Engineering Calculus Notes 415

Engineering Calculus Notes 415 - z > and on the...

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3.10. QUADRATIC CURVES AND SURFACES 403 origin. For example, the surface given by the “model equation” x 2 y 2 = z (3.45) intersects the horizontal plane z = k in a hyperbola opening in the direction of the x -axis ( resp . y -axis) for z > 0 ( resp . z < 0), and in the common asymptotes of all these hyperbolas when z = 0 (Figure 3.36 ). x y z = 0 z < 0 z < 0 z > 0 z > 0 Figure 3.36: Level sets of z = x 2 y 2 To see how these ±t together, we note that the vertices of the two branches lie on the curve z = x 2 in the xz -plane for
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Unformatted text preview: z &gt; and on the curve z = y 2 in the yz-plane for z &lt; (Figure 3.37 ). The ocial name of this surface is a hyperbolic paraboloid , but it is colloquially referred to as a saddle surface (Figure 3.38 ). 2. If the form Q ( x,y,z ) is denite, then all of the eigenvalues i have the same sign, and we can model the locus (up to rotation and displacement of the origin) by x 2 a 2 + y 2 b 2 + z 2 c 2 = k ;...
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