Talk:Lie derivative

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coordinate expression, and properties sections), and several book sources that are given as (unfootnoted) general references lack specific section or page numbers to guide readers to the material here that they source. Ideally, every separate claim in the article should have a specific footnote indicating where to find it in the sources, and each paragraph should have at least one footnote. Because it is so far from meeting criterion 2, I think this should be a quick fail, but (as usual) with no prejudice against re-nominating once this issue has been addressed.

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understanding, but their knowledge is incomplete, or they are searching for something related that was once known, but forgotten (e.g. the Picard-Lindelof or Frobenius theorem) -- thus links and mentions like this are not "clutter" but are actually useful. Similarly, references to books or other online "tutorial" sources are important. There is a fairly strong feeling within WP that it is not a book or a tutorial, but a reference, although who knows what the future may bring. You are invited to debate issues of style and content at

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namely that the vectors are to be treated as operators, specifically directional derivatives. The subsequent statement "the Lie derivative is the commutator" does not clarify this for the non-initiate, since the commutator does not define what product is involved; in this case it is evidently the operator composition. This can be especially confusing to someone who is thinking of vectors simply in terms of a basis, and is expecting, for example, tensor products, but is faced with an undefined juxtaposition in the article

129: 119: 98: 3485: 2795: 21: 3812:{\displaystyle {\mathcal {L}}_{X}Y:=\left.{\frac {\mathrm {d} ^{2}}{2\mathrm {d} ^{2}t}}\right|_{t=0}\mathrm {Fl} _{-t}^{Y}\circ \mathrm {Fl} _{-t}^{X}\circ \mathrm {Fl} _{t}^{Y}\circ \mathrm {Fl} _{t}^{X}=\left.{\frac {\mathrm {d} }{\mathrm {d} t}}\right|_{t=0}\mathrm {Fl} _{-{\sqrt {t}}}^{Y}\circ \mathrm {Fl} _{-{\sqrt {t}}}^{X}\circ \mathrm {Fl} _{\sqrt {t}}^{Y}\circ \mathrm {Fl} _{\sqrt {t}}^{X}} 3479: 2243: 67: 2472: 4758: 1503:

novices in the field find themselves rather confused by all the different derivatives, (Lie, exterior, partial and covariant) and have trouble sorting out the relationships between them. It is important for this article to succinctly state what these relationships are. That some of this info is also duplicated in other articles is acceptable.

780:. It doesn't matter whether you get a particular flow through the point in question, only that there exists one with the required first order behavior. Those who know about such things (myself included) are aware of these theorems, but they add needless clutter to the article. Simply things as far as possible, and no further. 3209: 1983: 2790:{\displaystyle ({\mathcal {L}}_{A}T)(\alpha ,\beta ,\ldots ,X,Y,\ldots )\equiv \nabla _{A}T(\alpha ,\beta ,\ldots ,X,Y,\ldots )-\nabla _{T(\cdot ,\beta ,\ldots ,X,Y,\ldots )}\alpha (A)-\ldots +T(\alpha ,\beta ,\ldots ,\nabla _{X}A,Y,\ldots )+\ldots +s(\nabla \cdot A)T(\alpha ,\beta ,\ldots ,X,Y,\ldots )} 1943: 4670:

The statement "The Lie derivative along a vector field is the evaluation of the vector field on functions" in the lead gives very little indication of how the vectors are to be considered, and in particular what the "evaluation" means. This assumes context that is simply missing from the description,

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techniques are perfectly OK. One person's helpful aside is another's buzzword generator, so I doubt that there is any absolute criterion of what to include. But the main thing is to aim at a comprehensive treatment of a topic. Well, that's in the Knowledge charter; of course little in mathematics can

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I'm sure it would only take a few words to mention the interpretation of vectors (or at least their evaluation over a scalar field) as operators (specifically directional derivatives) and that the commutator relates to their use and composition as operators (functionals). This is not my field, and I

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A short question from a reader of this page: The definition of the Lie derivative given here makes it look like it eats vector fields (in both slots), but the function f that is introduced later on in the definition of the Lie derivative, and which the Lie derivative eats, is a function from M to R.

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Yep, it needs to go. I can recommend two possible approaches. The first is to define the Lie derivative as a derivation on the full tensor bundle of a differentiable manifold. This has the advantage of fitting into the overall algebraic tone of the article. The second is to define it by means of

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Silly Rabbit, do note that the general consensus is that WP is a reference rather than a tutorial. To learn a new topic, a reader should read a book on the topic. In general, WP articles usually assume that the reader already has some general familiarity with the topic, possibly even a rather good

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It is more confusing that it begins with "One might..." which suggests that a line of faulty reasoning is about to be dismissed (c.f. covariant derivative vs simpler derivative and the need to compensate for metric variation). Why not simply say "One can... in several ways..." but choose carefully

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a function along a congruence, as this leads nicely into the definition of a Lie derivative (perhaps as a section further in the article, not necessarily in the intro.). Another suggestion is to have component expressions for the Lie derivative of a tensor field and explain why (or at least state)

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I just edited a bit of the "Lie derivative of tensor fields" section. I made a few clarifications in the first (and very long) sentence, and I converted the math to math-markup. I know this might be controversial, but in this case there is a really long tensorial equation at heart that is somewhat

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I am wondering what the original source of the equations on the second line is. The reference given at these equations (Kolar,Michor,Slovak) indeed provides a proof for it, but it doesn't give the origin of the equation either, nor does the second expression (with the square roots of t) appear in

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sections of Λ. I'm not too worried; which is which is should always be clear from context. There's not enough letters in the Greek alphabet. If its not clear from context, look for the phrase "where ω is a ...". I've never seen vol used for the volume form, that seems like a rather non-standard

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Sorry, but this article was obviously written by mathematicians.... It describes exactly what a Lie derivative is, but gives no clue whatsoever why it was defined or its uses. I've been seeing references to Lie derivatives in differential geometry and thought I could get an idea of why I should

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I have no explanation for the undo of the following lines. I will add them again. If somebody wants to erase them, first we should have a discussion. This notation exists and I personally believe it is much better. A good reference to find this notation is the following book (see introduction):

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Disagree. Note that different authors and different texts define the Lie derivative in different ways. Thus, all of these ways of approaching the definition should be described. No one particular definition should be elevated as "more right" than the others. A second important point is that many

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The section on Lie derivatives of tensor fields is total garbage, and needs to be completely re-written. The connection should be banished. Somewhere in this article we should make clear that the connection is not required for any of this stuff; and explain why; since I think the interplay of a

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I don't have any books that mention Lie dragging, so I wouldn't know how to write that up. If its placed in its own section, that would be fine to me. This article already mentions three different definitions for a Lie derivative; having a fourth one is fine, as long as this definition is not

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Unfortunately, despite what appears (on a very superficial reading, by a non-expert) to be a well-written description of this subject, it is a long way from meeting Good Article criterion 2, on sourcing. Most of the article has no inline sources (including the entire motivation, definition,

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care to learn about them from Knowledge, but instead got this dry definition more suitable for computers than humans. Could someone who knows more about the subject add sections on the history of the Lie derivative, its reasons for being singled out for definition by Lie, etc.

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Ok, I retract my anti-buzzword prejudice. But I still believe that the article could use some clarification and simplification. The algebraic methods are quite useful in practice, but they don't provide much insight into the meaning of the Lie derivative.

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Dear Linas, Alrighty then. I have extensively revised the article. In my opinion, it is now clearer, more complete, better organized, and contains fewer errors. I hope that my revisions satisfy you and the rest of the wikipedia math community. Best,

3474:{\displaystyle ({\mathcal {L}}_{X}Y)_{x}:=\lim _{t\to 0}(\mathrm {T} (\mathrm {Fl} _{-t}^{X})Y_{\mathrm {Fl} _{t}^{X}(x)}-Y_{x})/t=\left.{\frac {\mathrm {d} }{\mathrm {d} t}}\right|_{t=0}\mathrm {T} (\mathrm {Fl} _{-t}^{X})Y_{\mathrm {Fl} _{t}^{X}(x)}} 3097:

The definition in the case of tensor densities (or field) is the following ... In the following particular cases this boils down to simpler expressions: (i) Functions (that is, (0,0)-tensors):... (ii) Vector fields (that is (1,0)-tensors): ... And so

2238:{\displaystyle ({\mathcal {L}}_{A}T)(\alpha ,\beta ,\ldots ,X,Y,\ldots )\equiv \nabla _{A}T(\alpha ,\beta ,\ldots ,X,Y,\ldots )-\nabla _{T(\cdot ,\beta ,\ldots ,X,Y,\ldots )}\alpha (A)-\ldots +T(\alpha ,\beta ,\ldots ,\nabla _{X}A,Y,\ldots )+\ldots } 3058: 4719:

A few days ago I added a new subsection with some examples. I think they are very useful, especially for people who are now learning about the Lie derivative and since the article is definitely not one of the most clear ones on Knowledge.

