Engineering Calculus Notes 406

Engineering Calculus Notes 406 - 1 = u 1 x = cx + sy 2 = u...

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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 deFne 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: 1 = u 1 x = cx + sy 2 = u 2 x = sx + cy . You should check that these equations can by solved for x and y in terms of 1 and 2 : x = c 1 s 2 y = s 1 + c 2 so that p ( x,y ) can be rewritten as p ( x,y ) = Q ( x,y ) + Dx + Ey = 1 2 1 + 2 2 2 + 1 + 2 where = cD + sE = sD + cE. To Fnish our analysis, we distinguish two cases. By assumption, at least one of the eigenvalues is nonzero. or notational convenience, assume (renumbering if necessary) that | 1 | | 2 | ....
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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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