dif9-page4 - 44444...

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Lemma 2. If two functions f,g e M coincide on a neighborhood U of p e M ------------------------- and D is a derivation at p then D(f) = D(g). Proof. ------------------------- n Sublemma. If x e R and B( x , e ) is a fixed open ball about it, then there ---------------------------------------- n exists a smooth function h: R ----------L R such that h( y ) is equal to 1 if y e B( x , e /2), positive if y e B( x , e ) and zero if y m B( x , e ). Define the function h of the real variable t by the formula 0 2 -1 & -(1-t ) e if t e (-1,1) h (t) = { 0 7 0 otherwise. It is a good exercise to prove that h is a smooth function on R . Set 0 h ( y ) := h (4 N y N / e ), let c denote the characteristic function of the ball 1 0 B( x ,3 e /4), and define the function h as follows 2 i h ( y ) := c ( z )h ( y - z )d z . 2 j 1 n R If we put h( y ) = h ( y )/h ( x ) then we get a desirable function. 2 2 Now let us prove the lemma. Using the construction above we can define a smooth function h on M which is zero outside U and such that h(p) = 1. In this case h(f-g) is the constant 0 function on M. Thus we have 0 = D( 0 ) = D(h(f-g)) = D(h) (f(p)-g(p)) + h(p) D(f-g) = D(f) - D(g).
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Unformatted text preview: 44444 Remarks.-----------------------------------* i) The sublemma shows that the mapping i :M----------L F (M) above is indeed an inclusion. If p $ q are distinct points of M, then there is a smooth function h on M such that [ i (p)](h) = h(p) = 1 $ [ i (q)](h) = h(q) = 0. ii) We can extend a derivation D at a point p on functions f defined only in a neighborhood U of p by taking a smooth function h on M such that h is zero outside U and constant 1 in a neighborhood of p and putting D(f) := ~ D(f), where ( ~ f(x)h(x) for x e U f(x) = { for x m U. 9 By lemma 2 this extension of D is correctly defined. n Lemma 3. Let f:B----------L R be a smooth function defined on an open ball B C R-------------------------around the origin. Then there exist smooth functions g 1 < i < n on B such that i n f( x ) = f( ) + S x g ( x ) for x = (x ,. ..,x ) e B i i 1 n i=1 and d f g ( ) =--------------------( ). i d x i 4...
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This note was uploaded on 01/08/2012 for the course MATH 203 taught by Professor Sedita during the Spring '09 term at SUNY Stony Brook.

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