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Well, for somebody who is used to find clarity searching on wikipedia, this article is clearly written in mumbo jumbo language. So I've checked also the Wolfram mathworld for some clue, any clue...and there really is one:

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For me it's clear, see William M. Boothy "An introduction to differentiable manifolds and Riemannian Geometry", Academic Press, second edition, ISBN: 0-12-116052-1, Orlando 1986, p. 117, where one can read: "the function

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I made some major revisions to Lie derivatives of tensor fields. There was some good stuff in there which I had to delete involving Lie derivatives of tensor densities. I have preserved it below in this talk page.

1396: 4174:. I would suggest that we change the definition of Lie derivative according to Khalil (see H.K.Khalil, Nonlinear Systems, Prentice Hall editions, 3rd edition, ISBN: 0-13-067389-7, London 2002, pp.509-501). So, let 3124:. I've removed all of the ugly "One" language. I haven't added any extra content providing some insight/value for each definition, but hopefully the more straight-forward presentations of each definition help. — 1098: 1069: 5048: 351: 425: 675:

article and am writing stuff on the Lie derivative in that article. I have links to this (Lie derivative) article, but can I suggest that there should be more information here regarding the concept of

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that the Lie derivative does not need a connection for its definition (as is indicated by the fact that the covariant derivatives can be replaced by partials). Comments appreciated. Thanks. ---

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Separately from this, and not a reason for a quick fail: I think a subject as geometric as this one could benefit from some illustrations, such as we already have (for instance) at

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Well, if you've "been seeing references to Lie derivatives in differential geometry" then surely the setting within which these references arise are themselves motivation!

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No not exactly the push forward, T is the Tangent functional (aka derivative) but the overall effect is push forward. Seems a pretty obscure way to write it to me.

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it seems linas that you have used omega for a 1-form here. This clashes with the supposed standard use of omega as the volume form. Perhaps we should start using

1938:{\displaystyle T(f_{1}\alpha ,f_{2}\beta ,\ldots ,f_{p+1}X,f_{p+2}Y,\ldots )=f_{1}f_{2}\cdots f_{p+1}f_{p+2}\cdots f_{p+q}T(\alpha ,\beta ,\ldots ,X,Y,\ldots )} 1408: 4786: 5083: 175: 3957:

This is an issue that I have been aware of for a long time, but have never gotten around to fixing. It should (and will... someday) be fixed. Thanks,

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The explanation of the last two equations of section "The Lie derivative of a vector field" is completely missing. I talk about these equations:

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I will read it. Why did you remove it? It is from there that my use of the notation stems. It may not be standard, but it is certainly clearer. -

799: 151: 5078: 5073: 1323: 4835: 4768: 1198:{\displaystyle {\mathcal {L}}_{X}(\alpha \wedge \beta )=({\mathcal {L}}_{X}\alpha )\wedge \beta +\alpha \wedge ({\mathcal {L}}_{X}\beta )} 918:

The definition is extended so that it eats also functions. A function from M to R^n would still be a function and NOT a tangent vector. --

4539:. Your definition is "wrong". Remember, we are doing Differential Geometry. But I agree with you in that it is better to use the symbol 3986: 3921: 3856: 3079: 32: 4729: 52: 4608:(taking its flow), not with respect a function. Khalil's definition it's a particular one. It corresponds to taking the vector field 883:(P.S., I left the offending buzzwords intact, although I still believe that they ought to be removed or exported to an article like 672: 142: 103: 745:

is hopelessly confused! (For those of you readers who don't know what the Lie derivative of a function is, it's just the ordinary

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removed them without any discussion or explanation whatsoever though, so I reinstated them. Other articles like the one for the

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Second suggestion that we remove the definition and results concerning the interior product. These result are in the section

650:. Also, if there were any, they would not be more than one or two, so can be fixed by hand. Am I misunderstanding something? 3915:

Yes it's clear that it "Fl" is the flow and T is the pushforward. The only problem that this isn't mentioned in the article.

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have a small subsection with examples so it's obvious that having the examples really helps people understand the material.

963:. This is correct if a connection is defined. However the Lie derivative is defined even if the connection is not defined. 635: 570: 4611: 219:

I believe this topic has something to do with the intersection between algebra and differential geometry, am I correct?

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I see that several forms of the definition are mentioned. I think it would be less confusing to say something like:

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For a general differential form, the Lie derivative is likewise a contraction, taking into account the variation in

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Is there any source that uses this axiom to define Lie derivative? If not, what is the meaning of the expression

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is the result of a calculation. The Lie derivative of any object is always defined with respect a vector field

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No reference whatsoever to the Picard-Lindelof or Frobenius theorem is required. Please see the definition of

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What is "degree 0" derivation? This needs either an explanation, or a link to a page with an explanation.

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I'm not sure that's right. The prejudice is against getting confused between WP and a textbook. Various

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be changed? Then I will run the bot on the ones which should be changed, which are a majority. Thanks.

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Now that I've read it, I think the whole article should be scrapped. Even the Lie derivative of a

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on Knowledge. If you would like to participate, please visit the project page, where you can join

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I dunno, ω is used for many things; often it is a representative member of Ω which is the set of

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I'm familiar with the notation. "Fl" is indeed the flow, and T is the "tangent functor" or the

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In the section "The Lie derivative of a function" it is not clear what is meant by the symbols

4683: 4542: 4046: 3958: 3896: 3071: 2884: 1250:, the Lie derivative is just the contraction of the exterior derivative with the vector field 904: 877: 853: 781: 762: 727: 48: 4468: 4157: 2432: 1595: 1549: 1236: 1221: 684: 4925: 4675:. Juxtaposition of two vectors can, for example, be interpreted as the geometric product. 1600: 1483:{\displaystyle {\mathcal {L}}_{fX}\omega =f{\mathcal {L}}_{X}\omega +df\wedge i_{X}\omega } 565:

To fix redirects with a bot is easy. The hard problem is that not all the pages linking to

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I have never seen that notation before, so this is just a guess. "Fl" might be short for

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tedious to read as HTML and gains a lot of clarity in the TeXed version. Hope that's OK.

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Shouldn't this be a function from M to R^n, where n is the dimension of the manifold?

573:. In some cases the link should certainly be changed, in some cases, like the article 5067: 4957: 3940: 777: 758: 750: 614: 577:

I am not sure. Could anybody knowing these things make a list of articles linking to

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under the one-parameter group of diffeomorphisms generated by a vector field: the

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and Lie derivatives can then be summarized as follows. For an ordinary function

1224:(Or should be) and should not be reproduced here. The section would now become: 929:

Some suggestions about the list of axioms for Lie derivative of tensor fields.

919: 719: 681: 553: 511: 487: 264: 147: 4672: 2827: 1584: 1504: 803: 701: 647: 578: 566: 528: 500: 124: 4374:{\displaystyle (L_{f}g)(x)\triangleq ({\frac {\partial g}{\partial x}}f)(x)} 4817:. The edit link for this section can be used to add comments to the review. 2327:

and an infinitesimal generator of a diffeomorphism given by a vector field

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Removal or major edit of section "The Lie derivative of differential forms"

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Axiom 3: I propose replacing axiom 3 with two more "primitive axioms":

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notation. BTW, add your book reference to the article on Laplacian.

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connection is a stumbling block for mastering these concepts.

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and explain why each definition has particular value/insight.

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Now I am really confused. There are just a couple of links to

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Added a motivation paragraph to the beginning of the article

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induced by ψ. Then the Lie derivative of the tensor field

346:{\displaystyle :={\mathcal {L}}_{A}B=-{\mathcal {L}}_{B}A} 41:

at the time (December 11, 2016). There are suggestions on

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request someone more familiar with it to clarify this. —

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It would be necessary to explain at least the meaning of

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Lee. I have no reference to confirm this to you though. —

420:{\displaystyle :={\mathcal {L}}_{A}B-{\mathcal {L}}_{B}A} 259:

It seems that in the intro Lie derivate is used both for

726:. Both approaches should have a place in the article. 602:

