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: FORMS AND INTEGRATION I Differential forms: definitions Part I: Linear Theory Let V R n be a linear space: we avoid the symbol R n since the latter implicitly implies some coordinates. Definition. An exterior kform on V is a map : V V  {z } k times R , ( v 1 ,...,v k ) 7 ( v 1 ,...,v k ) , which is: linear in each argument, and antisymmetric: if S k is a permutation on k symbols, and   = 1 its parity, then ( v (1) ,...,v ( k ) ) = ( 1)   ( v 1 ,...,v k ) . The space of all kforms on V is denoted by k ( V * ): it is a linear space over R . / Example. Linear forms are 1forms: 1 ( V * ) = V * . Example. If dim V = k and a coordinate system in V is chosen, and v j = ( v j 1 ,...,v jk ), then ( v 1 ,...,v k ) = det fl fl fl fl fl fl v 11 ... v k 1 . . . . . . . . . v 1 k ... v kk fl fl fl fl fl fl is a kform. We denote it by det x , x explicitly indicating the coordinate system. Problem 1. Prove that for any u,v R 3 the two formulas, 2 = det x ( u, , ) , 1 = det x ( u,v, ) define 2 and 1forms respectively. In any coordinate system ( x 1 ,...,x n ) on V R n a kform can be associated with a tuple of reals: if : { 1 ,...,k } { 1 ,...,n } is an index map , and ( e 1 ,..., e n ) a basis in V , then we define a = ( e (1) ,..., e ( k ) ) and consider the collection { a } with ranging over all possible index maps.possible index maps....
View
Full
Document
 Spring '08
 PROTSAK
 Geometry

Click to edit the document details