Editing Cotangent space (section)

===Alternative definition===

In some cases, one might like to have a direct definition of the cotangent space without reference to the tangent space. Such a definition can be formulated in terms of [[equivalence class]]es of smooth functions on <math>\mathcal M</math>. Informally, we will say that two smooth functions ''f'' and ''g'' are equivalent at a point <math>x</math> if they have the same first-order behavior near <math>x</math>, analogous to their linear Taylor polynomials; two functions ''f'' and ''g'' have the same first order behavior near <math>x</math> if and only if the derivative of the function ''f'' − ''g'' vanishes at <math>x</math>. The cotangent space will then consist of all the possible first-order behaviors of a function near <math>x</math>.

Let <math>\mathcal M</math> be a smooth manifold and let <math>x</math> be a point in <math>\mathcal M</math>. Let <math>I_x</math>be the [[ideal (ring theory)|ideal]] of all functions in <math>C^\infty\! (\mathcal M)</math> vanishing at <math>x</math>, and let <math>I_x^2</math> be the set of functions of the form <math display="inline">\sum_i f_i g_i</math>, where <math>f_i, g_i \in I_x</math>. Then <math>I_x</math> and <math>I_x^2</math> are both real vector spaces and the cotangent space can be defined as the [[Quotient space (linear algebra)|quotient space]] <math>T^*_x\!\mathcal M = I_x/I^2_x</math> by showing that the two spaces are [[isomorphism|isomorphic]] to each other.

This formulation is analogous to the construction of the cotangent space to define the [[Zariski tangent space]] in algebraic geometry. The construction also generalizes to [[locally ringed space]]s.