} (''x''(''t''), ''y''(''t''))}} such that the first and second derivatives of ''x'' and ''y'' both exist and are continuous, and
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| :<math>\|\gamma'\|^2 = x'(t)^2 + y'(t)^2 \not= 0</math>
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| throughout the domain. For such a plane curve, there exists a reparametrization with respect to [[arc length]] ''s''. This is a parametrization of ''C'' such that
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| :<math>\|\gamma'\|^2 = x'(s)^2+y'(s)^2 = 1.</math><ref>{{citation
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| | url = https://sites.google.com/site/johnkennedyshome/home/class-downloads
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| | last1 = Kennedy | first1 = John
| |
| | title = The ArcLength Parametrization of a Curve
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| | year = 2011
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| }}</ref>
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| The velocity vector '''T'''(''s'') is the unit tangent vector. The unit normal vector '''N'''(''s''), the '''curvature''' ''κ''(''s''), the '''oriented''' or '''signed curvature''' ''k''(''s''), and the '''radius of curvature''' ''R(s)'' are given by
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| : <math> \mathbf{T}(s)=\gamma'(s),\quad \mathbf{T}'(s)=k(s)\mathbf{N}(s),\quad \kappa(s) = \|\mathbf{T}'(s)\| = \|\gamma''(s)\| = \left|k(s)\right|, \quad R(s)=\frac{1}{\kappa(s)}.</math>
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| Expressions for calculating the curvature in arbitrary coordinate systems are given below.
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| {{multiple image
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| | direction = vertical
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| | width = 260
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| | footer = Animations of the signed curvature and the acceleration vector <math>\mathbf{T}'(s)</math>
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| | image1 = Lissajous-Curve_nebeneinander_animated.gif
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| | image2 = Lemniscate_nebeneinander_animated.gif
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| }}
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| === Signed curvature ===
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| The sign of the signed curvature ''k'' indicates the direction in which the unit tangent vector rotates as a function of the parameter along the curve. If the unit tangent rotates counterclockwise, then ''k'' > 0. If it rotates clockwise, then ''k'' < 0.
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| The signed curvature depends on the particular parametrization chosen for a curve. For example the unit circle can be parametrised by {{nowrap|(cos(θ),sin(θ))}} (counterclockwise, with ''k'' > 0), or by {{nowrap|(cos(−θ),sin(−θ))}} (clockwise, with ''k'' < 0). More precisely, the signed curvature depends only on the choice of [[orientation (mathematics)|orientation]] of an [[immersion (mathematics)|immersed]] curve. Every immersed curve in the plane admits two possible orientations.
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| === Local expressions ===
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| {{see also|Centripetal force#Alternative approach}}
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| For a plane curve given parametrically in [[Cartesian coordinates]] as {{nowrap|γ(t) {{=}} (''x''(''t''),''y''(''t''))}}, the curvature is
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| :<math>\kappa = \frac{|x'y''-y'x''|}{(x'^2+y'^2)^{3/2}},</math>
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| where primes refer to derivatives with respect to parameter ''t'' . The signed curvature ''k'' is
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| :<math>k = \frac{x'y''-y'x''}{(x'^2+y'^2)^{3/2}}.</math>
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| These can be expressed in a coordinate-independent manner via
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| :<math>k = \frac{\det(\gamma',\gamma'')}{\|\gamma'\|^3},\ \ \ \kappa = \frac{|\det(\gamma',\gamma'')|}{\|\gamma'\|^3}.</math>
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| ===Curvature of a graph===
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| For the less general case of a plane curve given explicitly as <math>y=f(x)</math>, and now using primes for derivatives with respect to coordinate ''x'' , the curvature is
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| :<math>\kappa = \frac{|y''|}{(1+y'^2)^{3/2}}</math> , | |
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| and the signed curvature is
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| :<math>k = \frac{y''}{(1+y'^2)^{3/2}}</math> .
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| This quantity is common in [[physics]] and [[engineering]]; for example, in the [[Beam theory|equations]] of [[bending]] in beams, the 1D [[Wave equation|vibration]] of a tense string, approximations to the fluid flow around surfaces (in aeronautics), and the free surface boundary conditions in ocean waves. In such applications, the assumption is almost always made that the [[slope]] is small compared with unity, so that the approximation:
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| :<math>\kappa \approx \left|\frac{d^2y}{dx^2}\right|</math>
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| may be used. This approximation yields a straightforward linear equation describing the phenomenon.