I was very modestly thinking only of "all" the links on

2877:, as given above. Note that ψ is a local 1-parameter 1673:, …), such that for any collection of smooth functions 4981: 4928: 4888: 4614: 4592: 4567: 4545: 4500: 4471: 4415: 4390: 4301: 4279: 4259: 4220: 4180: 4160: 4140: 4089: 4049: 4011: 3828: 3488: 3212: 2961: 2938: 2918: 2887: 2863: 2839: 2806: 2475: 2440: 2417: 2387: 2353: 2333: 2313: 2290: 2270: 1986: 1963: 1721: 1644: 1603: 1557: 1534: 1411: 1326: 1263: 1101: 1077: 977: 939: 464: 361: 284: 4644:{\displaystyle X=f{\frac {\partial }{\partial x}}\,} 486:

for the volume form. I think it is less confusing. -

146:, a collaborative effort to improve the coverage of 429:The first equation is correct. The second is not. 5042: 4944: 4914: 4643: 4600: 4578: 4553: 4531: 4486: 4457: 4401: 4373: 4285: 4265: 4241: 4206: 4166: 4146: 4126: 4075: 4035: 3841: 3811: 3473: 3052: 2944: 2924: 2900: 2869: 2845: 2818: 2789: 2458: 2423: 2393: 2381:is nothing other than the infinitesimal change in 2373: 2339: 2319: 2296: 2276: 2237: 1969: 1937: 1653: 1619: 1575: 1540: 1482: 1390: 1302: 1197: 1083: 1063: 955: 478: 419: 345: 47:for improving the article. If you can improve it, 4083:. If it is simply meant that the defined entity ( 1231:. In this context, it is closely related to the 3247: 4882:"Another notation to denote the Lie derivative 4698:http://mathworld.wolfram.com/LieDerivative.html 4510: 4102: 3122:definition of the Lie derivative for a function 4878:http://matematicas.unex.es/~navarro/degree.pdf 4494:" and p. 226, where one can find the formula: 527:Has somebody a bot who could fix all links to 4971:The Lie derivative of a tensor field: axiom 3 694:presented as the first or primary definition. 8: 4651:, in a particular chart and representation. 3026: 4666:Non-intuitive definition in lead for layman 4242:{\displaystyle D\subseteq \mathbb {R} ^{n}} 3151:BTW - how is "Lie" pronounced? Lye or Lee? 2307:In other words, if you have a tensor field 1402:The derivative of products is distributed: 1303:{\displaystyle {\mathcal {L}}_{X}f=i_{X}df} 4746: 4127:{\displaystyle ({\mathcal {L}}_{\!X}f)(p)} 1227:The Lie derivative can also be defined on 92: 5028: 5009: 4990: 4984: 4983: 4980: 4933: 4927: 4897: 4891: 4890: 4887: 4624: 4613: 4591: 4566: 4544: 4509: 4503: 4502: 4499: 4476: 4470: 4444: 4414: 4389: 4336: 4309: 4300: 4278: 4258: 4233: 4229: 4228: 4219: 4200: 4199: 4179: 4159: 4139: 4101: 4095: 4094: 4088: 4048: 4016: 4010: 3829: 3827: 3803: 3796: 3788: 3778: 3771: 3763: 3753: 3745: 3741: 3733: 3723: 3715: 3711: 3703: 3690: 3675: 3669: 3667: 3654: 3649: 3641: 3631: 3626: 3618: 3608: 3600: 3592: 3582: 3574: 3566: 3553: 3537: 3532: 3521: 3516: 3513: 3497: 3491: 3490: 3487: 3454: 3449: 3441: 3439: 3426: 3418: 3410: 3401: 3389: 3374: 3368: 3366: 3351: 3342: 3318: 3313: 3305: 3303: 3290: 3282: 3274: 3265: 3250: 3237: 3224: 3218: 3217: 3211: 3029: 3011: 3006: 2982: 2970: 2964: 2963: 2960: 2937: 2917: 2892: 2886: 2862: 2838: 2805: 2694: 2600: 2545: 2487: 2481: 2480: 2474: 2439: 2416: 2386: 2362: 2356: 2355: 2352: 2332: 2312: 2289: 2269: 2205: 2111: 2056: 1998: 1992: 1991: 1985: 1962: 1881: 1862: 1846: 1833: 1823: 1792: 1770: 1748: 1732: 1720: 1643: 1608: 1602: 1556: 1533: 1471: 1446: 1440: 1439: 1420: 1414: 1413: 1410: 1376: 1351: 1335: 1329: 1328: 1325: 1288: 1272: 1266: 1265: 1262: 1183: 1177: 1176: 1145: 1139: 1138: 1110: 1104: 1103: 1100: 1076: 986: 980: 979: 976: 944: 938: 465: 463: 408: 402: 401: 388: 382: 381: 360: 334: 328: 327: 311: 305: 304: 283: 271:Where is the sign = ? where is the true ? 4532:{\displaystyle {\mathcal {L}}_{\!X}f=Xf} 2401:under the infinitesimal diffeomorphism. 1949:and if further we have a differentiable 4777: 4749: 4639: 4596: 4574: 4549: 4482: 4453: 4397: 4071: 4031: 3837: 1583:(which we consider as a differentiable 94: 64: 800:Knowledge talk:WikiProject Mathematics 4915:{\displaystyle {\mathcal {L}}_{X}(Y)} 4214:be sufficiently smooth functions and 4207:{\displaystyle f,g:D\to \mathbb {R} } 828:actually be treated that thoroughly. 20: 7: 2833:Alternately, given the vector field 263:and for elements of the range of . - 140:This article is within the scope of 2374:{\displaystyle {\mathcal {L}}_{U}T} 1514:Lie derivatives of tensor densities 83:It is of interest to the following 4630: 4626: 4347: 4339: 3833: 3830: 3792: 3789: 3767: 3764: 3737: 3734: 3707: 3704: 3676: 3670: 3645: 3642: 3622: 3619: 3596: 3593: 3570: 3567: 3533: 3517: 3445: 3442: 3414: 3411: 3402: 3375: 3369: 3309: 3306: 3278: 3275: 3266: 2853:, let ψ be the family of integral 2807: 2733: 2691: 2597: 2542: 2202: 2108: 2053: 941: 761:that things become interesting.) 472: 469: 466: 14: 5084:Mid-priority mathematics articles 3120:Please review the updates to the 673:mathematics of general relativity 160:Knowledge:WikiProject Mathematics 4458:{\displaystyle (Xf)(p)=X_{p}f\,} 275:(sorry for my very bad English) 203:Insufficiently explained concept 163:Template:WikiProject Mathematics 127: 117: 96: 65: 19: 3842:{\displaystyle \mathrm {Fl} \,} 2881:of local diffeomorphisms. Let 180:This article has been rated as 5037: 5034: 5002: 4996: 4909: 4903: 4434: 4428: 4425: 4416: 4368: 4362: 4359: 4333: 4327: 4321: 4318: 4302: 4196: 4121: 4115: 4112: 4090: 4068: 4062: 4059: 4050: 4028: 4022: 3466: 3460: 3432: 3406: 3348: 3330: 3324: 3296: 3270: 3262: 3254: 3234: 3213: 3104:23:44, 23 September 2007 (UTC) 3039: 3033: 2819:{\displaystyle \nabla \cdot A} 2784: 2748: 2742: 2730: 2715: 2669: 2654: 2648: 2640: 2604: 2590: 2554: 2535: 2499: 2496: 2476: 2453: 2441: 2226: 2180: 2165: 2159: 2151: 2115: 2101: 2065: 2046: 2010: 2007: 1987: 1953:(i.e. a smooth section of the 1932: 1896: 1813: 1725: 1570: 1558: 1385: 1369: 1192: 1172: 1154: 1134: 1128: 1116: 1058: 1055: 1043: 1040: 1031: 1028: 1022: 1016: 1007: 1001: 998: 992: 705:14:06, 29 September 2005 (UTC) 688:07:05, 29 September 2005 (UTC) 531:by adding (abstract algebra)? 