By the "algebraist's definition" of the tangent space of manifolds, can we say that the partial derivative $d/dx$ belongs to the the tangent space of $S^1$? It feels strange, but I can't see why it shouldn't be true.
Algebraist's definition of the tangent space of a manifold
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differential-topology
manifolds
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2I don't understand what you mean by the tangent space of a manifold. Do you mean the tangent space at a _point_, or do you mean the tangent _bundle_? – 2012-08-05