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Unformatted text preview: Math 425, Spring 2011 Jerry L. Kazdan Problem Set 8 DUE: Thursday March 31 [ Late papers will be accepted until 1:00 PM Friday ]. 1. a) Let A be an n n invertible real symmetric matrix, b R n and c R . For x R n consider the quadratic polynomial Q ( x ) = h x , Ax i + h b , x i + c . Show that by a translation by some vector v R n , so x = y + v in the new y variable the polynomial has the form Q ( y ) = h y , Ay i + for some real constant . HINT: Prove and use that for any vectors y and v we have h Ay , v i = h Av , y i . [This generalizes completing the square from high school algebra.] b) Let x , y R . Compute ZZ R 2 e- ( 2 x 2- 2 xy + 3 y 2 + x- 2 y- 3 ) dxdy c) Let h ( t ) be a given function and say you know that R h ( t ) dt = . If C be a positive definite real (symmetric) 2 2 matrix and x R 2 . Show that ZZ R 2 h ( h x , Cx i ) dA = det C and use this to compute ZZ R 2 dxdy ( 1 + x 2 + 2 xy + 5 y 2 ) 2 , where x , y R...
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