Timeline for Tangent Vectors as Infinitesimal Displacements
Current License: CC BY-SA 3.0
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| Sep 10, 2015 at 14:52 | history | tweeted | twitter.com/#!/StackPhysics/status/641987535040847872 | ||
| Sep 10, 2015 at 1:15 | comment | added | Selene Routley | .... Want to ride to $\exp_p(X(p)\,1)$ from $p$? $X$ is your ticket and Picard-Lindelöf is the unfailing locomotive for your train. You know you'll get there, it is well and uniquely defined(although often messy to compute). Actually, as you point out, infinitessimal length is a problem in a general manifold and can't be given a meaning unless you think of the manifold as an embedding in a higher dimensional Euclidean space (there is no notion of "chord" approximating tangent), so we must instead look at how derivations act on scalar fields on the manifold to get to a notion of a vector field. | |
| Sep 10, 2015 at 1:08 | comment | added | Selene Routley | It sounds as though you have a good grasp of that is going on. I think perhaps what might have said that the machinery of tangent bundles, vector fields as sections thereof and then flows induced by the vector fields is what stands as a rigorous alternative to the Leibnitz conception. The implicit, seldom stated but always present, link between vector field and flow is the existence and uniqueness theorem (Picard-Lindelöf for $C^2$ manifold) in one direction, and the operation of differentiation (derivation) in the other. You don't need to imagine summing up "infinitessimal lengths" .... | |
| Sep 10, 2015 at 0:23 | history | edited | Qmechanic♦ |
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| Sep 10, 2015 at 0:07 | history | edited | user153582 |
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| Sep 10, 2015 at 0:02 | history | asked | user153582 | CC BY-SA 3.0 |