自相交曲线 交点处存在两类不同的切矢量，它们中的每一类都独自可以张出一个一维线性空间，该线性空间中的矢量都是曲线切矢（没记错的话

如图所示的三个一维图形，都不是一维流形，因为交点处及其邻域不与R1同胚。 －－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－ 以前在学张量分析的时候，曾经遇到这么一句话：曲线坐标不自身相交。一直没弄懂自身相交是个什么概念，现在初学流形，也遇到了类似的问题，请高手详细讲解，不胜感激！
	顶 0
2010-1-5 20:23 回复

超级云K CloudK 489位粉丝	2楼 不自相交严格说就是连续曲线上的任何一点的tangent vector构成一个流形上的一个矢量场.通俗说就是曲线自己和自己不能相交~~
2010-1-5 20:49 回复

210.45.66.*	3楼 谢谢楼上的回答。但是自己和自己不能相交有歧义，例如封闭曲线，所以还是不太明白
2010-1-5 20:57 回复

超级云K CloudK 489位粉丝	4楼 额~你把2L的加上个“非闭合曲线”就行了~
2010-1-5 21:10 回复

yxtgh 2位粉丝	5楼 回复：4楼 你那个严格说已经足够了啊 对于封闭曲线切失显然构成矢量场。 对于自相交曲线交点处有两个切失 这个定义可以往高维推广
2010-1-5 22:20 回复

超级云K CloudK 489位粉丝	6楼 额~2L那个按理说还不够严格，简单例子对于R^2上的圆，每一点都可以有一个确定的tangent vector，可惜这种曲线确实属于闭合的，和自相交曲线脱离不了干系~
2010-1-5 22:27 回复

超级云K CloudK 489位粉丝	8楼 映射角度不太好说~~如果从区间I∈R到流形M上的一个映射是同胚映射，那么曲线就是个非自相交的（也不是闭合的），不过这样没法挑出来自相交和封闭曲线的区别~
2010-1-5 22:40 回复

Schrodinger 1111位粉丝	9楼 曲线坐标不自身相交。 -- 给定一个坐标系，任意一点的坐标就唯一确定了下来。任一点都不会存在2个坐标值。
2010-1-5 22:43 回复

Schrodinger 1111位粉丝	10楼 回复：8楼 7楼没什么意义，所以我自己删除了...
2010-1-5 22:43 回复

Schrodinger 1111位粉丝	11楼 不过，浸入子流形和嵌入子流形（一个有自相交，一个没有）就是根据映射是否单一来区别的。所以看到自相交我首先想到了映射。
2010-1-5 22:45 回复

超级云K CloudK 489位粉丝	12楼 额~其实还就得用映射来区分所谓的自相交和闭合曲线，光用切矢量定义还真挺麻烦~~
2010-1-5 23:00 回复

118.203.29.*	13楼 是不是因为坐标曲线应该是同胚与欧式空间中的开集的，而闭合曲线不能同胚与欧式空间的开集，至于相交的曲线也是不能同胚与欧式空间里的开集的。
2010-1-12 20:52 回复

LostAbaddon 102位粉丝

14楼 交点处存在两类不同的切矢量，它们中的每一类都独自可以张出一个一维线性空间，该线性空间中的矢量都是曲线切矢（没记错的话，这就构成了一个芽空间，不同的矢量就是不同参数化曲线的切矢）。但问题是，这两类矢量的混合不再是交点处曲线的切矢了。 当然，这个说法本身也是有问题的。比如两个圆在某点处相切，切点处的两个芽空间是完全一样的。这里就需要更加细致地去分析了。可以考虑建立节丛，那么对于上述问题（相切的圆），至少他们的一阶节丛是不同的。如果变态一点，那至少总存在一个自然数n，使得其n阶节丛是不同的，也即两个在一起无法构成一个恰当的曲线n阶节丛空间。 如果这个n是无穷大，那么就是说：这两条光滑曲线完全相同，就是一条曲线。 闭合曲线的特点就在于次，它从切丛到无穷大阶节丛都是相同的，都是R1，所以啥都别说了，是同一个东西。

