differential geometry w notes from teacher_Part_14

differential geometry w notes from teacher_Part_14 - 2.1....

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Unformatted text preview: 2.1. COVECTORS AND RIEMANNIAN METRIC 27 1. 2.1.2 • Differential of a Function Definition 2.1.3 Let M be a manifold and p ∈ M be a point in M. The ∗ space T p M dual to the tangent space T p M at p is called the cotangent space. Definition 2.1.4 Let M be a manifold and f : M → R be a real valued smooth function on M. Let p ∈ M be a point in M. The differential ∗ d f ∈ T p M of f at p is the linear functional • d f : Tp → R defined by d f (v) = v p ( f ) . • In local coordinates x j the differential is defined by n d f (v) = v j ( x) j=1 In particular, ∂f ∂x j ∂ ∂f = j. ∂x j ∂x df Thus dxi ∂ = δij . j ∂x and dxi (v) = vi . ∗ • The differentials {dxi } form a basis for the cotangent space T p M called the coordinate basis. • Therefore, every linear functional has the form n α= a j dx j . j =1 diffgeom.tex; April 12, 2006; 17:59; p. 30 28 CHAPTER 2. TENSORS • That is why, the linear functionals are also called differential forms, or 1-forms, or covectors, or covariant vectors. • Definition 2.1.5 A covector field α is a differentiable assignment of ∗ a covector α p ∈ T p M to each point p of a manifold. This means that the components of the covector field are differentiable functions of local coordinates. • Therefore, a covector field has the form n α= a j ( x)dx j j =1 j j • Under a change of local coordinates xα = xα ( xβ ) the differentials transform according to n j ∂ xα i j dxα = dx . i ∂ xβ β i=1 • Therefore, the components of a covector transform as n aα = i j =1 2.1.3 j ∂ xα β a. i ∂ xβ j Inner Product • Let E be a n-dimensional real vector space. • The inner product (or scalar product) on E is a symmetric bilinear nondegenerate functional on E × E , that is, it is a map , : E × E → R such that 1. ∀v, w ∈ E , v, w = w, v 2. ∀v, w, u ∈ E , ∀a, b ∈ R av + bu, w = a v, w + b u, w 3. ∀v ∈ E , if v, w = 0, ∀w ∈ E , then v = 0 . diffgeom.tex; April 12, 2006; 17:59; p. 31 ...
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This note was uploaded on 11/26/2011 for the course MAT 4821 taught by Professor Wong during the Spring '10 term at FSU.

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differential geometry w notes from teacher_Part_14 - 2.1....

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