Calculus: Analytic Geometry and Calculus, with Vectors

406 Cones and conics The graph of the equation x2 V2 16.66) a2 b2 =z is an elliptic paraboloid. It intersects the plane having t ...

6.6 Quadric surfaces 407 examples of surfaces known as saddle surfaces. The origin is a saddle point, and a huge surprise awaits ...

Curves, r] lengths, Curves, lengths, and curvatures 7.1 Curves and lengths One of the main purposes of this chapter may seem at ...

7.1 Curves and lengths 409 For an accurate treatment of matters relating to lengths of curves, we need more information about th ...

410 Curves, lengths, and curvatures as we think of them asbeing traversed once in the positive (counter- clockwise) or negative ...

7.1 Curves and lengths 411 that (x1, f (xi)) precedes (X2, f (X2)) when x1 < x2. Then C has finite length ;CI and (7.171) If( ...

412 Curves, lengths, and curvatures 3 Show that the projection on the xy plane of a curve in Es is a curve in E2. 4 Discover con ...

7.1 Curves and lengths 413 the earth is much like a spherical ball, but is not a spherical ball because moun- tains and valleys ...

414 Curves, lengths, and curvatures terminology, the semicubical parabola is a graph (not necessarily a curve) which is the poin ...

7.1 Curves and lengths 415 there being one such point for each t for which t -1. One who has consid- erable time to study the fu ...

416 Curves, lengths, and curvatures are, when r > 0, continuous but notpiecewise monotone over 0 5 x < 1. Sup- posing that ...

7.2 Lengths and integrals 417 17 Sketch some figures to obtain preliminary ideas about matters relating to the following definit ...

418 Curves, lengths, and curvatures could be supposed that our main interest is in lengths of curves and that we should get our ...

7.2 Lengths and integrals 419 This is, in fact, a correct and very satisfying formula which treats all coordinates alike and ena ...

420 Curves, lengths, and curvatures being no reason for hope that the three numbers tk *, and t***are the same. Thus (7.252) JCJ ...

7.2 Lengths and integrals 421 all have absolute values less than e' when I u - oil < S. When JP1 < S, the right member of ...

422 Curves, lengths, and curvatures and calculate the velocity v and the speed (vI from the formulas (7.284) v = x'(t)i + y'(t)j ...

7.2 Lengths and integrals 423 5 When 0 < b < a and a circle (or cog wheel) of radius b rolls without slipping inside a cir ...

424 Curves, lengths, and curvatures the curve C to be the ordered set of points P(t) = P(u,v,w) for which u, v, w are functions ...

7.2 Lengths and integrals 425 11 Let units of measurement be so employed that the potential at Po(xo,yo,zo) due to a particle of ...