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I'm having trouble understanding why is tangent space at point p on a smooth manifold, not embedded in any ambient euclidean sapce, has to be defined as, for example, set of all directional derivatives at that point.

To my understanding, the goal of defining tangent space is to provide linear approximation of a manifold near certain point. Why not to just say that tangent space of n-manifold at point p is R^n? In the end, the set of all directional derivatives is isomorphic to it anyway. And after all, manifolds key characteristic is that it is localy similar to R^n, so why to not use it?

But there has to be some reason it is defined the way it is, since everybody is using it, so I just wonder what I'm missing here...

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# Definition of tangent space on smooth manifolds

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