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Engineering Calculus Notes 406

Engineering Calculus Notes 406 - ΞΎ 1 = ββ u 1 Β ββ...

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394 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION By Remark 3.9.4 , there is an orthonormal basis for R 2 consisting of two unit eigenvectors −→ u 1 and −→ u 2 (with eigenvalues λ 1 , λ 2 ) for A . Note that the negative of an eigenvector is also an eigenvector, so we can assume that −→ u 2 is the result of rotating −→ u 1 counterclockwise by a right angle. Thus we can write −→ u 1 = ( c,s ) −→ u 2 = ( s,c ) where c = cos θ s = sin θ ( θ is the angle between −→ u 1 and the positive x -axis). These vectors define a rectangular coordinate system (with coordinate axes rotated counterclockwise from the standard axes) in which the point with standard coordinates ( x,y ) has coordinates in the new system
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Unformatted text preview: ΞΎ 1 = ββ u 1 Β· ββ x = cx + sy ΞΎ 2 = ββ u 2 Β· ββ x = β sx + cy . You should check that these equations can by solved for x and y in terms of ΞΎ 1 and ΞΎ 2 : x = cΞΎ 1 β sΞΎ 2 y = sΞΎ 1 + cΞΎ 2 so that p ( x,y ) can be rewritten as p ( x,y ) = Q ( x,y ) + Dx + Ey = Ξ» 1 ΞΎ 2 1 + Ξ» 2 ΞΎ 2 2 + Ξ±ΞΎ 1 + Ξ²ΞΎ 2 where Ξ± = cD + sE Ξ² = β sD + cE. To Fnish our analysis, we distinguish two cases. By assumption, at least one of the eigenvalues is nonzero. Β±or notational convenience, assume (renumbering if necessary) that | Ξ» 1 | β₯ | Ξ» 2 | ....
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