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LectureNotes0U - Math 4441 1 Differential Geometry Fall...

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Unformatted text preview: Math 4441 Aug 21, 2007 1 Differential Geometry Fall 2007, Georgia Tech Lecture Notes 0 Basics of Euclidean Geometry By R we shall always mean the set of real numbers. The set of all n-tuples of real numbers R n := { ( p 1 , . . . , p n ) | p i ∈ R } is called the Euclidean n-space . So we have p ∈ R n ⇐⇒ p = ( p 1 , . . . , p n ) , p i ∈ R . Let p and q be a pair of points (or vectors) in R n . We define p + q := ( p 1 + q 1 , . . . , p n + q n ). Further, for any scalar r ∈ R , we define rp := ( rp 1 , . . . , rp n ). It is easy to show that the operations of addition and scalar multiplication that we have defined turn R n into a vector space over the field of real num- bers. Next we define the standard inner product on R n by h p, q i = p 1 q 1 + . . . + p n q n . Note that the mapping h· , ·i : R n × R n → R is linear in each variable and is symmetric. The standard inner product induces a norm on R n defined by k p k := h p, p i 1 2 . If p ∈ R , we usually write | p | instead of k p k . The first nontrivial fact in Euclidean geometry, is the following important result which had numerous applications: Theorem 1. (The Cauchy-Schwartz inequality) For all p and q in R n |h p, q i| 6 k p kk q k ....
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This note was uploaded on 08/25/2011 for the course MATH 6456 taught by Professor Staff during the Spring '08 term at Georgia Tech.

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LectureNotes0U - Math 4441 1 Differential Geometry Fall...

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