 Such a definition can be formulated in terms of equivalence classes of smooth functions on "formula_2".Informally, we will say that two smooth functions "f" and "g" are equivalent at a point "formula_1" if they have the same first-order behavior near "formula_1", analogous to their linear Taylor polynomials; two functions "f" and "g" have the same first order behavior near "formula_1" if and only if the derivative of the function "f"-"g" vanishes at "formula_1".The cotangent space will then consist of all the possible first-order behaviors of a function near "formula_1".