ONT Re: Differential Geometry for Engineers
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Note 9
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| 3. Tangent Spaces
|
| The previous chapter defines manifolds and gives several examples of them.
| This chapter considers a basic construction of one manifold from another.
| While the method of construction itself is of interest insofar as it
| illustrates general procedures of modern differential geometry,
| the particular result, the tangent space, is an object of
| great importance. It is by way of the tangent space
| that calculus can be done in general situations.
|
| To gain familiarity with the idea of a tangent space, it is worthwhile to
| spend some time with an example, that of the tangent space to the sphere.
| The information in the previous section concerning charts for the sphere
| allows charts to be constructed for this new space. The atlas resulting
| from the construction is examined in the light of the earlier definitions
| to see that this tangent space forms a manifold. The example is useful, too,
| for giving insight into such things as the dimensionality of a tangent space
| and the fact that its maps preserve its linear and differentiable structure.
| Part of the problem of constructing an atlas is that a map must be inverted
| and that its composition with another map be a diffeomorphism. Reducing our
| example from a sphere to a circle simplifies this calculation considerably.
|
| Next, preparatory to considering the general construction of a tangent space,
| the notion of equivalence classes of curves on a manifold, and their addition
| and scalar multiplication is explored. This study provides the guide to the
| constructions that follow, and to the confirmation that the tangent space is
| a manifold.
|
| The rest of the chapter is devoted to the tangent space in general.
| It is seen to be a manifold whose charts and chart maps are derived
| from those of the underlying manifold. It is seen to have vector
| space properties. Similar properties of maps between tangent
| manifolds are examined. The differentiating properties of
| these induced maps are noted.
|
| Doolin & Martin, DGFE, pages 23-24.
|
| Brian F. Doolin & Clyde F. Martin,
|'Introduction to Differential Geometry for Engineers',
| Marcel Dekker, New York, NY, 1990.
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