479:{\displaystyle \mathrm {vol} } 374: 362: 297: 285: 1: 5060:03:54, 25 November 2020 (UTC) 4866:22:46, 11 December 2016 (UTC) 4840:22:46, 11 December 2016 (UTC) 3084:03:03, 11 November 2005 (UTC) 2404:More generally, if we have a 1508:21:33, 16 December 2005 (UTC) 1497:12:36, 16 December 2005 (UTC) 1211:13:17, 15 December 2005 (UTC) 908:03:00, 11 November 2005 (UTC) 881:06:10, 11 November 2005 (UTC) 636:derivation (abstract algebra) 571:derivation (abstract algebra) 154:and see a list of open tasks. 5079:B-Class mathematics articles 5074:Former good article nominees 4966:13:49, 16 January 2018 (UTC) 4954:Juan Bautista Sancho Guimerá 4738:10:03, 16 January 2015 (UTC) 4710:20:15, 29 January 2013 (UTC) 4661:17:45, 3 November 2011 (UTC) 3995:23:48, 1 December 2012 (UTC) 3875:which is a related concept. 2260:. This tensor is called the 956:{\displaystyle \nabla _{Y}f} 923:13:17, 31 October 2005 (UTC) 857:02:07, 9 November 2005 (UTC) 833:15:45, 8 November 2005 (UTC) 807:14:50, 8 November 2005 (UTC) 785:06:45, 8 November 2005 (UTC) 766:06:22, 8 November 2005 (UTC) 731:06:09, 8 November 2005 (UTC) 671:I'm currently rewriting the 4922:, although less common, is 4147:{\displaystyle \triangleq } 3973:21:31, 11 August 2008 (UTC) 3949:21:16, 12 August 2008 (UTC) 3930:21:22, 11 August 2008 (UTC) 3911:12:47, 11 August 2008 (UTC) 3885:12:27, 11 August 2008 (UTC) 3865:06:17, 11 August 2008 (UTC) 646:. I don't see any links to 5100: 4036:{\displaystyle X_{p}(f)\,} 3893:pushforward (differential) 233:12:07, 5 August 2008 (UTC) 37:, but it did not meet the 4871:Sancho Guimerá's notation 4854:as accessible as possible 4686:11:32, 24 June 2012 (UTC) 4554:{\displaystyle \equiv \,} 4076:{\displaystyle (Xf)(p)\,} 3188:18:35, 22 July 2009 (UTC) 3161:15:36, 22 July 2009 (UTC) 3144:18:35, 22 July 2009 (UTC) 2901:{\displaystyle \psi ^{*}} 2252:∇ used; as long as it is 1242:The relationship between 490:15:56, 27 Apr 2005 (UTC) 450:- Thomas, 131.111.231.27 434:05:23, 22 July 2006 (UTC) 267:14:19, 27 Apr 2005 (UTC) 179: 112: 91: 4870: 3109:for more than one reason 933:Axiom 1 should not have 749:. It's when you get to 655:21:59, 11 May 2005 (UTC) 618:21:48, 11 May 2005 (UTC) 590:20:17, 11 May 2005 (UTC) 557:14:43, 11 May 2005 (UTC) 543:22:20, 10 May 2005 (UTC) 503:00:33, 29 Apr 2005 (UTC) 439:Using math sections etc. 251:17:26, 5 June 2016 (UTC) 186:project's priority scale 33:Mathematics good article 5050:on the left-hand side? 4815:Talk:Lie derivative/GA1 4487:{\displaystyle C^{r}\,} 4167:{\displaystyle \equiv } 3091:Definition is confusing 1977:, then the linear map 523:Fix links to derivation 515:13:36, 2 May 2005 (UTC) 143:WikiProject Mathematics 5044: 4946: 4945:{\displaystyle X^{L}Y} 4916: 4645: 4602: 4580: 4555: 4533: 4488: 4459: 4403: 4375: 4287: 4267: 4243: 4208: 4168: 4148: 4128: 4077: 4037: 3843: 3813: 3475: 3054: 2946: 2926: 2902: 2871: 2847: 2820: 2791: 2460: 2425: 2395: 2375: 2341: 2321: 2298: 2278: 2248:is independent of the 2239: 1971: 1939: 1655: 1621: 1620:{\displaystyle T^{*}M} 1577: 1542: 1484: 1392: 1304: 1199: 1085: 1065: 957: 747:directional derivative 480: 421: 347: 73:This article is rated 5045: 4947: 4917: 4646: 4603: 4581: 4556: 4534: 4489: 4460: 4404: 4376: 4288: 4268: 4244: 4209: 4169: 4149: 4129: 4078: 4038: 3844: 3814: 3476: 3196:Unexplained equations 3055: 2947: 2927: 2903: 2872: 2848: 2821: 2792: 2461: 2459:{\displaystyle (p,q)} 2426: 2396: 2376: 2342: 2322: 2299: 2279: 2240: 1972: 1940: 1656: 1622: 1578: 1576:{\displaystyle (p,q)} 1543: 1520:differential geometry 1485: 1393: 1305: 1200: 1086: 1066: 958: 667:Proposals/suggestions 481: 422: 348: 39:good article criteria 4979: 4926: 4886: 4726:covariant derivative 4612: 4590: 4579:{\displaystyle Xf\,} 4565: 4543: 4498: 4469: 4413: 4402:{\displaystyle Xf\,} 4388: 4299: 4277: 4257: 4218: 4178: 4158: 4138: 4087: 4047: 4009: 3826: 3486: 3210: 2959: 2936: 2916: 2885: 2861: 2837: 2804: 2473: 2466:and density s, then 2438: 2415: 2385: 2351: 2331: 2311: 2288: 2268: 2256:, and in fact, is a 1984: 1961: 1719: 1642: 1601: 1555: 1532: 1409: 1324: 1261: 1244:exterior derivatives 1099: 1084:{\displaystyle \xi } 1075: 975: 937: 550:User:Oleg Alexandrov 462: 359: 282: 166:mathematics articles 4601:{\displaystyle X\,} 3808: 3783: 3758: 3728: 3659: 3636: 3613: 3587: 3459: 3431: 3323: 3295: 3016: 1233:exterior derivative 644:Talk:Lie derivative 29:was nominated as a 5040: 4942: 4912: 4641: 4640: 4598: 4597: 4576: 4575: 4551: 4550: 4529: 4511: 4484: 4483: 4455: 4454: 4399: 4398: 4371: 4283: 4263: 4239: 4204: 4164: 4144: 4124: 4103: 4073: 4072: 4033: 4032: 4001:Ambiguous notation 3873:Flow (mathematics) 3839: 3838: 3809: 3787: 3762: 3732: 3702: 3640: 3617: 3591: 3565: 3471: 3440: 3409: 3304: 3273: 3261: 3050: 3002: 2942: 2922: 2898: 2867: 2843: 2816: 2787: 2456: 2421: 2391: 2371: 2337: 2317: 2294: 2274: 2235: 1967: 1935: 1654:{\displaystyle TM} 1651: 1617: 1573: 1538: 1480: 1388: 1300: 1229:differential forms 1195: 1081: 1061: 953: 755:differential forms 642:, and a couple on 476: 417: 343: 135:Mathematics portal 79:content assessment 4805: 4804: 4637: 4354: 4286:{\displaystyle f} 4266:{\displaystyle g} 3985:comment added by 3867: 3855:comment added by 3801: 3776: 3750: 3720: 3684: 3547: 3383: 3246: 3153:Julian I Do Stuff 3087: 3070:comment added by 2995: 2945:{\displaystyle p} 2925:{\displaystyle T} 2870:{\displaystyle U} 2846:{\displaystyle U} 2424:{\displaystyle T} 2394:{\displaystyle T} 2340:{\displaystyle U} 2320:{\displaystyle T} 2297:{\displaystyle A} 2277:{\displaystyle T} 1970:{\displaystyle A} 1541:{\displaystyle T} 552:, he has a bot. - 211:Motivation needed 200: 199: 196: 195: 192: 191: 59: 58: 51:; it may then be 5091: 5052:StrokeOfMidnight 5049: 5047: 5046: 5041: 5033: 5032: 5014: 5013: 4995: 4994: 4989: 4988: 4951: 4949: 4948: 4943: 