lecture24

# lecture24 - Lecture 24 We review the pullback operation...

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Lecture 24 We review the pullback operation from last lecture. Let U be open in R m and let V be open in R n . Let f : U V be a C map, and let f ( p ) = q . From the map df p : T p R m T q R n , (4.212) we obtain the pullback map ( df p ) : Λ k ( T q ) Λ k ( T p ) ω Ω k ( V ) f ω Ω k ( U ) . (4.213) We define, f ω p = ( df p ) ω q , when ω q Λ k ( T q ). The pullback operation has some useful properties: 1. If ω i Ω k i ( V ) , i = 1 , 2, then f ( ω 1 ω 2 ) = f ω 1 f ω 2 . (4.214) 2. If ω Ω k ( V ), then df ω = f dω. (4.215) We prove some other useful properties of the pullback operation. Claim. For all ω Ω k ( W ) , f g ω = ( g f ) ω. (4.216) Proof. Let f ( p ) = q and g ( q ) = w . We have the pullback maps ( df p ) k ( T q ) Λ k ( T p ) (4.217) ( dg q ) k ( T w ) Λ k ( T q ) (4.218) ( g f ) k ( T w ) Λ k ( T p ) . (4.219) The chain rule says that ( dg f ) p = ( dg ) q ( df ) p , (4.220) so d ( g f ) p = ( df p ) ( dg q ) . (4.221) Let U, V be open sets in R n , and let f : U V be a C map. pullback operation on n -forms ω Ω n ( V ). Let f (0) = q . Then We consider the ( dx i ) p , i = 1 , . . . , n, is a basis of T p , and (4.222) ( dx i ) q , i = 1 , . . . , n, is a basis of T q . (4.223) 1

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Using f i = x i f , ( df p ) ( dx i ) q = ( df i ) p ∂f i (4.224) = ( p )( dx j ) p .
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