differential geometry w notes from teacher_Part_1

differential geometry w notes from teacher_Part_1 - Chapter...

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Unformatted text preview: 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 ) = v t n X k = 1 ( x k- y k ) 2 . The open ball of radius centered at x is the set of points defined by B ( x ) = { x R n | d ( x , x ) < } . A neighborhood of a point x is the set of points that contain an open ball around it. 1 2 CHAPTER 1. MANIFOLDS Let x R n be a fixed point with Cartesian coordinates x i , i = 1 , . . . , n , in the Euclidean space and...
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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_1 - Chapter...

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