Timeline for Why is the Leibniz rule a definition for derivations?
Current License: CC BY-SA 3.0
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| Dec 31, 2012 at 10:00 | comment | added | Dror Atariah | For the sake of reference: W. Kühnel provides in Differential Geometry - Curves, Surfaces, Manifolds (in 5B) three possible ways to defined the tangent space. The one given here is one of them. | |
| Dec 28, 2012 at 5:09 | history | edited | David Corwin | CC BY-SA 3.0 |
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| Dec 28, 2012 at 5:03 | history | edited | David Corwin | CC BY-SA 3.0 |
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| Dec 28, 2012 at 4:53 | comment | added | Eric Wofsey | This is a great answer. I would only add that if you don't like thinking about the collection of all smooth paths, you can simply think of vectors based at a point in any coordinate chart. Any such vector gives a directional derivative you can apply to any smooth function at the point, and you consider two such vectors to be the same if their directoinal derivatives happen to agree on all functions. This turns out to be equivalent to saying that the (total) derivative of the change-of-coordinates diffeomorphism between the two charts maps one vector to the other. | |
| Dec 28, 2012 at 4:15 | comment | added | Michael Murray | +1 for using tangency classes of paths instead of derivation as the definition of tangent vectors. Personally I think it makes calculations easier but that's a subjective judgement not a mathematical one. | |
| Dec 28, 2012 at 1:53 | comment | added | Qiaochu Yuan | This is only a good definition in situations where you expect tangent vectors to actually extend to paths in some reasonable sense. In algebraic geometry, for example, this is often false. | |
| Dec 28, 2012 at 1:39 | history | answered | David Corwin | CC BY-SA 3.0 |