4938: 4937: 4921: 4919: 4918: 4913: 4902: 4901: 4896: 4895: 4759:Copyvio detector 4747: 4722:User:Mgvongoeden 4650: 4648: 4647: 4642: 4638: 4636: 4625: 4607: 4605: 4604: 4599: 4585: 4583: 4582: 4577: 4560: 4558: 4557: 4552: 4538: 4536: 4535: 4530: 4516: 4515: 4508: 4507: 4493: 4491: 4490: 4485: 4481: 4480: 4464: 4462: 4461: 4456: 4449: 4448: 4408: 4406: 4405: 4400: 4380: 4378: 4377: 4372: 4355: 4353: 4345: 4337: 4314: 4313: 4292: 4290: 4289: 4284: 4272: 4270: 4269: 4264: 4249:. We define the 4248: 4246: 4245: 4240: 4238: 4237: 4232: 4213: 4211: 4210: 4205: 4203: 4173: 4171: 4170: 4165: 4153: 4151: 4150: 4145: 4133: 4131: 4130: 4125: 4108: 4107: 4100: 4099: 4082: 4080: 4079: 4074: 4042: 4040: 4039: 4034: 4021: 4020: 3997: 3969: 3968: 3961: 3907: 3906: 3899: 3850: 3848: 3846: 3845: 3840: 3836: 3818: 3816: 3815: 3810: 3807: 3802: 3797: 3795: 3782: 3777: 3772: 3770: 3757: 3752: 3751: 3746: 3740: 3727: 3722: 3721: 3716: 3710: 3701: 3700: 3689: 3685: 3683: 3679: 3673: 3668: 3658: 3653: 3648: 3635: 3630: 3625: 3612: 3607: 3599: 3586: 3581: 3573: 3564: 3563: 3552: 3548: 3546: 3542: 3541: 3536: 3526: 3525: 3520: 3514: 3502: 3501: 3496: 3495: 3480: 3478: 3477: 3472: 3470: 3469: 3458: 3453: 3448: 3430: 3425: 3417: 3405: 3400: 3399: 3388: 3384: 3382: 3378: 3372: 3367: 3355: 3347: 3346: 3334: 3333: 3322: 3317: 3312: 3294: 3289: 3281: 3269: 3260: 3242: 3241: 3229: 3228: 3223: 3222: 3086: 3064: 3059: 3057: 3056: 3051: 3049: 3048: 3024: 3020: 3015: 3010: 2996: 2994: 2983: 2975: 2974: 2969: 2968: 2951: 2949: 2948: 2943: 2931: 2929: 2928: 2923: 2907: 2905: 2904: 2899: 2897: 2896: 2876: 2874: 2873: 2868: 2852: 2850: 2849: 2844: 2825: 2823: 2822: 2817: 2796: 2794: 2793: 2788: 2699: 2698: 2644: 2643: 2550: 2549: 2492: 2491: 2486: 2485: 2465: 2463: 2462: 2457: 2430: 2428: 2427: 2422: 2400: 2398: 2397: 2392: 2380: 2378: 2377: 2372: 2367: 2366: 2361: 2360: 2346: 2344: 2343: 2338: 2326: 2324: 2323: 2318: 2303: 2301: 2300: 2295: 2284:with respect to 2283: 2281: 2280: 2275: 2244: 2242: 2241: 2236: 2210: 2209: 2155: 2154: 2061: 2060: 2003: 2002: 1997: 1996: 1976: 1974: 1973: 1968: 1944: 1942: 1941: 1936: 1892: 1891: 1873: 1872: 1857: 1856: 1838: 1837: 1828: 1827: 1803: 1802: 1781: 1780: 1753: 1752: 1737: 1736: 1660: 1658: 1657: 1652: 1627:and of sections 1626: 1624: 1623: 1618: 1613: 1612: 1596:cotangent bundle 1582: 1580: 1579: 1574: 1547: 1545: 1544: 1539: 1489: 1487: 1486: 1481: 1476: 1475: 1451: 1450: 1445: 1444: 1428: 1427: 1419: 1418: 1397: 1395: 1394: 1389: 1381: 1380: 1356: 1355: 1340: 1339: 1334: 1333: 1309: 1307: 1306: 1301: 1293: 1292: 1277: 1276: 1271: 1270: 1237:interior product 1222:interior product 1204: 1202: 1201: 1196: 1188: 1187: 1182: 1181: 1150: 1149: 1144: 1143: 1115: 1114: 1109: 1108: 1090: 1088: 1087: 1082: 1071:for a co-vector 1070: 1068: 1067: 1062: 991: 990: 985: 984: 962: 960: 959: 954: 949: 948: 830:Charles Matthews 569:need to link to 485: 483: 482: 477: 475: 426: 424: 423: 418: 413: 412: 407: 406: 393: 392: 387: 386: 352: 350: 349: 344: 339: 338: 333: 332: 316: 315: 310: 309: 168: 167: 164: 161: 158: 137: 132: 131: 121: 114: 113: 108: 100: 93: 76: 70: 69: 61: 23: 22: 16: 5099: 5098: 5094: 5093: 5092: 5090: 5089: 5088: 5064: 5063: 5024: 5005: 4982: 4977: 4976: 4973: 4929: 4924: 4923: 4889: 4884: 4883: 4873: 4809:This review is 4801: 4773: 4745: 4717: 4693: 4668: 4629: 4610: 4609: 4588: 4587: 4563: 4562: 4541: 4540: 4501: 4496: 4495: 4472: 4467: 4466: 4440: 4411: 4410: 4386: 4385: 4346: 4338: 4305: 4297: 4296: 4275: 4274: 4255: 4254: 4227: 4216: 4215: 4176: 4175: 4156: 4155: 4154:one should use 4136: 4135: 4093: 4085: 4084: 4045: 4044: 4012: 4007: 4006: 4003: 3980: 3966: 3965: 3959: 3904: 3903: 3897: 3824: 3823: 3674: 3664: 3663: 3531: 3527: 3515: 3510: 3509: 3489: 3484: 3483: 3435: 3373: 3363: 3362: 3338: 3299: 3233: 3216: 3208: 3207: 3198: 3111: 3101:140.180.171.121 3093: 3065: 3025: 3001: 2997: 2987: 2962: 2957: 2956: 2934: 2933: 2914: 2913: 2888: 2883: 2882: 2859: 2858: 2835: 2834: 2802: 2801: 2690: 2596: 2541: 2479: 2471: 2470: 2436: 2435: 2413: 2412: 2383: 2382: 2354: 2349: 2348: 2329: 2328: 2309: 2308: 2286: 2285: 2266: 2265: 2201: 2107: 2052: 1990: 1982: 1981: 1959: 1958: 1877: 1858: 1842: 1829: 1819: 1788: 1766: 1744: 1728: 1717: 1716: 1711: 1698: 1688: 1679: 1640: 1639: 1604: 1599: 1598: 1594:α, β, … of the 1553: 1552: 1530: 1529: 1522:, if we have a 1516: 1467: 1438: 1412: 1407: 1406: 1372: 1347: 1327: 1322: 1321: 1284: 1264: 1259: 1258: 1218: 1175: 1137: 1102: 1097: 1096: 1073: 1072: 978: 973: 972: 940: 935: 934: 669: 652:Oleg Alexandrov 587:Oleg Alexandrov 525: 460: 459: 456: 441: 400: 380: 357: 356: 326: 303: 280: 279: 273: 213: 205: 165: 162: 159: 156: 155: 133: 126: 106: 77:on Knowledge's 74: 44:the review page 12: 11: 5: 5097: 5095: 5087: 5086: 5081: 5076: 5066: 5065: 5039: 5036: 5031: 5027: 5023: 5020: 5017: 5012: 5008: 5004: 5001: 4998: 4993: 4987: 4972: 4969: 4941: 4936: 4932: 4911: 4908: 4905: 4900: 4894: 4872: 4869: 4858:David Eppstein 4826:David Eppstein 4820: 4819: 4803: 4802: 4800: 4799: 4794: 4789: 4783: 4780: 4779: 4775: 4774: 4772: 4771: 4769:External links 4766: 4761: 4755: 4752: 4751: 4744: 4741: 4716: 4713: 4692: 4691:Mumbo jumbo... 