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237q3s - 1(a[7 Find the constant k 0 such that(r k = 0...

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1. (a) [7] Find the constant such that , where 0 k 0 ) ( = k r r = r , r . 0 = ) , , ( z y x = = = ) ( ) ( ) ( 2 1 r r k k k r k r r k r 2 3 2 ) 1 ( ] ) 2 ( ) 3 ( [ + = + k k k r k k r r k r k r r = 0 for . Here the identity 5.28 from the text was used to shorten calculations. 0 1 = k (b) [8] Evaluate , where C d curl x F ) ( ) , , ( ) , , ( z yz y xz z y x + = F and C is the segment of the curve of intersection of the plane x y = and the paraboloid from (0,0,0) to (1,1,2). 2 2 y x z + = Direct calculation shows that ) 1 , , ( = x y curl F . Setting x = t , we get y = t and , 2 2 t z = hence the parametrization of C is g and we get 1 0 ), 2 , , ( ) ( 2 = t t t t t . 1 | ) 2 ( ) 4 , 1 , 1 ( ) 1 , , ( ) ( 1 0 1 0 2 1 0 = = = = t dt t dt t t t d curl C x F 2

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2. (a) [3] Is the set a regular region in } 4 0 : ) , {( 2 2 + < = y x y x S 2 R ? Justify your answer. NO. S is regular if it is compact and it is the closure of its interior. S is not compact since it is not closed: S k k k = ) 1 , 1 ( a , but S k k = ) 0 , 0 ( lim a . Obviously it is also not a closure of . int S (b) [10] Evaluate dy x y y x
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