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differential geometry w notes from teacher_Part_1

# differential geometry w notes from teacher_Part_1 - Chapter...

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Chapter 1 Manifolds 1.1 Submanifolds of Euclidean Space Idea: Manifold is a general space that looks locally like a Euclidean space of the same dimension. This allows to develop the di ff erential and integral calculus. Let n N be a positive integer. The Euclidean space R n is a set of points x described by ordered n -tuples ( x 1 , . . . , x n ) or real numbers. The numbers x i R , i = 1 , . . . , n , are called the Cartesian coordinates of the point x . The integer n is the dimension of the Euclidean space. The distance between two points of the Euclidean space is defined by d ( x , y ) = n k = 1 ( x k - y k ) 2 . The open ball of radius ε centered at x 0 is the set of points defined by B ε ( x 0 ) = { x R n | d ( x , x 0 ) < ε } . A neighborhood of a point x 0 is the set of points that contain an open ball around it. 1

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2 CHAPTER 1. MANIFOLDS Let x 0 R n be a fixed point with Cartesian coordinates x i , i = 1 , . . . , n , in the Euclidean space and S R n be a neighborhood of x 0 . An injective (one-to-one) map f : S R n defined by y i = f i ( x 1 , . . . , x n ) , i = 1 , . . . , n , where f i ( x ) are smooth functions, is called a coordinate system in S . The map f is injective if for any point x in S det f i x j 0 . 1.1.1 Submanifolds of
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differential geometry w notes from teacher_Part_1 - Chapter...

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