# hw8 - Math 425 Spring 2011 Jerry L Kazdan Problem Set 8 DUE...

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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 + γ for some real constant γ . 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 = α . 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 = πα √ 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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hw8 - Math 425 Spring 2011 Jerry L Kazdan Problem Set 8 DUE...

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