Engineering Calculus Notes 374

Engineering Calculus Notes 374 - 362 CHAPTER 3. REAL-VALUED...

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Unformatted text preview: 362 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION is definite only if ∆2 := ac − b2 > 0; it is positive definite if in addition ∆1 := a > 0 and negative definite if ∆1 < 0. → If ∆2 < 0, then Q(− ) takes both (strictly) positive and (strictly) negative x values. Let us see what this tells us about the forms we introduced at the beginning of this section: 1. Q(x, y ) = x2 + y 2 has A = [Q] = 10 01 so ∆1 = 1 > 0 ∆2 = 1 > 0 and Q is positive definite. 2. Q(x, y ) = −x2 − 2y 2 has A = [Q] = −1 0 0 −2 so ∆1 = − 1 < 0 ∆2 = 2 > 0 and Q is negative definite. ...
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