Why are the tangent vectors of smooth manifolds defined as mappings [itex]C^{\infty}(p)\to\mathbb{R}[/itex] that have the similar properties as derivations?(adsbygoogle = window.adsbygoogle || []).push({});

If a manifold is defined as a subset of some larger euclidean space, then the tangent spaces are simply affine subspaces of the larger space, but if the manifold is instead defined without the underlying larger euclidean space, then I don't understand what the tangent spaces even should be like.

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# Why are tangent vectors like this?

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