所有跟帖： 

• 局部性质的讨论只需要关心简单的正则曲线即可;从这个意义上说,象自相交这种性质是曲线的大范围性质,也就是整体性质 -marketreflections- ♂ (18688 bytes) () 07/19/2011 postreply 14:24:15

• 自相交曲线 交点处存在两类不同的切矢量，它们中的每一类都独自可以张出一个一维线性空间，该线性空间中的矢量都是曲线切矢（没记错的话 -marketreflections- ♂ (302 bytes) () 07/31/2011 postreply 15:57:28

• 人体的窗口---视觉系统 ;拉格朗日法是以流场中每一流体质点作为描述对象的方法:一个质点不可能同时有两个速度矢量;曲率连续的曲面 -marketreflections- ♂ (6055 bytes) () 07/31/2011 postreply 16:04:10

• 在理论方面，曲线曲面的连续性〔13〕通常分为两种形式:参数连续性和几何连续性。曲线在连接点处具有相等的n阶连续导矢，则称之为具有 -marketreflections- ♂ (49083 bytes) () 07/31/2011 postreply 16:13:19

• 解析几何中，空间曲线上—点p 的每个坐标被表示成某个参数u 的函数,x＝x(u)，y＝y(u)，s＝z(M)。把三个方程合写一起 -marketreflections- ♂ (7751 bytes) () 07/31/2011 postreply 16:20:53

• 曲线01：自相交曲线 交点处存在两类不同的切矢量，它们中的每一类都独自可以张出一个一维线性空间，该线性空间中的矢量都是曲线切矢（ -marketreflections- ♂ (14908 bytes) () 07/31/2011 postreply 16:34:07

• kevin kelky 网络测地线,网络中大部分关系之间都可以通过3 步或4 步到达,很少一部分需要7 步 -marketreflections- ♂ (11115 bytes) () 07/31/2011 postreply 17:39:33

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• 正则01 形体在计算机内的表示 非正则形体 正则形体 二维流形:所谓二维流形是指这样一些面，其上任一点都存在一个充分小的邻域，该 -marketreflections- ♂ (2359 bytes) () 07/31/2011 postreply 17:56:56

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• check01 vix001 http://www.diffresearch.com/# Daily Recap -marketreflections- ♂ (6553 bytes) () 08/01/2011 postreply 14:25:23

• vix001 08012011 www.diffresearch.com/ -marketreflections- ♂ (188 bytes) () 08/01/2011 postreply 17:52:34

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• vix001 The last time the market didn’t buy a sell off, but the V -marketreflections- ♂ (5211 bytes) () 08/02/2011 postreply 22:46:44

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• mkt01 portfolio01 minimize cyclical exposures aapl bidu pot -marketreflections- ♂ (10366 bytes) () 08/03/2011 postreply 21:00:54

• 全息原理01：自相交曲线 交点处存在两类不同的切矢量 一团空间内所能包含的最大熵，或者说，这团空间所能记录的信息量，存在上限。对 -marketreflections- ♂ (29387 bytes) () 08/01/2011 postreply 21:45:41

• 全息原理01：自相交曲线 "蚂蚁信息系统" -marketreflections- ♂ (101893 bytes) () 08/04/2011 postreply 18:22:58

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• 警报器也使用白噪声,對所有時間其平均值為零的隨機過程，並且它的自相關函數是狄拉克δ函數，有無限大的功率,信號在各個频段上的功率是 -marketreflections- ♂ (76962 bytes) () 08/06/2011 postreply 16:43:53

• 狄拉克δ函数01 连续谱的本征波函数可以规格化为8函数 -marketreflections- ♂ (2275 bytes) () 08/06/2011 postreply 16:53:47

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• 平面波01. 本征值p r. 连续取值，构成连续谱:归到Dirac的δ 函数（即δ 归一化，也可叫“规格化”）. 考虑积分： . -marketreflections- ♂ (4644 bytes) () 08/06/2011 postreply 17:05:59

