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Unformatted text preview: 1.3. TANGENT VECTORS AND MAPPINGS 15 Definition 1.3.1 A tangent vector at a point p M of a manifold M is a map that assigns to each coordinate chart ( U , x ) about p an ordered n-tuple ( X 1 , . . . , X n ) such that X i = n X j = 1 x i x j ( p ) X j . 1.3.2 Vectors as Di ff erential Operators Let f : M R be a real-valued function on M . Let p M be a point on M and X be a tangent vector at p . Let ( U , x ) be a coordinate chart about p . The (directional) derivative of f with respect to X at p (or along X , or in the direction of X ) is defined by X p ( f ) = D X ( f ) = n X j = 1 f x j ! ( p ) X j . Theorem 1.3.1 D X ( f ) does not depend on the local coordinate system. Proof : 1. Chain rule. The intrinsic properties are invariant under a change of coordinate system....
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