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The article is going to have to define what J(X) means. Charles Matthews 22:07, 27 March 2007 (UTC)
I think J(X) means the Jacobian matrix of partial derivatives of X in coordinates. This matrix is then applied to a column vector Y to obtain what I would call dX(Y). The difference between the two terms is then the Lie bracket of vector fields. Geometry guy 10:40, 17 June 2007 (UTC)
-- Coordinate chart
I guess the x^j are the coordinate chart talked about? —Preceding unsigned comment added by 88.77.250.91 ( talk) 13:57, 5 January 2008 (UTC)
The Lie Bracket, although used in differential geometry, is a construction based in differential topology. It has no dependance on a metric / norm / or any other type of geometry. Rybu ( talk) 05:16, 10 September 2008 (UTC)
The theorem: " iff their corresponding flows commute (i.e. )." is problematic, since there exist counter examples: Let be the manifold were the vector fields and are defined. (z sort of measures the change of angle while following the flow of X or Y.) Now we have but .
It looks like 3.15 Corollary (page 21) is the corresponding statement of this theorem in the book "Natural operations in differential geometry" (one of the references of the article). I wrote to the author of the book that I have the impression that the statement of the conditions under which this statement holds (3.15 Corollary says "wherever defined") is not restrictive enough (or misleading). He answered: "you are right. wherever defined means here: if one (equiv. both) side is defined on , i.e., on the rectangle from to . OR: it holds locally and globally when lifted to simply connected coverings. This is clear form the proof of 3.15."
It might make sense to mention at least that there are counter examples and provide a reference that is sufficient explicit about this issue. Jakito ( talk) 09:52, 3 July 2009 (UTC)
This appears in the Properties section:
To a new reader, this is incomprehensible. What are the domain and range of the functions? What is the action fX? What is the action X(g)Y? I made several naive speculations about possible meanings, but none of them have been plausible so far. Rschwieb ( talk) 13:38, 10 April 2013 (UTC)
The reference 'Lewis, Andrew D., Notes on (Nonlinear) Control Theory (PDF)' no longer works. GilR 11:41, 19 May 2015 (UTC) — Preceding unsigned comment added by Gilbert.Rooke ( talk • contribs)
How does X(Y^i) or Y(X^i) make any sense when X and Y are function from M to TM and Y^i and X^i are function from M to R? — Preceding unsigned comment added by 92.200.61.85 ( talk) 15:21, 2 April 2017 (UTC)