4689: 4667: 4664: 4635: 4632: 4628: 4623: 4620: 4617: 4595: 4573: 4570: 4548: 4528: 4525: 4522: 4519: 4514: 4506: 4479: 4475: 4465:, is of class 4452: 4447: 4443: 4439: 4436: 4433: 4430: 4427: 4424: 4421: 4418: 4409:, defined by 4396: 4393: 4370: 4367: 4364: 4361: 4358: 4352: 4349: 4344: 4341: 4335: 4332: 4329: 4326: 4323: 4320: 4317: 4312: 4308: 4304: 4282: 4262: 4251:Lie derivative 4236: 4231: 4226: 4223: 4202: 4198: 4195: 4192: 4189: 4186: 4183: 4163: 4143: 4123: 4120: 4117: 4114: 4111: 4106: 4098: 4092: 4070: 4067: 4064: 4061: 4058: 4055: 4052: 4030: 4027: 4024: 4019: 4015: 4002: 3999: 3976: 3975: 3955: 3954: 3953: 3952: 3951: 3919: 3918: 3917: 3916: 3888: 3887: 3835: 3832: 3820: 3819: 3806: 3800: 3794: 3791: 3786: 3781: 3775: 3769: 3766: 3761: 3756: 3749: 3744: 3739: 3736: 3731: 3726: 3719: 3714: 3709: 3706: 3699: 3696: 3693: 3688: 3682: 3678: 3672: 3666: 3662: 3657: 3652: 3647: 3644: 3639: 3634: 3629: 3624: 3621: 3616: 3611: 3606: 3603: 3598: 3595: 3590: 3585: 3580: 3577: 3572: 3569: 3562: 3559: 3556: 3551: 3545: 3540: 3535: 3530: 3524: 3519: 3512: 3508: 3505: 3500: 3494: 3481: 3468: 3465: 3462: 3457: 3452: 3447: 3444: 3438: 3434: 3429: 3424: 3421: 3416: 3413: 3408: 3404: 3398: 3395: 3392: 3387: 3381: 3377: 3371: 3365: 3361: 3358: 3354: 3350: 3345: 3341: 3337: 3332: 3329: 3326: 3321: 3316: 3311: 3308: 3302: 3298: 3293: 3288: 3285: 3280: 3277: 3272: 3268: 3264: 3259: 3256: 3253: 3249: 3245: 3240: 3236: 3232: 3227: 3221: 3215: 3203: 3197: 3194: 3193: 3192: 3191: 3190: 3149: 3148: 3147: 3146: 3110: 3107: 3092: 3089: 3062: 3061: 3047: 3044: 3041: 3038: 3035: 3032: 3028: 3023: 3019: 3014: 3009: 3005: 3000: 2993: 2990: 2986: 2981: 2978: 2973: 2967: 2941: 2921: 2895: 2891: 2866: 2842: 2815: 2812: 2809: 2798: 2797: 2786: 2783: 2780: 2777: 2774: 2771: 2768: 2765: 2762: 2759: 2756: 2753: 2750: 2747: 2744: 2741: 2738: 2735: 2732: 2729: 2726: 2723: 2720: 2717: 2714: 2711: 2708: 2705: 2702: 2697: 2693: 2689: 2686: 2683: 2680: 2677: 2674: 2671: 2668: 2665: 2662: 2659: 2656: 2653: 2650: 2647: 2642: 2639: 2636: 2633: 2630: 2627: 2624: 2621: 2618: 2615: 2612: 2609: 2606: 2603: 2599: 2595: 2592: 2589: 2586: 2583: 2580: 2577: 2574: 2571: 2568: 2565: 2562: 2559: 2556: 2553: 2548: 2544: 2540: 2537: 2534: 2531: 2528: 2525: 2522: 2519: 2516: 2513: 2510: 2507: 2504: 2501: 2498: 2495: 2490: 2484: 2478: 2455: 2452: 2449: 2446: 2443: 2420: 2409:tensor density 2406:differentiable 2390: 2370: 2365: 2359: 2336: 2316: 2293: 2273: 2262:Lie derivative 2246: 2245: 2234: 2231: 2228: 2225: 2222: 2219: 2216: 2213: 2208: 2204: 2200: 2197: 2194: 2191: 2188: 2185: 2182: 2179: 2176: 2173: 2170: 2167: 2164: 2161: 2158: 2153: 2150: 2147: 2144: 2141: 2138: 2135: 2132: 2129: 2126: 2123: 2120: 2117: 2114: 2110: 2106: 2103: 2100: 2097: 2094: 2091: 2088: 2085: 2082: 2079: 2076: 2073: 2070: 2067: 2064: 2059: 2055: 2051: 2048: 2045: 2042: 2039: 2036: 2033: 2030: 2027: 2024: 2021: 2018: 2015: 2012: 2009: 2006: 2001: 1995: 1989: 1966: 1955:tangent bundle 1947: 1946: 1934: 1931: 1928: 1925: 1922: 1919: 1916: 1913: 1910: 1907: 1904: 1901: 1898: 1895: 1890: 1887: 1884: 1880: 1876: 1871: 1868: 1865: 1861: 1855: 1852: 1849: 1845: 1841: 1836: 1832: 1826: 1822: 1818: 1815: 1812: 1809: 1806: 1801: 1798: 1795: 1791: 1787: 1784: 1779: 1776: 1773: 1769: 1765: 1762: 1759: 1756: 1751: 1747: 1743: 1740: 1735: 1731: 1727: 1724: 1703: 1693: 1684: 1677: 1650: 1647: 1637:tangent bundle 1616: 1611: 1607: 1572: 1569: 1566: 1563: 1560: 1537: 1524:differentiable 1515: 1512: 1511: 1510: 1491: 1490: 1479: 1474: 1470: 1466: 1463: 1460: 1457: 1454: 1449: 1443: 1437: 1434: 1431: 1426: 1423: 1417: 1400: 1399: 1387: 1384: 1379: 1375: 1371: 1368: 1365: 1362: 1359: 1354: 1350: 1346: 1343: 1338: 1332: 1311: 1310: 1299: 1296: 1291: 1287: 1283: 1280: 1275: 1269: 1217: 1214: 1194: 1191: 1186: 1180: 1174: 1171: 1168: 1165: 1162: 1159: 1156: 1153: 1148: 1142: 1136: 1133: 1130: 1127: 1124: 1121: 1118: 1113: 1107: 1080: 1060: 1057: 1054: 1051: 1048: 1045: 1042: 1039: 1036: 1033: 1030: 1027: 1024: 1021: 1018: 1015: 1012: 1009: 1006: 1003: 1000: 997: 994: 989: 983: 970: 969: 965: 964: 952: 947: 943: 928: 926: 925: 911: 910: 899: 898: 897: 896: 895: 894: 893: 892: 885:tangent bundle 866: 865: 864: 863: 862: 861: 860: 859: 842: 841: 840: 839: 838: 837: 836: 835: 814: 813: 812: 811: 810: 809: 790: 789: 788: 787: 771: 770: 769: 768: 736: 735: 734: 733: 708: 707: 696: 695: 668: 665: 664: 663: 662: 661: 660: 659: 658: 657: 640:Lie derivative 625: 624: 623: 622: 621: 620: 595: 594: 593: 592: 560: 559: 524: 521: 520: 519: 518: 517: 505: 504: 474: 471: 468: 455: 452: 440: 437: 416: 411: 405: 399: 396: 391: 385: 379: 376: 373: 370: 367: 364: 342: 337: 331: 325: 322: 319: 314: 308: 302: 299: 296: 293: 290: 287: 272: 269: 257: 256: 255: 254: 253: 236: 235: 212: 209: 204: 201: 198: 197: 194: 193: 190: 189: 178: 172: 171: 169: 152:the discussion 139: 138: 122: 110: 109: 101: 89: 88: 82: 71: 57: 56: 27:Lie derivative 24: 13: 10: 9: 6: 4: 3: 2: 5096: 5085: 5082: 5080: 5077: 5075: 5072: 5071: 5069: 5062: 5061: 5057: 5053: 5029: 5025: 5021: 5018: 5015: 5010: 5006: 4999: 4991: 4970: 4968: 4967: 4963: 4959: 4955: 4939: 4934: 4930: 4906: 4898: 4880: 4879: 4868: 4867: 4863: 4859: 4855: 4851: 4846: 4842: 4841: 4837: 4834: 4831: 4827: 4824: 4818: 4816: 4812: 4807: 4806: 4798: 4795: 4793: 4790: 4788: 4785: 4784: 4782: 4781: 4776: 4770: 4767: 4765: 4762: 4760: 4757: 4756: 4754: 4753: 4748: 4742: 4740: 4739: 4735: 4731: 4727: 4723: 4714: 4712: 4711: 4707: 4703: 4702:Theodore Yoda 4699: 4690: 4688: 4687: 4684: 4682: 4676: 4674: 4665: 4663: 4662: 4658: 4654: 4633: 4621: 4618: 4615: 4593: 4571: 4568: 4546: 4526: 4523: 4520: 4517: 4512: 4477: 4473: 4450: 4445: 4441: 4437: 4431: 4422: 4419: 4394: 4391: 4381: 4365: 4356: 4350: 4342: 4330: 4324: 4315: 4310: 4306: 4294: 4280: 4260: 4252: 4234: 4224: 4221: 4193: 4190: 4187: 4184: 4181: 4161: 4141: 4118: 4109: 4104: 4065: 4056: 4053: 4025: 4017: 4013: 4000: 3998: 3996: 3992: 3988: 3987:94.224.97.175 3984: 3974: 3970: 3962: 3956: 3950: 3946: 3942: 3938: 3937: 3936: 3935: 3934: 3933: 3932: 3931: 3927: 3923: 3922:89.135.19.155 3914: 3913: 3912: 3908: 3900: 3894: 3890: 3889: 3886: 3882: 3878: 3874: 3870: 3869: 3868: 3866: 3862: 3858: 3857:89.135.19.155 3854: 3804: 3798: 3784: 3779: 3773: 3759: 3754: 3747: 3742: 3729: 3724: 3717: 3712: 3697: 3694: 3691: 3686: 3680: 3660: 3655: 3650: 3637: 3632: 3627: 3614: 3609: 3604: 3601: 3588: 3583: 3578: 3575: 3560: 3557: 3554: 3549: 3543: 3538: 3528: 3522: 3506: 3503: 3498: 3482: 3463: 3455: 3450: 3436: 3427: 3422: 3419: 3396: 