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| If a curve is defined in polar coordinates as <math>r(\theta)</math>, then its curvature is
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| :<math>\kappa(\theta) = \frac{|r^2 + 2r'^2 - r r''|}{\left(r^2+r'^2 \right)^{3/2}}</math>
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| where here the prime now refers to differentiation with respect to <math>\theta</math>.
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| === Example ===
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| Consider the [[parabola]] {{nowrap|''y'' {{=}} ''x''<sup>2</sup>}}. We can parametrize the curve simply as {{nowrap|γ(''t'') {{=}} (''t'',''t''<sup>2</sup>) {{=}} (''x'',''y'')}}. If we use primes for derivatives with respect to parameter ''t'' , then
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| : <math>x'= 1,\quad x''=0,\quad y'= 2t,\quad y''=2.</math>
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| Substituting and dropping unnecessary absolute values, get
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| :<math>\kappa(t)= \left|\frac{x'y''-y'x''}{({x'^2+y'^2})^{3/2}}\right|= {1\cdot 2-(2t)(0) \over (1+(2t)^2)^{3/2} }={2 \over (1+4t^2)^{3/2}}.</math> | |
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| ==Curvature of space curves==
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| [[File:Torus-Knot uebereinander animated.gif|thumb|upright|Animation of the curvature and the acceleration vector <math>\mathbf{T}'(s)</math>]]
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| As in the case of curves in two dimensions, the curvature of a regular [[space curve]] ''C'' in three dimensions (and higher) is the magnitude of the acceleration of a particle moving with unit speed along a curve. Thus if γ(''s'') is the arclength parametrization of ''C'' then the unit tangent vector '''T'''(''s'') is given by
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| :<math>\mathbf{T}(s) = \gamma'(s)</math>
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| and the curvature is the magnitude of the acceleration:
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| :<math>\kappa(s) = \|\mathbf{T}'(s)\| = \|\gamma''(s)\|.</math>
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| The direction of the acceleration is the unit normal vector '''N'''(''s''), which is defined by
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| :<math>\mathbf{N}(s) = \frac{\mathbf{T}'(s)}{\|\mathbf{T}'(s)\|}.</math>
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| The plane containing the two vectors '''T'''(''s'') and '''N'''(''s'') is called the osculating plane to the curve at γ(''s''). The curvature has the following geometrical interpretation. There exists a circle in the osculating plane tangent to γ(''s'') whose Taylor series to second order at the point of contact agrees with that of ''γ''(''s''). This is the osculating circle to the curve. The radius of the circle ''R''(''s'') is called the radius of curvature, and the curvature is the reciprocal of the radius of curvature:
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| :<math>\kappa(s) = \frac{1}{R(s)}.</math>
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| The tangent, curvature, and normal vector together describe the second-order behavior of a curve near a point. In three-dimensions, the third order behavior of a curve is described by a related notion of [[torsion of curves|torsion]], which measures the extent to which a curve tends to perform a corkscrew in space. The torsion and curvature are related by the [[Frenet–Serret formulas]] (in three dimensions) and [[differential geometry of curves|their generalization]] (in higher dimensions).
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| ===Local expressions===
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| For a parametrically defined space curve in three-dimensions given in Cartesian coordinates by {{nowrap|γ(''t'') {{=}} (''x''(''t''),''y''(''t''),''z''(''t''))}},
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| the curvature is
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| :<math>\kappa=\frac{\sqrt{(z''y'-y''z')^2+(x''z'-z''x')^2+(y''x'-x''y')^2}}{(x'^2+y'^2+z'^2)^{3/2}}.</math>
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| where the prime denotes differentiation with respect to the parameter ''t''. This can be expressed independently of the coordinate system by means of the formula
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| :<math>\kappa = \frac{|\gamma' \times \gamma''|}{|\gamma'|^3}</math>
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| where <math>\times</math> is the [[cross product|vector cross product]]. Equivalently,
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| :<math>\kappa = \frac{\sqrt{\det\left( (\gamma',\gamma'')^t(\gamma',\gamma'') \right)} }{\|\gamma'\|^3}.</math> | |
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| Here the ''t'' denotes the [[matrix transpose]]. This last formula is also valid for the curvature of curves in a Euclidean space of any dimension.
