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differential geometry w notes from teacher_Part_8

# differential geometry w notes from teacher_Part_8 - 15 1.3...

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1.3. TANGENT VECTORS AND MAPPINGS 15 Definition 1.3.1 A tangent vector at a point p 0 M of a manifold M is a map that assigns to each coordinate chart ( U α , x α ) about p 0 an ordered n-tuple ( X 1 α , . . . , X n α ) such that X i β = n j = 1 x i β x j α ( p 0 ) X j α . 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 α , x α ) 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 j = 1 f x j α ( p ) X j α . 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. They should not depend on the choice of the local chart. There is a one-to-one correspondence between tangent vectors to M at p and first-order di ff

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differential geometry w notes from teacher_Part_8 - 15 1.3...

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