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Added to tangent bundle the discussion in the context of synthetic differential geometry.
In that context I also restructured a bit and expanded the introduction slightly.
(…almost 8 years later…)
I have started to fill in at Definitions in ordinary differential geometry – Geometric definition details of the classical construction.
Continued to spell out traditional elementary detail at Geometric definition. In particular more of a proof now that the tangent bundle of a differentiable manifold is itself a manifold.
I have re-arranged the sections at tangent bundle:
made all the various sections that existed subsections of the “Definition”-section (because all discuss alternative definitions)
merged what used to be three sections for “algebraic”, “geometric” and “physics” definition (this was not my idea) into a single section “Traditional definition”
(the “algebraic definition” via derivations is one of vector fields, not of the tangent bundle itself, hence hardly an alternative definition; and the “physics” definition via gluing is really the only definition there is: even if one describes the topology on $T X$ in a way that it does not explicitly mention the gluing construction, it is the corresponding quotient topology and the gluing construction is arguably the most transparent way to understand that topology )
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