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| ===Curvature from arc and chord length===
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| Given two points ''P'' and ''Q'' on ''C'', let ''s''(''P'',''Q'') be the arc length of the portion of the curve between ''P'' and ''Q'' and let ''d''(''P'',''Q'') denote the length of the line segment from ''P'' to ''Q''. The curvature of ''C'' at ''P'' is given by the limit{{citation needed|date=December 2010}}
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| :<math>\kappa(P) = \lim_{Q\to P}\sqrt{\frac{24\left(s(P,Q)-d(P,Q)\right)}{s(P,Q)^3}}</math>
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| where the limit is taken as the point ''Q'' approaches ''P'' on ''C''. The denominator can equally well be taken to be ''d''(''P'',''Q'')<sup>3</sup>. The formula is valid in any dimension. Furthermore, by considering the limit independently on either side of ''P'', this definition of the curvature can sometimes accommodate a singularity at ''P''. The formula follows by verifying it for the osculating circle.
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| == Curves on surfaces == | |
| When a one dimensional curve lies on a two dimensional surface embedded in three dimensions [[Euclidean space|'''R'''<sup>3</sup>]], further measures of curvature are
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| available, which take the surface's unit-[[normal vector]], '''u''' into account. These are the [[normal curvature]], [[geodesic curvature]] and [[geodesic torsion]].
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| Any non-singular curve on a smooth surface will have its tangent vector '''T''' lying in the [[tangent plane]] of the surface orthogonal
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| to the normal vector. The '''normal curvature''', ''k<sub>n</sub>'', is the curvature of the curve projected onto the plane containing the curve's tangent '''T''' and the surface normal '''u'''; the '''geodesic curvature''', ''k<sub>g</sub>'', is the curvature of the curve projected onto the
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| surface's tangent plane; and the '''geodesic torsion''' (or '''relative torsion'''), ''τ<sub>r</sub>'', measures the rate of change of the surface normal around the curve's tangent.
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| Let the curve be a unit speed curve and let '''t''' = '''u''' × '''T''' so that '''T''', '''u''', '''t''' form an [[orthonormal basis]]: the '''[[Darboux frame]]'''. The above quantities are related by:
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| :<math>\begin{pmatrix}
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| \mathbf{T'}\\
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| \mathbf{t'}\\
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| \mathbf{u'}
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| \end{pmatrix}
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| = | |
| \begin{pmatrix}
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| 0&\kappa_g&\kappa_n\\
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| -\kappa_g&0&\tau_r\\
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| -\kappa_n&-\tau_r&0
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| \end{pmatrix}
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| \begin{pmatrix}
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| \mathbf{T}\\
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| \mathbf{t}\\
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| \mathbf{u}
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| \end{pmatrix}
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| </math>
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| ===Principal curvature===
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| [[Image:Minimal surface curvature planes-en.svg|thumb|300px|right|[[Saddle surface]] with normal planes in directions of principal curvatures]]
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| All curves with the same tangent vector will have the same normal curvature, which is the same as the curvature of the curve obtained by intersecting the surface with the plane containing '''T''' and '''u'''. Taking all possible tangent vectors
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| then the maximum and minimum values of the normal curvature at a point are called the '''[[principal curvature]]s''', ''k''<sub>1</sub> and ''k''<sub>2</sub>, and the directions of the corresponding tangent vectors are called '''principal directions'''.
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| ==Curvature of surfaces==
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| {{main|Differential geometry of surfaces}}
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| ===Gaussian curvature===
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| {{main|Gaussian curvature}}
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| In contrast to curves, which do not have intrinsic curvature, but do have extrinsic curvature (they only have a curvature given an embedding), surfaces can have intrinsic curvature, independent of an embedding. The [[Gaussian curvature]], named after [[Carl Friedrich Gauss]], is equal to the product of the principal curvatures, ''k''<sub>1</sub>''k''<sub>2</sub>. It has the dimension of 1/length<sup>2</sup> and is positive for [[sphere]]s, negative for one-sheet [[hyperboloid]]s and zero for planes. It determines whether a surface is [[locally]] [[:wikt:convex|convex]] (when it is positive) or locally saddle (when it is negative).
