Engineering Calculus Notes 408

Engineering Calculus Notes 408 - 396 CHAPTER 3. REAL-VALUED...

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Unformatted text preview: 396 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION and x+y ξ1 = √ 2 −x + y ξ2 = √ 2 while 1 √ (−6) + 2 √ = −2 2 1 √ 2 α= 1 −√ 2 √ = 4 2. 1 √ 2 (−6) + β= (2) (2) This leads to the equation (in ξ1 and ξ2 ) 1 2 ξ1 − √ 2 2 − 2 ξ2 − √ 2 2 = 4 + 1 − 4 = 1. We recognize this as a hyperbola with asymptotes 1 ξ2 = ξ1 + √ 2 3 ξ2 = − ξ1 + √ 2 or, in terms of x and y , x=− 1 2 3 y= . 2 (See Figure 3.30.) As a second example, consider the curve given by x2 − 2xy + y 2 + 3x − 5y + 5 = 0. The quadratic form Q(x, y ) = x2 − 2xy + y 2 ...
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