 Let "M" be a smooth manifold and let "x" be a point in "formula_2".Let formula_26be the ideal of all functions in formula_27 vanishing at "formula_1", and let formula_29 be the set of functions of the form formula_30, where formula_31.Then "formula_26" and formula_29 are real vector spaces and the cotangent space is defined as the quotient space formula_34.