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| This definition of Gaussian curvature is ''extrinsic'' in that it uses the surface's [[embedding]] in '''R'''<sup>3</sup>, normal vectors, external planes etc. Gaussian curvature is however in fact an ''intrinsic'' property of the surface, meaning it does not depend on the particular [[embedding]] of the surface; intuitively, this means that ants living on the surface could determine the Gaussian curvature. For example, an ant living on a sphere could measure the sum of the interior angles of a triangle and determine that it was greater than 180 degrees, implying that the space it inhabited had positive curvature. On the other hand, an ant living on a cylinder would not detect any such departure from [[Euclidean geometry]]; in particular the ant could not detect that the two surfaces have different mean curvatures (see below), which is a purely extrinsic type of curvature.
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| Formally, Gaussian curvature only depends on the [[Riemannian metric]] of the surface. This is [[Carl Friedrich Gauss|Gauss]]'s celebrated [[Theorema Egregium]], which he found while concerned with geographic surveys and mapmaking.
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| An intrinsic definition of the Gaussian curvature at a point ''P'' is the following: imagine an ant which is tied to ''P'' with a short thread of length ''r''. She runs around ''P'' while the thread is completely stretched and measures the length C(''r'') of one complete trip around ''P''. If the surface were flat, she would find C(''r'') = 2π''r''. On curved surfaces, the formula for C(''r'') will be different, and the Gaussian curvature ''K'' at the point ''P'' can be computed by the [[Bertrand–Diquet–Puiseux theorem]] as
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| :<math> K = \lim_{r\to 0^+} 3\frac{2\pi r-C(r)}{\pi r^3}.</math>
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| The [[integral]] of the Gaussian curvature over the whole surface is closely related to the surface's [[Euler characteristic]]; see the [[Gauss–Bonnet theorem]].
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| The discrete analog of curvature, corresponding to curvature being concentrated at a point and particularly useful for [[polyhedra]], is the [[defect (geometry)|(angular) defect]]; the analog for the [[Gauss–Bonnet theorem]] is [[Defect (geometry)#Descartes.27 theorem|Descartes' theorem on total angular defect]].
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| Because (Gaussian) curvature can be defined without reference to an embedding space, it is not necessary that a surface be embedded in a higher dimensional space in order to be curved. Such an intrinsically curved two-dimensional surface is a simple example of a [[Riemannian manifold]].
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| ===Mean curvature===
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| The [[mean curvature]] is equal to half the sum of the [[principal curvature]]s, (''k''<sub>1</sub>+''k''<sub>2</sub>)/2. It has the dimension of 1/length. Mean curvature is closely related to the first variation of [[surface area]], in particular a [[minimal surface]] such as a [[soap film]], has mean curvature zero and a [[soap bubble]] has constant mean curvature. Unlike Gauss curvature, the mean curvature is extrinsic and depends on the embedding, for instance, a [[cylinder (geometry)|cylinder]] and a plane are locally [[isometry|isometric]] but the mean curvature of a plane is zero while that of a cylinder is nonzero.
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| === Second fundamental form ===
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| The intrinsic and extrinsic curvature of a surface can be combined in the [[second fundamental form]]. This is a [[quadratic form]] in the tangent plane to the surface at a point whose value at a particular tangent vector ''X'' to the surface is the normal component of the acceleration of a curve along the surface tangent to ''X''; that is, it is the normal curvature to a curve tangent to ''X'' (see [[#Curves on surfaces|above]]). Symbolically,
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| :<math>I\!I(X,X) = N\cdot (\nabla_X X)</math>
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| where ''N'' is the unit normal to the surface. For unit tangent vectors ''X'', the second fundamental form assumes the maximum value ''k''<sub>1</sub> and minimum value ''k''<sub>2</sub>, which occur in the principal directions ''u''<sub>1</sub> and ''u''<sub>2</sub>, respectively. Thus, by the [[principal axis theorem]], the second fundamental form is
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| :<math>I\!I(X,X) = k_1(X\cdot u_1)^2 + k_2(X\cdot u_2)^2.</math>
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| Thus the second fundamental form encodes both the intrinsic and extrinsic curvatures.
