Chapter 12 Vectors and the Geometry of Space Calculus 5e Early Transcendentals Multivariable James Stewart

© 2003 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning ™ is a trademark used herein under license. 2 Follow the link to the slide. Then click on the figure to play the animation. A Animations Ellipsoid Elliptic Paraboloid Sec Table 1 Hyperbolic Paraboloid Figure Cone Hyperboloid of One Sheet Hyperboloid of Two Sheets Figure

© 2003 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning ™ is a trademark used herein under license. 3 Section 12.3  Figure 2 The Dot Product A

© 2003 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning ™ is a trademark used herein under license. 4 Section 12.6  Figures 6, 7 Traces of

© 2003 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning ™ is a trademark used herein under license. 5 A Section 12.6  Figure 8 Graph of is the hyperbolic paraboloid

© 2003 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning ™ is a trademark used herein under license. 6 Section 12.6  Table 1a Ellipsoid All traces are ellipses. If a = b = c, the ellipsoid is a sphere. A

© 2003 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning ™ is a trademark used herein under license. 7 Section 12.6  Table 1b Elliptic Paraboloid Horizontal traces are ellipses. Vertical traces are parabolas. The variable raised to the first power indicates the axis of the paraboloid. A

© 2003 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning ™ is a trademark used herein under license. 8 Section 12.6  Table 1c Hyperbolic Paraboloid Horizontal traces are hyperbolas. Vertical traces are parabolas. The case where c < 0 is illustrated. A

© 2003 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning ™ is a trademark used herein under license. 9 Section 12.6  Table 1d Cone Horizontal traces are ellipses. Vertical traces in the planes x = k and y = k are hyperbolas if k ­ 0 but are pairs of lines if k = 0. A

© 2003 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning ™ is a trademark used herein under license. 10 Section 12.6  Table 1e Hyperboloid of One Sheet Horizontal traces are ellipses. Vertical traces are hyperbolas. The axis of symmetry corresponds to the variable whose coefficient is negative. A

© 2003 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning ™ is a trademark used herein under license. 11 Section 12.6  Table 1f Hyperboloid of Two Sheets Horizontal traces in z = k are ellipses if k > c or k <  c. Vertical traces are hyperbolas. The two minus signs indicate two sheets. A