237q2s - 1. [5 marks] Suppose f : R 3 R is of class C 2 ( R...

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1. [5 marks] Suppose is of class and is defined by R R f 3 : ) ( 3 2 R C 3 2 : R R g . Compute ) 1 2 , 1 , 3 ( ) , ( 2 + + = y x y x y x g 2 2 y ω where ) , )( ( y x f g o = in terms of the derivatives of . To have the same notations in all papers, please f consider where . ) , , ( w v u f 1 2 , 1 , 3 2 + = + = = y w x v y x u = y u f y u + v f y v + w f y w = ) 3 ( u f + 2 w f 2 2 y = ) ( y y = 2 2 )[ 3 ( u f y u + u v f 2 y v + u w f 2 y w ] + 2[ w u f 2 [ y u + w v f 2 y v + 2 2 w f y w ] = 6 9 2 2 u f u w f 2 w u f 2 6+ 4 2 2 w f Since f is of class the mixed partials are equal we get 2 C 2 2 y = 2 2 9 u f w u f 2 12 + 4 2 2 w f 2. [5 marks] Let be defined by . Near which points R R F 2 : ) 1 ( ) , ( 2 + + = y x x y x F of the set } 0 ) , ( : ) , {( = = y x F y x S is S the graph of a function or 1 C ) ( x f y = ? Justify your answer. ) ( y g x = F is of class on 1 C 2 R and for 0 + + + = ] , ) 1 ( 2 [ 2 2 x x y x x F 0 x That is by the theorem S is the graph of a function 1 C ) ( x f y = or ) ( y g x = near all points where . 2 ) , ( R y x 0 x If , S is the graph of a function 0 = x 1 C ) ( y g x = only ( = 0 ) near all points except the point which is the point of intersection of the lines and ) ( y g 2 ) , 0 ( R y ) 1 , 0 ( 0 = x 0 1 = + + y x . At the point , S is neither a graph of ) 1 , 0 ( ) ( x f y = nor . ) ( y g x = 2
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3. (a) [2 marks] Under what assumptions on is locally a graph of a 3 2 : R R f f im smooth surface?
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This note was uploaded on 10/21/2010 for the course MATHEMATIC MAT237Y1 taught by Professor Romauldstanczak during the Fall '09 term at University of Toronto.

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237q2s - 1. [5 marks] Suppose f : R 3 R is of class C 2 ( R...

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