Engineering Calculus Notes 398

# Engineering Calculus Notes 398 - Q = 1 − 1 − 1 and it...

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386 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION from which we easily get the following test for a quadratic form in three variables to be negative deFnite: Corollary 3.9.7 (Determinant Test for Negative DeFnite ±orms in R 3 ) . The quadratic form Q in three variables is negative deFnite precisely if its matrix representative A = [ Q ] satisFes ( 1) k Δ k > 0 for k = 1 , 2 , 3 where Δ k are the determinants given in Proposition 3.9.6 . Let us see how this test works on the examples studied in detail earlier in this section. 1. The form Q ( x,y,z ) = x 2 y 2 z 2 has matrix representative
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Unformatted text preview: [ Q ] = 1 − 1 − 1 and it is easy to calculate that Δ 1 = det[ Q ] = (1)( − 1)( − 1) = 1 > ...which, so far, tells us that the form is not negative deFnite. .. Δ 2 = (1)( − 1) = − 1 < so that Q is also not positive deFnite. There is no further information to be gained from calculating Δ 1 = 1 . 2. The form Q ( x,y,z ) = 2 xy + 2 xz + 2 yz has matrix representative [ Q ] = 0 1 1 1 0 1 1 1 0...
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## This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Spring '08 term at University of Florida.

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