This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 ntuples ( 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...
View
Full
Document
This note was uploaded on 11/26/2011 for the course MAT 4821 taught by Professor Wong during the Spring '10 term at FSU.
 Spring '10
 Wong
 Calculus, Geometry

Click to edit the document details