The tangent space at a point p in an abstract manifold M can be described without the use of embeddings or coordinate charts. The elements of the tangent space are called tangent vectors, and the collection of tangent spaces forms the tangent bundle. One description is to put an equivalence relation on smooth paths through the point p. More precisely, consider all smooth maps f:I->M where I = (-1, 1) and f(0) = p. We say that two maps f and g are equivalent if they agree to first order. That is, in any coordinate chart around p, f'(0) = g'(0). If they are similar in one chart then they are similar in any other chart, by the chain rule. The notion of agreeing to first order depends on coordinate charts, but this cannot be completely eliminated since that is how manifolds are defined.
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