Engineering Calculus Notes 373

Engineering Calculus Notes 373 - of Q . If Δ 2 > 0,...

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3.8. LOCAL EXTREMA 361 Quadratic Forms in R 2 To take advantage of Proposition 3.8.3 we need a way to decide whether or not a given quadratic form Q is positive deFnite, negative deFnite, or neither. In the planar case, there is an easy and direct way to decide this. If we write Q ( x 1 ,x 2 ) in the form Q ( x 1 ,x 2 ) = ax 2 1 + 2 bx 1 x 2 + cx 2 2 then we can factor out “ a ” from the Frst two terms and complete the square: Q ( x 1 ,x 2 ) = a p ( x 1 + b a x 2 ) 2 b 2 a 2 P + cx 2 2 = a p x 1 + b a x 2 P 2 + p c b 2 a Pp x 2 P 2 . Thus, Q is deFnite provided the two coe±cients in the last line have the same sign, or equivalently, if their product is positive: 16 ( a ) p c b 2 a P = ac b 2 > 0 . The quantity in this inequality will be denoted Δ 2 ; it can be written as the determinant of the matrix [ Q ] = b a b b c B which is the matrix representative
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Unformatted text preview: of Q . If Δ 2 > 0, then Q is defnite , which is to say the two coe±cients in the expression for Q ( x 1 ,x 2 ) have the same sign; to tell whether it is positive deFnite or negative deFnite, we need to decide if this sign is positive or negative, and this is most easily seen by looking at the sign of a , which we will denote Δ 1 . The signiFcance of this notation will become clear later. With this notation, we have Proposition 3.8.5. A quadratic Form Q ( x 1 ,x 2 ) = ax 2 1 + 2 bx 1 x 2 + cx 2 2 16 Note that if either coeFcient is zero, then there is a whole line along which Q = 0, so it is not de±nite....
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

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