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• 三種顏色均重疊的部分則呈現白色:人眼视觉特性 从空间频率域来看，人眼是一个低通型线性系统，分辨景物的能力是有限的。由于瞳孔有一定 -marketreflections- ♂ (6777 bytes) () 08/06/2011 postreply 17:18:39

• 通信原理经典讲义01 白高斯噪声通过某带通或低通型线性系统H ( f ) 后的输出是0 ,均值平稳高斯过程,功率（方差）为σ2 -marketreflections- ♂ (1555 bytes) () 08/06/2011 postreply 17:29:33

• whitenoise01 随机过程一般都是二阶矩过程，这种随机过程一般都只涉及其均值函数和相关函数的讨论，白噪声的相关函数只有在 -marketreflections- ♂ (18217 bytes) () 08/06/2011 postreply 17:40:27

• 随机过程01 研究随机过程的一个重要切入点是研究一个随机过程的数字特征（有限维分布一般很难求得），数字特征主要包括数学期望、相关 -marketreflections- ♂ (313 bytes) () 08/06/2011 postreply 17:42:52

• 中国科学院随机过程讲义 严平稳随机过程，只有它二阶矩存在时，它才是宽平稳过程 -marketreflections- ♂ (1757 bytes) () 08/06/2011 postreply 17:44:30

• 神经网络简介 积分的下极限有效的取为 ，使得Dirac delta 函数 的拉普拉斯变换等于1;白噪声,自相關函數是狄拉克δ函數 -marketreflections- ♂ (19329 bytes) () 08/06/2011 postreply 17:48:54

• 希尔伯特条件 多层分割的区域上寻求这样一种分区全纯函数，它在一部分边界上满足黎曼条件，而在另外的边界上满足希尔伯特条件 -marketreflections- ♂ (4488 bytes) () 08/06/2011 postreply 18:03:23

• 积分的“全空间”本身就是时间t的函数的话，那就有一个问题，即如果空间的大小是时间的函数的话，那空间的度规也必然是时间t的函数了吧 -marketreflections- ♂ (36606 bytes) () 08/06/2011 postreply 18:27:44

• 散射都是二階段過程（second-order process）──先吸收光元躍遷到一中介能階（所謂「虛能 -marketreflections- ♂ (2200 bytes) () 08/06/2011 postreply 20:30:34

• 量子电动力学 由于不确定关系，真空不是对应“零”，而是“正”和“负”之间“来回振荡”，其平均值为零，但方差均值并不为零，通常所说 -marketreflections- ♂ (16941 bytes) () 08/06/2011 postreply 20:37:21

• 江兴流提出的涡旋动力学（Vortex Dynamics）模型。涡旋动力学模型的主要论点在于：局域瞬态非平衡体系产生涡旋，而涡旋的 -marketreflections- ♂ (1901 bytes) () 08/06/2011 postreply 20:53:13

• 量子场论01 作用量 粒子世界线在时空中的长度，具有庞加莱对称性和参数变换对称性;从现在的观点来说，只有两种对象：有限自由度的系 -marketreflections- ♂ (28209 bytes) () 08/06/2011 postreply 21:14:35

• 质量必定受到对称性的约束,质量公式中的参数和微扰质量矩阵的表示。可是可是像往常所有时候一样，留有参数待定。至于这些待定的参数缘何 -marketreflections- ♂ (5366 bytes) () 08/06/2011 postreply 21:31:16

• 张茉楠：旧增长模式失灵 全球变革已无退路 growthmodel01 -marketreflections- ♂ (6237 bytes) () 08/07/2011 postreply 11:05:05

• ecbmodel01 1998年成立的欧洲央行其首要的货币政策目标是维持物价稳定，次要目标才是促进高水平就业和可持续、非通胀性的 -marketreflections- ♂ (4558 bytes) () 08/07/2011 postreply 11:10:42

