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several.pdf - REAL ANALYSIS FUNCTIONS OF SEVERAL VARIABLES...

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REAL ANALYSIS FUNCTIONS OF SEVERAL VARIABLES We denote ( x 1 . . . x 2 ) by X and f ( x 1 . . . x n ) by f ( X ). We may think of X as a vector or a point. If A = ( a 1 . . . a n ) B = ( b 1 . . . B - n ) then A - B = ( a 1 - b 1 . . . a n - b n ) A + B = ( a 1 + b 1 . . . a n + b n ) A.B = a 1 b 1 + . . . + a n b n - a scalar || A || = q a 2 1 + . . . + a 2 n norm of A || A - B || = q (1 a - a b ) 2 + . . . + ( a n - b n ) 2 is the distance AB | A.B | ≤ || A || || B || Cauchy’s inequality. Suppose we have m functions of n variables (1) f ( X ) , (2) f ( X ) . . . ( m ) F ( X ). We shall denote by the vector function F ( X ) = ( (1) f ( x 1 . . . x n ) , . . . ( m ) f ( x 1 . . . x n )) Theorem 1 If f ( X ) g ( X ) are continuous at A relative to S then so are f ( X ) ± g ( X ) , f ( X ) g ( X ) and, if g ( A ) 6 = 0 , f ( X ) g ( X ) . Theorem 2 Suppose that the components (1) f ( X ) . . . ( m ) f ( X ) of the vector function F ( X ) are continuous at A relative to S . Let B = F ( A ) and let T be the set of all points F ( X ) with X in S . Then if g ( Y ) = g ( y 1 . . . y m ) is continuous at B relative to T , it follows that g ( F ( x )) is continuous at A relative to S . Differentiability f ( X ) is differentiable at X + A ⇔ ∃ a vector G | f ( X ) - f ( A ) - G ( X - A ) || X - A || 0 as X A . If f is differentiable then ∂f ∂x 1 . . . ∂f ∂x n all exist and the vector G is ∂f ∂x 1 . . . ∂f ∂x n · . We call this ( grad f ( X )) X = A or ( f ) X = A . Thus f ( X ) is differentiable δf -∇ fδX || δX || 0 as δX 0. Theorem 3 If ∂f ∂x 1 , . . . ∂f ∂f ∂x n are continuous at X = A , then f ( X ) is differ- entiable at X = A . 1
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Proof Suppose H 6 = 0 and || H || is sufficiently small. Consider 1 || H || fl fl fl fl fl n X r =1 { f ( a 1 + h 1 . . . , a r + h r , a r +1 . . . a n ) - f ( a 1 + h 1 . . . a r - 1 + h r - 1 a r . . . a n ) - h r f r ( A ) }| (This is 1 || H || { f ( A + h ) - f ( A ) - A f } ) 1 || H || fl fl fl fl fl n X r =1 [ { f r ( a 1 + h 1 . . . a r - 1 + h r - 1 a r + θ r f r 1 a r +1 - a n ) - f r ( A ) } h r ] fl fl fl fl fl 0 < θ r < 1 Let V be the vector with components f r ( a 1 + h 1 , . . . a r θ f h r , a r +1 . . . a n ) - f r A ( r = 1 , 2 , . . . n )
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