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| A related notion of curvature is the [[shape operator]], which is a [[linear operator]] from the tangent plane to itself. When applied to a tangent vector ''X'' to the surface, the shape operator is the tangential component of the rate of change of the normal vector when moved along a curve on the surface tangent to ''X''. The principal curvatures are the eigenvalues of the shape operator, and in fact the shape operator and second fundamental form have the same matrix representation with respect to a pair of orthonormal vectors of the tangent plane. The Gauss curvature is thus the [[determinant]] of the shape tensor and the mean curvature is half its [[trace (linear algebra)|trace]].
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| ==Higher dimensions: Curvature of space==
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| {{Main|Curvature of Riemannian manifolds}}
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| By extension of the former argument, a space of three or more dimensions can be intrinsically curved. The curvature is ''intrinsic'' in the sense that it is a property defined at every point in the space, rather than a property defined with respect to a larger space that contains it. In general, a curved space may or may not be conceived as being embedded in a higher-dimensional [[ambient space]]; if not then its curvature can only be defined intrinsically.
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| After the discovery of the intrinsic definition of curvature, which is closely connected with [[non-Euclidean geometry]], many mathematicians and scientists questioned whether ordinary physical space might be curved, although the success of Euclidean geometry up to that time meant that the radius of curvature must be astronomically large. In the theory of [[general relativity]], which describes [[gravity]] and [[physical cosmology|cosmology]], the idea is slightly generalised to the "curvature of [[space-time]]"; in relativity theory space-time is a [[pseudo-Riemannian manifold]]. Once a time coordinate is defined, the three-dimensional space corresponding to a particular time is generally a curved Riemannian manifold; but since the time coordinate choice is largely arbitrary, it is the underlying space-time curvature that is physically significant.
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| Although an arbitrarily curved space is very complex to describe, the curvature of a space which is locally [[isotropic]] and [[Homogeneous space|homogeneous]] is described by a single Gaussian curvature, as for a surface; mathematically these are strong conditions, but they correspond to reasonable physical assumptions (all points and all directions are indistinguishable). A positive curvature corresponds to the inverse square radius of curvature; an example is a sphere or [[hypersphere]]. An example of negatively curved space is [[hyperbolic geometry]]. A space or space-time with zero curvature is called '''flat'''. For example, [[Euclidean space]] is an example of a flat space, and [[Minkowski space]] is an example of a flat space-time. There are other examples of flat geometries in both settings, though. A [[torus]] or a [[Cylinder (geometry)|cylinder]] can both be given flat metrics, but differ in their [[topology]]. Other topologies are also possible for curved space. See also [[shape of the universe]].
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| ==Generalizations==
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| [[Image:Parallel transport.png|thumb|Parallel transporting a vector from ''A'' → ''N'' → ''B'' → ''A'' yields a different vector. This failure to return to the initial vector is measured by the holonomy of the surface.]]
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| The mathematical notion of ''curvature'' is also defined in much more general contexts.<ref>See e.g. [[Shoshichi Kobayashi|S.Kobayashi]] and [[Katsumi Nomizu|K.Nomizu]], [[Foundations of Differential Geometry]], Chapters 2 and 3, Vol.I, [[Wiley Interscience]].</ref> Many of these generalizations emphasize different aspects of the curvature as it is understood in lower dimensions.
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| One such generalization is kinematic. The curvature of a curve can naturally be considered as a kinematic quantity, representing the force felt by a certain observer moving along the curve; analogously, curvature in higher dimensions can be regarded as a kind of [[tidal force]] (this is one way of thinking of the [[sectional curvature]]). This generalization of curvature depends on how nearby test particles diverge or converge when they are allowed to move freely in the space; see [[Jacobi field]].
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| Another broad generalization of curvature comes from the study of [[parallel transport]] on a surface. For instance, if a vector is moved around a loop on the surface of a sphere keeping parallel throughout the motion, then the final position of the vector may not be the same as the initial position of the vector. This phenomenon is known as [[holonomy]]. Various generalizations capture in an abstract form this idea of curvature as a measure of holonomy; see [[curvature form]]. A closely related notion of curvature comes from [[gauge theory]] in physics, where the curvature represents a field and a [[vector potential]] for the field is a quantity that is in general path-dependent: it may change if an observer moves around a loop.