3393: 3390: 3385: 3379: 3359: 3356: 3352: 3343: 3339: 3335: 3327: 3319: 3314: 3300: 3291: 3286: 3283: 3257: 3251: 3243: 3238: 3230: 3225: 3206: 3205: 3204: 3201: 3195: 3189: 3185: 3181: 3177: 3173: 3172: 3167: 3166: 3165: 3164: 3163: 3162: 3158: 3154: 3145: 3141: 3137: 3133: 3129: 3128: 3123: 3119: 3118: 3117: 3116: 3115: 3108: 3106: 3105: 3102: 3099: 3090: 3088: 3085: 3081: 3077: 3073: 3072:Silly rabbit 3069: 3045: 3042: 3036: 3030: 3021: 3017: 3012: 3007: 3003: 2998: 2991: 2988: 2984: 2979: 2976: 2971: 2955: 2954: 2953: 2939: 2932:at the point 2919: 2911: 2893: 2889: 2880: 2864: 2856: 2840: 2831: 2829: 2813: 2810: 2781: 2778: 2775: 2772: 2769: 2766: 2763: 2760: 2757: 2754: 2751: 2745: 2739: 2736: 2727: 2724: 2721: 2718: 2712: 2709: 2706: 2703: 2700: 2695: 2687: 2684: 2681: 2678: 2675: 2672: 2666: 2663: 2660: 2657: 2651: 2645: 2637: 2634: 2631: 2628: 2625: 2622: 2619: 2616: 2613: 2610: 2607: 2601: 2593: 2587: 2584: 2581: 2578: 2575: 2572: 2569: 2566: 2563: 2560: 2557: 2551: 2546: 2538: 2532: 2529: 2526: 2523: 2520: 2517: 2514: 2511: 2508: 2505: 2502: 2493: 2488: 2469: 2468: 2467: 2450: 2447: 2444: 2434: 2418: 2410: 2407: 2402: 2388: 2368: 2363: 2334: 2314: 2305: 2291: 2271: 2263: 2259: 2255: 2251: 2232: 2229: 2223: 2220: 2217: 2214: 2211: 2206: 2198: 2195: 2192: 2189: 2186: 2183: 2177: 2174: 2171: 2168: 2162: 2156: 2148: 2145: 2142: 2139: 2136: 2133: 2130: 2127: 2124: 2121: 2118: 2112: 2104: 2098: 2095: 2092: 2089: 2086: 2083: 2080: 2077: 2074: 2071: 2068: 2062: 2057: 2049: 2043: 2040: 2037: 2034: 2031: 2028: 2025: 2022: 2019: 2016: 2013: 2004: 1999: 1980: 1979: 1978: 1964: 1956: 1952: 1929: 1926: 1923: 1920: 1917: 1914: 1911: 1908: 1905: 1902: 1899: 1893: 1888: 1885: 1882: 1878: 1874: 1869: 1866: 1863: 1859: 1853: 1850: 1847: 1843: 1839: 1834: 1830: 1824: 1820: 1816: 1810: 1807: 1804: 1799: 1796: 1793: 1789: 1785: 1782: 1777: 1774: 1771: 1767: 1763: 1760: 1757: 1754: 1749: 1745: 1741: 1738: 1733: 1729: 1722: 1715: 1714: 1713: 1710: 1706: 1702: 1696: 1692: 1687: 1683: 1676: 1672: 1668: 1664: 1648: 1645: 1638: 1634: 1630: 1614: 1609: 1605: 1597: 1593: 1590: 1586: 1567: 1564: 1561: 1551: 1535: 1528: 1525: 1521: 1513: 1509: 1506: 1501: 1500: 1499: 1498: 1495: 1477: 1472: 1468: 1464: 1461: 1458: 1455: 1452: 1447: 1435: 1432: 1429: 1424: 1421: 1405: 1404: 1403: 1382: 1377: 1373: 1366: 1363: 1360: 1357: 1352: 1348: 1344: 1341: 1336: 1320: 1319: 1318: 1316: 1297: 1294: 1289: 1285: 1281: 1278: 1273: 1257: 1256: 1255: 1253: 1249: 1245: 1240: 1238: 1234: 1230: 1225: 1223: 1215: 1213: 1212: 1209: 1205: 1189: 1184: 1169: 1166: 1163: 1160: 1157: 1151: 1146: 1131: 1125: 1122: 1119: 1111: 1094: 1091: 1078: 1052: 1049: 1046: 1037: 1034: 1025: 1019: 1013: 1010: 1004: 995: 987: 967: 966: 950: 945: 932: 931: 930: 924: 921: 917: 916: 915: 909: 906: 901: 900: 890: 886: 882: 879: 874: 873: 872: 871: 870: 869: 868: 867: 858: 855: 850: 849: 848: 847: 846: 845: 844: 843: 834: 831: 826: 822: 821: 820: 819: 818: 817: 816: 815: 808: 805: 801: 796: 795: 794: 793: 792: 791: 786: 783: 779: 778:tangent space 775: 774: 773: 772: 767: 764: 760: 759:tensor fields 756: 752: 751:vector fields 748: 744: 740: 739: 738: 737: 732: 729: 725: 721: 717: 712: 711: 710: 709: 706: 703: 698: 697: 692: 691: 690: 689: 686: 683: 678: 674: 666: 656: 653: 649: 645: 641: 637: 633: 632: 631: 630: 629: 628: 627: 626: 619: 616: 612: 608: 605: 601: 600: 599: 598: 597: 596: 591: 588: 584: 581:which should 580: 576: 572: 568: 564: 563: 562: 561: 558: 555: 551: 547: 546: 545: 544: 541: 537: 533: 530: 522: 516: 513: 509: 508: 507: 506: 502: 497: 493: 492: 491: 489: 453: 451: 448: 444: 438: 436: 435: 432: 431:69.129.194.27 427: 414: 409: 397: 394: 389: 377: 371: 368: 365: 353: 340: 335: 323: 320: 317: 312: 300: 294: 291: 288: 276: 270: 268: 266: 262: 252: 248: 244: 240: 239: 238: 237: 234: 230: 226: 225:Dharma6662000 222: 221: 220: 217: 210: 208: 202: 187: 183: 177: 174: 173: 170: 153: 149: 145: 144: 136: 130: 125: 123: 120: 116: 115: 111: 105: 102: 99: 95: 90: 86: 80: 72: 68: 63: 62: 54: 50: 46: 45: 40: 36: 35: 34: 28: 25: 18: 17: 4974: 4881: 4874: 4850:vector field 4847: 4843: 4832: 4822: 4821: 4808: 4797:Instructions 4730:129.194.8.73 4718: 4694: 4677: 4669: 4382: 4295: 4250: 4004: 3981:— Preceding 3979:this book. 3977: 3960:siℓℓy rabbit 3920: 3898:siℓℓy rabbit 3821: 3202: 3199: 3169: 3150: 3125: 3112: 3096: 3094: 3066:— Preceding 3063: 2952:is given by 2832: 2799: 2403: 2306: 2261: 2254:torsion-free 2247: 1951:vector field 1948: 1708: 1704: 1700: 1694: 1690: 1685: 1681: 1674: 1670: 1666: 1662: 1632: 1628: 1527:tensor field 1517: 1492: 1401: 1314: 1312: 1251: 1247: 1241: 1226: 1219: 1206: 1095: 1092: 971: 927: 912: 905:Silly rabbit 889:vector field 878:Silly rabbit 854:Silly rabbit 824: 782:Silly rabbit 763:Silly rabbit 742: 728:Silly rabbit 723: 677:Lie dragging 676: 670: 643: 603: 582: 575:dual numbers 526: 495: 457: 449: 445: 442: 428: 354: 277: 274: 260: 258: 218: 214: 206: 182:Mid-priority 181: 141: 107:Mid‑priority 85:WikiProjects 42: 31: 30: 26: 4811:transcluded 4653:Mgvongoeden 3851:—Preceding 1635:, … of the 720:vector flow 157:Mathematics 148:mathematics 104:Mathematics 53:renominated 5068:Categories 4764:Authorship 4750:GA toolbox 4673:commutator 4561:, because 2828:divergence 2250:connection 1665:(α, β, …, 1661:, written 1585:linear map 648:derivation 579:derivation 567:derivation 529:derivation 4823:Reviewer: 4787:Templates 4778:Reviewing 4743:GA Review 3877:JRSpriggs 3171:TedPavlic 3127:TedPavlic 243:David9550 49:please do 4958:JBecerra 4836:contribs 4792:Criteria 4715:Examples 3983:unsigned 3941:Billlion 3853:unsigned 3080:contribs 3068:unsigned 2910:pullback 1712:we have 1592:sections 1494:Mungbean 1235:and the 1208:Mungbean 825:tutorial 743:function 716:pullback 4681:Quondum 3180:contrib 3136:contrib 2908:be the 2826:is the 2347:, then 443:Hello, 184:on the 75:B-class 4273:along 2855:curves 2800:where 2411:field 2258:tensor 1589:smooth 920:MarSch 757:, and 685:(talk) 682:Mpatel 606:page. 554:MarSch 512:MarSch 496:smooth 488:MarSch 265:MarSch 81:scale. 