differential geometry w notes from teacher_Part_14

differential geometry w notes from teacher_Part_14 - 2.1...

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2.1. COVECTORS AND RIEMANNIAN METRIC 27 1. 2.1.2 Di ff erential 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 di ff erential d f T * p M of f at p is the linear functional d f : T p R defined by d f ( v ) = v p ( f ) . In local coordinates x j the di ff erential is defined by d f ( v ) = n j = 1 v j ( x ) f x j In particular, d f x j = f x j . Thus dx i x j = δ i j . and dx i ( v ) = v i . The di ff erentials { dx i } form a basis for the cotangent space T * p M called the coordinate basis . Therefore, every linear functional has the form α = n j = 1 a j dx j . di ff geom.tex; April 12, 2006; 17:59; p. 30
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28 CHAPTER 2. TENSORS That is why, the linear functionals are also called di ff erential forms , or 1 -forms , or covectors , or covariant vectors. Definition 2.1.5 A covector field α is a di ff erentiable 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 di
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