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| Two more generalizations of curvature are the [[scalar curvature]] and [[Ricci curvature]]. In a curved surface such as the sphere, the area of a disc on the surface differs from the area of a disc of the same radius in flat space. This difference (in a suitable limit) is measured by the scalar curvature. The difference in area of a sector of the disc is measured by the Ricci curvature. Each of the scalar curvature and Ricci curvature are defined in analogous ways in three and higher dimensions. They are particularly important in relativity theory, where they both appear on the side of [[Einstein's field equations]] that represents the geometry of spacetime (the other side of which represents the presence of matter and energy). These generalizations of curvature underlie, for instance, the notion that curvature can be a property of a [[measure (mathematics)|measure]]; see [[curvature of a measure]].
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| Another generalization of curvature relies on the ability to [[comparison theorem|compare]] a curved space with another space that has ''constant'' curvature. Often this is done with triangles in the spaces. The notion of a triangle makes senses in [[metric space]]s, and this gives rise to [[CAT(k) space]]s.
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| ==See also==
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| * [[Curvature form]] for the appropriate notion of curvature for [[vector bundle]]s and [[principal bundle]]s with [[Connection (mathematics)|connection]]
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| * [[Curvature of a measure]] for a notion of curvature in [[measure theory]]
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| * [[Parametric_surface#Curvature|Curvature of parametric surfaces]]
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| * [[Curvature of Riemannian manifolds]] for generalizations of Gauss curvature to higher-dimensional [[Riemannian manifold]]s
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| * [[Curvature vector]] and [[geodesic curvature]] for appropriate notions of curvature of ''curves in'' Riemannian manifolds, of any dimension
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| * [[Curve]]
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| * [[Degree of curvature]]
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| * [[Differential geometry of curves]] for a full treatment of curves embedded in a Euclidean space of arbitrary dimension
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| * [[Dioptre]], a measurement of curvature used in optics
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| * [[Gauss-Bonnet theorem]] for an elementary application of curvature
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| * [[Gauss map]] for more geometric properties of Gauss curvature
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| * [[Hertz's principle of least curvature]], an expression of the [[Principle of Least Action]]
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| * [[Mean curvature]] at one point on a surface
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| * [[Minimum railway curve radius]]
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| * [[Radius of curvature]]
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| * [[Second fundamental form]] for the extrinsic curvature of hypersurfaces in general
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| * [[Torsion of a curve]]
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| *Coolidge, J.L. "The Unsatisfactory Story of Curvature". The American Mathematical Monthly, Vol. 59, No. 6 (Jun. - Jul., 1952), pp. 375–379
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| *{{SpringerEOM| title=Curvature | id=Curvature | oldid=12026 | first=D.D. | last=Sokolov }}
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| *[[Morris Kline]]: ''Calculus: An Intuitive and Physical Approach''. Dover 1998, ISBN 978-0-486-40453-0, p. 457-461 ({{Google books|YdjK_rD7BEkC|restricted online copy|page=457}})
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| *A. Albert Klaf: ''Calculus Refresher''. Dover 1956, ISBN 978-0-486-20370-6, p. 151-168 ({{Google books|NR6ZuvBP3zwC|restricted online copy|page=151}})
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| *James Casey: ''Exploring Curvature''. Vieweg+Teubner Verlag 1996, ISBN 978-3-528-06475-4
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| ==External links==
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| {{Commons category|Illustrations for curvature and torsion of curves|Graphical illustrations of the curvature of curves}}
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| {{Wiktionary|curvature}}
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| *[http://www.math.uni-muenster.de/u/urs.hartl/gifs/CurvatureAndTorsionOfCurves.mw Create your own animated illustrations of moving Frenet-Serret frames and curvature] ([[Maple (software)|Maple]]-Worksheet)
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| * [http://www3.villanova.edu/maple/misc/history_of_curvature/k.htm The History of Curvature]
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| * [http://www.mathpages.com/rr/s5-03/5-03.htm Curvature, Intrinsic and Extrinsic] at MathPages
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| {{curvature}}
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| [[Category:Curvature (mathematics)| ]]
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| [[Category:Multivariable calculus]]
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