4813:from 2879:group 1699:, …, 1680:, …, 1505:linas 804:linas 724:means 702:linas 501:linas 454:omega 355:+ : 278:- : 5056:talk 4962:talk 4956:"". 4862:talk 4830:talk 4734:talk 4706:talk 4657:talk 4293:as: 4043:and 3991:talk 3967:talk 3945:talk 3926:talk 3905:talk 3881:talk 3861:talk 3176:talk 3157:talk 3132:talk 3076:talk 2433:rank 1550:rank 1239:. 1093:and 615:Talk 604:this 548:ask 540:Talk 247:talk 229:talk 4253:of 3849:. 3248:lim 3098:on. 3082:) 2857:of 2431:of 2264:of 1697:+ 1 1587:of 1548:of 1518:In 887:or 638:on 611:MFH 583:not 536:MFH 176:Mid 5070:: 5058:) 5019:… 4964:) 4864:) 4838:) 4736:) 4708:) 4659:) 4631:∂ 4627:∂ 4547:≡ 4348:∂ 4340:∂ 4331:≜ 4225:⊆ 4197:→ 4162:≡ 4142:≜ 3993:) 3971:) 3947:) 3928:) 3909:) 3895:. 3883:) 3863:) 3785:∘ 3760:∘ 3743:− 3730:∘ 3713:− 3638:∘ 3615:∘ 3602:− 3589:∘ 3576:− 3507::= 3420:− 3336:− 3284:− 3255:→ 3244::= 3186:) 3159:) 3142:) 3078:• 3031:ψ 3013:∗ 3004:ψ 2894:∗ 2890:ψ 2830:. 2811:⋅ 2808:∇ 2782:… 2764:… 2758:β 2752:α 2737:⋅ 2734:∇ 2722:… 2713:… 2692:∇ 2685:… 2679:β 2673:α 2661:… 2658:− 2646:α 2638:… 2620:… 2614:β 2608:⋅ 2598:∇ 2594:− 2588:… 2570:… 2564:β 2558:α 2543:∇ 2539:≡ 2533:… 2515:… 2509:β 2503:α 2304:. 2233:… 2224:… 2203:∇ 2196:… 2190:β 2184:α 2172:… 2169:− 2157:α 2149:… 2131:… 2125:β 2119:⋅ 2109:∇ 2105:− 2099:… 2081:… 2075:β 2069:α 2054:∇ 2050:≡ 2044:… 2026:… 2020:β 2014:α 1957:) 1930:… 1912:… 1906:β 1900:α 1875:⋯ 1840:⋯ 1811:… 1761:… 1755:β 1739:α 1707:+ 1689:, 1669:, 1631:, 1610:∗ 1478:ω 1465:∧ 1453:ω 1430:ω 1383:ω 1361:ω 1342:ω 1317:: 1254:: 1190:β 1170:∧ 1167:α 1161:β 1158:∧ 1152:α 1126:β 1123:∧ 1120:α 1079:ξ 1038:ξ 1035:− 1020:ξ 996:ξ 942:∇ 891:.) 802:. 753:, 613:: 609:— 538:: 534:— 398:− 378::= 324:− 301::= 249:) 231:) 5054:( 5038:) 5035:) 5030:n 5026:T 5022:, 5016:, 5011:1 5007:Y 5003:( 5000:T 4997:( 4992:X 4986:L 4960:( 4940:Y 4935:L 4931:X 4910:) 4907:Y 4904:( 4899:X 4893:L 4860:( 4833:· 4828:( 4732:( 4704:( 4655:( 4634:x 4622:f 4619:= 4616:X 4594:X 4572:f 4569:X 4527:f 4524:X 4521:= 4518:f 4513:X 4505:L 4478:r 4474:C 4451:f 4446:p 4442:X 4438:= 4435:) 4432:p 4429:( 4426:) 4423:f 4420:X 4417:( 4395:f 4392:X 4369:) 4366:x 4363:( 4360:) 4357:f 4351:x 4343:g 4334:( 4328:) 4325:x 4322:( 4319:) 4316:g 4311:f 4307:L 4303:( 4281:f 4261:g 4235:n 4230:R 4222:D 4201:R 4194:D 4191:: 4188:g 4185:, 4182:f 4122:) 4119:p 4116:( 4113:) 4110:f 4105:X 4097:L 4091:( 4069:) 4066:p 4063:( 4060:) 4057:f 4054:X 4051:( 4029:) 4026:f 4023:( 4018:p 4014:X 3989:( 3963:( 3943:( 3924:( 3901:( 3879:( 3859:( 3834:l 3831:F 3805:X 3799:t 3793:l 3790:F 3780:Y 3774:t 3768:l 3765:F 3755:X 3748:t 3738:l 3735:F 3725:Y 3718:t 3708:l 3705:F 3698:0 3695:= 3692:t 3687:| 3681:t 3677:d 3671:d 3661:= 3656:X 3651:t 3646:l 3643:F 3633:Y 3628:t 3623:l 3620:F 3610:X 3605:t 3597:l 3594:F 3584:Y 3579:t 3571:l 3568:F 3561:0 3558:= 3555:t 3550:| 3544:t 3539:2 3534:d 3529:2 3523:2 3518:d 3504:Y 3499:X 3493:L 3467:) 3464:x 3461:( 3456:X 3451:t 3446:l 3443:F 3437:Y 3433:) 3428:X 3423:t 3415:l 3412:F 3407:( 3403:T 3397:0 3394:= 3391:t 3386:| 3380:t 3376:d 3370:d 3360:= 3357:t 3353:/ 3349:) 3344:x 3340:Y 3331:) 3328:x 3325:( 3320:X 3315:t 3310:l 3307:F 3301:Y 3297:) 3292:X 3287:t 3279:l 3276:F 3271:( 3267:T 3263:( 3258:0 3252:t 3239:x 3235:) 3231:Y 3226:X 3220:L 3214:( 3184:@ 3182:/ 3178:/ 3174:( 3155:( 3140:@ 3138:/ 3134:/ 3130:( 3074:( 3060:. 3046:p 3043:= 3040:) 3037:t 3034:( 3027:| 3022:) 3018:T 3008:t 2999:( 2992:t 2989:d 2985:d 2980:= 2977:T 2972:U 2966:L 2940:p 2920:T 2865:U 2841:U 2814:A 2785:) 2779:, 2776:Y 2773:, 2770:X 2767:, 2761:, 2755:, 2749:( 2746:T 2743:) 2740:A 2731:( 2728:s 2725:+ 2719:+ 2716:) 2710:, 2707:Y 2704:, 2701:A 2696:X 2688:, 2682:, 2676:, 2670:( 2667:T 2664:+ 2655:) 2652:A 2649:( 2641:) 2635:, 2632:Y 2629:, 2626:X 2623:, 2617:, 2611:, 2605:( 2602:T 2591:) 2585:, 2582:Y 2579:, 2576:X 2573:, 2567:, 2561:, 2555:( 2552:T 2547:A 2536:) 2530:, 2527:Y 2524:, 2521:X 2518:, 2512:, 2506:, 2500:( 2497:) 2494:T 2489:A 2483:L 2477:( 2454:) 2451:q 2448:, 2445:p 2442:( 2419:T 2389:T 2369:T 2364:U 2358:L 2335:U 2315:T 2292:A 2272:T 2230:+ 2227:) 2221:, 2218:Y 2215:, 2212:A 2207:X 2199:, 2193:, 2187:, 2181:( 2178:T 2175:+ 2166:) 2163:A 2160:( 2152:) 2146:, 2143:Y 2140:, 2137:X 2134:, 2128:, 2122:, 2116:( 2113:T 2102:) 2096:, 2093:Y 2090:, 2087:X 2084:, 2078:, 2072:, 2066:( 2063:T 2058:A 2047:) 2041:, 2038:Y 2035:, 2032:X 2029:, 2023:, 2017:, 2011:( 2008:) 2005:T 2000:A 1994:L 1988:( 1965:A 1945:, 1933:) 1927:, 1924:Y 1921:, 1918:X 1915:, 1909:, 1903:, 1897:( 1894:T 1889:q 1886:+ 1883:p 1879:f 1870:2 1867:+ 1864:p 1860:f 1854:1 1851:+ 1848:p 1844:f 1835:2 1831:f 1825:1 1821:f 1817:= 1814:) 1808:, 1805:Y 1800:2 1797:+ 1794:p 1790:f 1786:, 1783:X 1778:1 1775:+ 1772:p 1768:f 1764:, 1758:, 1750:2 1746:f 1742:, 1734:1 1730:f 1726:( 1723:T 1709:q 1705:p 1701:f 1695:p 1691:f 1686:p 1682:f 1678:1 1675:f 1671:Y 1667:X 1663:T 1649:M 1646:T 1633:Y 1629:X 1615:M 1606:T 1571:) 1568:q 1565:, 1562:p 1559:( 1536:T 1473:X 1469:i 1462:f 1459:d 1456:+ 1448:X 1442:L 1436:f 1433:= 1425:X 1422:f 1416:L 1398:. 1386:) 1378:X 1374:i 1370:( 1367:d 1364:+ 1358:d 1353:X 1349:i 1345:= 1337:X 1331:L 1315:X 1298:f 1295:d 1290:X 1286:i 1282:= 1279:f 1274:X 1268:L 1252:X 1248:f 1193:) 1185:X 1179:L 1173:( 1164:+ 1155:) 1147:X 1141:L 1135:( 1132:= 1129:) 1117:( 1112:X 1106:L 1059:) 1056:] 1053:Y 1050:, 1047:X 1044:[ 1041:( 1032:) 1029:) 1026:Y 1023:( 1017:( 1014:X 1011:= 1008:) 1005:Y 1002:( 999:) 993:( 988:X 982:L 951:f 946:Y 473:l 470:o 467:v 415:A 410:B 404:L 395:B 390:A 384:L 375:] 372:B 369:, 366:A 363:[ 341:A 336:B 330:L 321:= 318:B 313:A 307:L 298:] 295:B 292:, 289:A 286:[ 261:L 245:( 227:( 188:. 87:: 55:.

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Index