31. This yields the correspondence between the tangent space defined via derivations and the tangent space defined via the cotangent space. 32. A smooth function has ( at every point ) the differential, a linear functional on the tangent space . 33. (Note : The right hand side of the above may not lie in the tangent space to the manifold. 34. It is one half of the value obtained without regard for the tangent space orientation, but with opposite sign. 35. This tangent space generates a ( unit ) pseudoscalar which is a function of the points of the vector manifold. 36. Likewise, cotangent space is a contravariant functor, essentially the composition of the tangent space with the dual space above. 37. For example, the tangent bundle consists of the collection of tangent spaces parametrized by the points of a differentiable manifold. 38. Therefore, all tangent vectors in a point p span a linear space, called the tangent space at point p. 39. This means that the tangent spaces at each point are canonically identified with each other and with the vector space itself. 40. Pick a point \ mu \ in S ( X ) and consider the tangent space T _ \ mu S.
