The Cotangent Bundle
In this chapter we introduce a construction that is not typically seen in elementary calculus: tangent covectors, which are linear functionals on a tangent space to a smooth manifold M. The space of all covectors at p∈M is a vector space called the cotang
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The Cotangent Bundle
In this chapter we introduce a construction that is not typically seen in elementary calculus: tangent covectors, which are linear functionals on the tangent space at a point p 2 M . The space of all covectors at p is a vector space called the cotangent space at p; in linear-algebraic terms, it is the dual space to Tp M . The union of all cotangent spaces at all points of M is a vector bundle called the cotangent bundle. Whereas tangent vectors give us a coordinate-free interpretation of derivatives of curves, it turns out that derivatives of real-valued functions on a manifold are most naturally interpreted as tangent covectors. Thus we define the differential of a realvalued function as a covector field (a smooth section of the cotangent bundle); it is a coordinate-independent analogue of the gradient. We then explore the behavior of covector fields under smooth maps, and show that covector fields on the codomain of a smooth map always pull back to covector fields on the domain. In the second half of the chapter we introduce line integrals of covector fields, which are the natural generalization of the line integrals of elementary calculus. Then we explore the relationships among three closely related types of covector fields: exact (those that are the differentials of functions), conservative (those whose line integrals around closed curves are zero), and closed (those that satisfy a certain differential equation in coordinates). This leads to a far-reaching generalization of the fundamental theorem of calculus to line integrals on manifolds.
Covectors Let V be a finite-dimensional vector space. (As usual, all of our vector spaces are assumed to be real.) We define a covector on V to be a real-valued linear functional on V , that is, a linear map ! W V ! R. The space of all covectors on V is itself a real vector space under the obvious operations of pointwise addition and scalar multiplication. It is denoted by V and called the dual space of V . The next proposition expresses the most important fact about V in the finitedimensional case. Recall from Exercise B.13 that a linear map is uniquely determined by specifying its values on the elements of any basis. J.M. Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics 218, 272 DOI 10.1007/978-1-4419-9982-5_11, © Springer Science+Business Media New York 2013
Covectors
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Proposition 11.1. Let V be a finite-dimensional vector space. Given any basis .E1 ; : : : ; En / for V , let "1 ; : : : ; "n 2 V be the covectors defined by "i .Ej / D ıji ; where ıji is the Kronecker delta symbol defined by (4.4). Then "1 ; : : : ; "n is a basis for V , called the dual basis to .E j /. Therefore, dim V D dim V . I Exercise 11.2. Prove Proposition 11.1. n For example, we can apply this to the standard basis .e1 ; : : : ; en / for R . The dual 1 n basis is denoted by e ; : : : ; e (note the upper indices), and is called the standard dual basis. These basis covectors are the linear functionals on Rn given by e i .v/ D e i
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