the cotangent bundle

If a tangent vector at p is a velocity — a direction you can move — then a cotangent vector at p is a linear gauge that reads off a number from each velocity, for instance 'how fast is this function increasing as I move in that direction'. The cotangent bundle T*M collects all such gauges at all points into one smooth manifold, exactly dual to the tangent bundle.

At each p the cotangent space T*_p M is the dual vector space (T_p M)*, the linear functionals on T_p M. The prototype cotangent vector is the differential df_p of a smooth function: it sends a tangent vector v to the directional derivative v(f). In a chart with coordinates x^1, ..., x^n, the differentials dx^1_p, ..., dx^n_p form the basis of T*_p M dual to the coordinate basis of T_p M, so a general covector is a sum of a_i times dx^i. The bundle T*M is assembled just like TM, with the crucial difference that under a change of coordinates the components a_i transform by the inverse-transpose of the Jacobian — they are covariant, not contravariant — which is exactly what makes the pairing of a covector with a vector coordinate-independent.

Sections of T*M are 1-forms (covector fields), the first rung of the de Rham complex, though differential forms proper belong to a later field. T*M carries one piece of structure that TM lacks: a canonical 1-form (the tautological or Liouville form) and hence a canonical symplectic structure, making every cotangent bundle the natural phase space of classical mechanics. A frequent confusion is to treat 'the gradient' as a vector — without a metric the derivative of a function is a covector df living in T*M, and only after choosing a metric do you raise its index to get a gradient vector field in TM.