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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.

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    I 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

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