• 回复：whitenoise01 随机过程一般都是二阶矩过程，这种随机过程一般都只涉及其均值函数和相关函数的讨论，白噪声的相关函数 -marketreflections- ♂ (6791 bytes) () 08/07/2011 postreply 11:21:32

• 高通或低通滤波器无法轻易滤除的噪声很多，最常见的就是白噪声。白噪声在整个频谱内每个频点的能量为常数，且基本恒定，不管对信号进行低 -marketreflections- ♂ (5671 bytes) () 08/07/2011 postreply 11:41:35

• gs01 on italy ecb -marketreflections- ♂ (54229 bytes) () 08/07/2011 postreply 12:10:48

• check01 morelivers.blogspot.com/ 從量子物理的觀點，光的散射都是二階段過程（second-ord -marketreflections- ♂ (1810 bytes) () 08/07/2011 postreply 12:42:36

• physics01 www.scu.edu.tw/physics/teacher/ytliu/book/quantum/7-2. -marketreflections- ♂ (2119 bytes) () 08/07/2011 postreply 12:46:51

• Niederhoffer computational neuroscience We see what’s happening -marketreflections- ♂ (8196 bytes) () 08/07/2011 postreply 16:51:09

• risk01 a general increase in risk premia and volatility -marketreflections- ♂ (5753 bytes) () 08/07/2011 postreply 17:06:39

• risk01 liken cash to a call option on uncertainty whether our es -marketreflections- ♂ (4415 bytes) () 08/07/2011 postreply 17:20:51

• check01 www.economonitor.com/ -marketreflections- ♂ (78 bytes) () 08/07/2011 postreply 21:00:28

• whitenoise01 VIX popped more than 6.5% in only a few minutes. A -marketreflections- ♂ (6115 bytes) () 08/07/2011 postreply 21:10:19

• whitenoise01 Adam Warner 08072011 August futures (bet on vix cas -marketreflections- ♂ (3893 bytes) () 08/07/2011 postreply 21:27:58

• whitenoise01 The VIX hit 80 during the peak of the 2008 financia -marketreflections- ♂ (2355 bytes) () 08/07/2011 postreply 21:36:28

• vix01 STEVEN M. SEARS Michael Schwartz, Oppenheimer & Co.'s chie -marketreflections- ♂ (14716 bytes) () 08/07/2011 postreply 21:44:55

• whitenoise01 vix=48 arrived at last year, during the 21 May 2010 -marketreflections- ♂ (3398 bytes) () 08/07/2011 postreply 21:58:22

• whitenoise01 Monthly Trading Report (July 2011) -marketreflections- ♂ (188 bytes) () 08/07/2011 postreply 22:08:44

• 一阶和二阶过程的原理: 从数学上讲，一个连续过程的阶次等同于该过程的控制方程中微分算子的个数 -marketreflections- ♂ (4819 bytes) () 08/08/2011 postreply 10:57:33

• 一阶、二阶系统的瞬态响应 -marketreflections- ♂ (1987 bytes) () 08/08/2011 postreply 11:06:18

• 典型的二阶系统过渡过程通常表现为欠阻尼、. 过阻尼以及临界阻尼三种形式。我们可以通过改变. 系统的参数M, C, K 调节底层运 -marketreflections- ♂ (1372 bytes) () 08/08/2011 postreply 12:27:11

• check01 Peter Tchir 阻尼 在电学中，差不多就是响应时间的意思 ;在机械物理学中，系统的能量的减小——阻尼振动 -marketreflections- ♂ (7237 bytes) () 08/08/2011 postreply 16:03:01

• check01 Peter Tchir even down 14 points on the SPX, the market f -marketreflections- ♂ (7759 bytes) () 08/08/2011 postreply 16:12:11

• check01 Reggie Middleton Armageddon Put Trade watching for a Fre -marketreflections- ♂ (25808 bytes) () 08/08/2011 postreply 16:21:00

