inverse[1] - MATH 23b, SPRING 2005 THEORETICAL LINEAR...

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MATH 23b, SPRING 2005 THEORETICAL LINEAR ALGEBRA AND MULTIVARIABLE CALCULUS The Inverse Function Theorem The Inverse Function Theorem. Let f : R n -→ R n be continuously differentiable on some open set containing a , and suppose det Jf ( a ) 6 = 0. Then there is some open set V containing a and an open W containing f ( a ) such that f : V W has a continuous inverse f - 1 : W V which is differentiable for all y W . Note: As matrices, J ( f - 1 )( y ) = [( Jf )( f - 1 ( y ))] - 1 . Lemma: Let A R n be an open rectangle, and suppose f : A -→ R n is continuously differentiable. If there is some M > 0 such that ± ± ± ± ∂f i ∂x j ( x ) ± ± ± ± M, x A , then || f ( y ) - f ( z ) || ≤ n 2 · M · || y - z || , y , z A . Proof: We write f i ( y ) - f i ( z ) = f i ( y 1 , . . . , y n ) - f i ( z 1 , . . . , z n ) = n X j =1 [ f ( y 1 , . . . , y j , z j +1 , . . . , z n ) - f ( y 1 , . . . , y j - 1 , z j , z j +1 , . . . , z n )] = n X j =1 ∂f i ∂x j ( x ij )( y j - z j ) for some x ij = ( y 1 , . . . , y j - 1 , c j , z j +1 , . . . , z n ) where, for each j = 1 , . . . , n , we have c j is in the interval ( y j , z j ), by the single-variable Mean Value Theorem. Then || f ( y ) - f ( z ) || ≤ n X i =1 || f i ( y ) - f i ( z ) || = n X i =1 n X j =1 ± ± ± ± ∂f i ∂x j ( x ij ) ± ± ± ± · | y j - z j | n X i =1 n X j =1 M · || y - z || = n 2 · M · || y - z || ±
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Proof of the Inverse Function Theorem: (borrowed principally from Spivak’s Calculus on Manifolds ) Let L = Jf ( a ). Then det( L ) 6 = 0, and so L - 1 exists. Consider the com- posite function L - 1 f : R n R n . Then: J ( L - 1 f )( a ) = J ( L - 1 )( f ( a )) Jf
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This note was uploaded on 02/27/2012 for the course MATH 250A taught by Professor N.r.wallach during the Fall '10 term at Colorado.

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inverse[1] - MATH 23b, SPRING 2005 THEORETICAL LINEAR...

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