Tangent Vectors
The central idea of calculus is linear approximation. In order to make sense of calculus on manifolds, we need to introduce the tangent space to a manifold at a point, which we can think of as a sort of “linear model” for the manifold near the point. Moti
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Tangent Vectors
The central idea of calculus is linear approximation. This arises repeatedly in the study of calculus in Euclidean spaces, where, for example, a function of one variable can be approximated by its tangent line, a parametrized curve in Rn by its velocity vector, a surface in R3 by its tangent plane, or a map from Rn to Rm by its total derivative (see Appendix C). In order to make sense of calculus on manifolds, we need to introduce the tangent space to a manifold at a point, which we can think of as a sort of “linear model” for the manifold near the point. Because of the abstractness of the definition of a smooth manifold, this takes some work, which we carry out in this chapter. We begin by studying much more concrete objects: geometric tangent vectors in Rn , which can be visualized as “arrows” attached to points. Because the definition of smooth manifolds is built around the idea of identifying which functions are smooth, the property of a geometric tangent vector that is amenable to generalization is its action on smooth functions as a “directional derivative.” The key observation, which we prove in the first section of this chapter, is that the process of taking directional derivatives gives a natural one-to-one correspondence between geometric tangent vectors and linear maps from C 1 .Rn / to R satisfying the product rule. (Such maps are called derivations.) With this as motivation, we then define a tangent vector on a smooth manifold as a derivation of C 1 .M / at a point. In the second section of the chapter, we show how a smooth map between manifolds yields a linear map between tangent spaces, called the differential of the map, which generalizes the total derivative of a map between Euclidean spaces. This allows us to connect the abstract definition of tangent vectors to our concrete geometric picture by showing that any smooth coordinate chart .U; '/ gives a natural isomorphism from the space of tangent vectors to M at p to the space of tangent vectors to Rn at '.p/, which in turn is isomorphic to the space of geometric tangent vectors at '.p/. Thus, any smooth coordinate chart yields a basis for each tangent space. Using this isomorphism, we describe how to do concrete computations in such a basis. Based on these coordinate computations, we show how the union of all the tangent spaces at all points of a smooth manifold can be “glued together” to form a new manifold, called the tangent bundle of the original manifold. J.M. Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics 218, DOI 10.1007/978-1-4419-9982-5_3, © Springer Science+Business Media New York 2013
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Tangent Vectors
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Next we show how a smooth curve determines a tangent vector at each point, called its velocity, which can be regarded as the derivation of C 1 .M / that takes the derivative of each function along the curve. In the final two sections we discuss and compare several other approaches to defining tangent spaces, and give a brief overview of the terminology of category theory, which puts the tang
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