• check001 boombustblog.com/BoomBustBlog/Trading-the-US-Treasury-D -marketreflections- ♂ (214 bytes) () 08/08/2011 postreply 16:29:49

• check001 ReggieMiddleton there’s still a plausible case for 10Y -marketreflections- ♂ (15182 bytes) () 08/08/2011 postreply 16:35:07

• check01 BarclayHedge/TrimTabs www.indexuniverse.com/sections/fea -marketreflections- ♂ (6890 bytes) () 08/08/2011 postreply 21:08:35

• hedge01 Spengler I consider this a liquidity event like 1987 rat -marketreflections- ♂ (15075 bytes) () 08/08/2011 postreply 21:21:37

• check01 riskpremium01 doug kass the ratio between normalized ea -marketreflections- ♂ (16220 bytes) () 08/09/2011 postreply 17:29:25

• check01 MacroStory.com The pattern is a series of rallies some r -marketreflections- ♂ (5656 bytes) () 08/08/2011 postreply 21:53:28

• They have succeeded in repealing the Glass-Steagall Act on Nov 1 -marketreflections- ♂ (9216 bytes) () 08/09/2011 postreply 08:45:09

• gold parabolic: Greenspan Gold stands in the way of this insidio -marketreflections- ♂ (10568 bytes) () 08/09/2011 postreply 13:50:07

• Frank Holmes gold love trade east trade fear trade -marketreflections- ♂ (6375 bytes) () 08/09/2011 postreply 14:03:05

• check01 Russ Certo creditors are thinking as they are spending r -marketreflections- ♂ (3822 bytes) () 08/09/2011 postreply 14:26:38

• physical economy, slow growth; financial economy broken,The mil -marketreflections- ♂ (5262 bytes) () 08/09/2011 postreply 14:49:08

• air pocket or gap: no resistence or support, thin trade:index ha -marketreflections- ♂ (30004 bytes) () 08/09/2011 postreply 15:20:42

• For the moment the best measure of the cost of capital is gold. -marketreflections- ♂ (180587 bytes) () 08/09/2011 postreply 15:40:10

• short complacency, and long energy/geopolitical plays,which I do -marketreflections- ♂ (13666 bytes) () 08/09/2011 postreply 15:55:07

• check01 Bob Pisani Also: the Fed has "punted" stimulus discussio -marketreflections- ♂ (10638 bytes) () 08/09/2011 postreply 17:15:41

• check01 Simon Hallett mutual-fund investors have had a split per -marketreflections- ♂ (6613 bytes) () 08/09/2011 postreply 17:40:03

• check01 jim bianco John Mauldin Global Capital Shift Is Accelera -marketreflections- ♂ (2788 bytes) () 08/09/2011 postreply 17:43:26

• check01 paul.kedrosky.com ecb balance sheet not expanded, compar -marketreflections- ♂ (16228 bytes) () 08/09/2011 postreply 18:30:26

• check01 kitco.com CFTC Commodity Futures Trading Commission’s we -marketreflections- ♂ (5833 bytes) () 08/09/2011 postreply 18:38:09

• gold Christopher Barker Jim Sinclair -marketreflections- ♂ (16336 bytes) () 08/09/2011 postreply 18:50:17

• gold01 usd01 ratio of paper gold to physical gold (45:1) to the -marketreflections- ♂ (7532 bytes) () 08/10/2011 postreply 14:40:56

• in 2008, we would note a significant difference: gold was fallin -marketreflections- ♂ (9069 bytes) () 08/10/2011 postreply 15:11:46

• Blow-off top? Swissy Goes Parabolic -marketreflections- ♂ (334 bytes) () 08/10/2011 postreply 15:20:31

• Flakmeister $2500 may be for real, but there is a host of new l -marketreflections- ♂ (638 bytes) () 08/10/2011 postreply 17:45:09

• TA01 reached the 20-day moving average on the first attempt, but -marketreflections- ♂ (2335 bytes) () 08/11/2011 postreply 10:50:20

