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