• “Gold VIX” (GVZ) “benchmark barometer of the U.S. Debt Super Com -marketreflections- ♂ (21032 bytes) () 08/11/2011 postreply 05:13:01

• vix01 expiration quickly approaching for VIX options and future -marketreflections- ♂ (889 bytes) () 08/11/2011 postreply 08:06:20

• Martin Fridson，美国高收益债券市场现在把衰退的几率定在了30%，而且还在上升中,没有可用的杠杆能够改变这个几率 -marketreflections- ♂ (2171 bytes) () 08/10/2011 postreply 14:49:44

• capital01 An introduction to capital markets: products, strategi -marketreflections- ♂ (656 bytes) () 08/11/2011 postreply 09:24:51

• eurobank01 european (especially French) banks pretty much found -marketreflections- ♂ (62240 bytes) () 08/11/2011 postreply 09:38:51

• credit01 Michael Shaoul,Credit cycles have tremendous momentum, -marketreflections- ♂ (580 bytes) () 08/11/2011 postreply 09:58:48

• gold01 traderrog.wordpress.com/ -marketreflections- ♂ (230 bytes) () 08/10/2011 postreply 15:00:09

• gold01 www.jsmineset.com www.TraderDan.net -marketreflections- ♂ (14706 bytes) () 08/09/2011 postreply 19:00:14

• trader001 Kathy Boyle Nicholas Colas suckerrally01 Once it rall -marketreflections- ♂ (15797 bytes) () 08/10/2011 postreply 12:07:35

• check01 "tobias levkovich cost of capital" -marketreflections- ♂ (193 bytes) () 08/10/2011 postreply 12:33:45

• bank01 Fred Cannon Richard Bove low rate structure 'helicopter B -marketreflections- ♂ (28306 bytes) () 08/10/2011 postreply 14:20:59

• eurobanks01 Steve Slater The ECB's use of repurchase operations, -marketreflections- ♂ (19974 bytes) () 08/10/2011 postreply 15:28:34

• gold01 gold to get back to prior highs in inflation- adjusted t -marketreflections- ♂ (35664 bytes) () 08/10/2011 postreply 15:44:09

• check01 ibankcoin.com/ -marketreflections- ♂ (64 bytes) () 08/10/2011 postreply 16:18:37

• eurobank01 Bruce Krasting Chris Whalen french 15% of us mmf -marketreflections- ♂ (83354 bytes) () 08/10/2011 postreply 16:28:51

• eurobank01 www.voxeu.org The debt fears reach the core -marketreflections- ♂ (8787 bytes) () 08/11/2011 postreply 12:07:55

• economist01 Stephen King Mike Dolan macro01 Bretton Woods II. -marketreflections- ♂ (7375 bytes) () 08/11/2011 postreply 05:29:19

• fx01 Robert Sinche The volatility will move on to the equities -marketreflections- ♂ (23798 bytes) () 08/11/2011 postreply 06:11:55

• fx01 Adarsh Sinha When the slowdown spreads to the rest of the w -marketreflections- ♂ (14248 bytes) () 08/11/2011 postreply 06:22:07

• 掌控华尔街交易的计算机算法 -marketreflections- ♂ (11995 bytes) () 08/01/2011 postreply 21:57:25

• 非均匀流中的流线有弯曲或流线之间有夹角,均匀流中的流线为一平行的直线簇 -marketreflections- ♂ (30047 bytes) () 07/31/2011 postreply 17:03:39

• 有旋流与通常的旋转（图３ －１８ｂ）不同，其区别在于有旋流的流体质点绕自身任意轴旋转,有旋流一般存在于有粘性的实际流体中 -marketreflections- ♂ (57810 bytes) () 07/31/2011 postreply 17:10:09

• 流体的连续性微分方程的一般形式,在流场内取一微元六面体为控制体,设中心点O′的流速为ux 、uy 、uz 。以x 轴方向为例，分 -marketreflections- ♂ (14837 bytes) () 07/31/2011 postreply 17:13:26