. Differential and integral calculus, an introductory course for colleges and engineering schools. and the conjugate axis is in the :n/-plane. Whenk = b, the section is a pair of right lines intersecting in the remarks apply to the sections made by the planesx = k. Let the student give the complete discussion of thesesections. 4. The Hyperboloid of Two Sheets, x2 _ y2 _ z2 a2 b2 c2 §185 THE POINT, THE PLANE, AND THE SURFACE 275 Writing this in the form t. _l zl = *L. i -« -,2 1: b2 C* a1 it is seen that when | x \ | a J), cut it in ellipses. The solid bounded by the surface may
. Differential and integral calculus, an introductory course for colleges and engineering schools. and the conjugate axis is in the :n/-plane. Whenk = b, the section is a pair of right lines intersecting in the remarks apply to the sections made by the planesx = k. Let the student give the complete discussion of thesesections. 4. The Hyperboloid of Two Sheets, x2 _ y2 _ z2 a2 b2 c2 §185 THE POINT, THE PLANE, AND THE SURFACE 275 Writing this in the form t. _l zl = *L. i -« -,2 1: b2 C* a1 it is seen that when | x \ | a J), cut it in ellipses. The solid bounded by the surface may be conceived as made up of ellipses strung on the x-axis. 5. The Elliptic Paraboloid,. When z is —, x and y cannot both be real, which means that thesurface has no real points below the xy-plane. It is readily foundthat the traces in the xy-plsme and xz-p\sme are parabolas turnedupward, while the trace in the xy-ipl&ne is the origin. The planesz = k cut the surface in ellipses which increase in size as k in-creases. The surface may be conceived as built up of ellipsesstrung on the 2-axis. Let the student complete the discussionand draw the figure of this surface. 6. The Elliptic Wedge, x2z2 + a2y2 = the equation in the form ay = ±x Vb2 — z2, it appears that the surface lies entirely between the planes z = — b,z = +6. When z = d=6, y = 0, and therefore the trace in thezz-plane consists of two parallel lines z=—b, z = -\-b. Everyplane parallel to the xy-plane, z = ±k, cuts the surface in a pair 276 GEOMETRY OF THREE DIMENSIONS 186 of right lines, ay = zLx Vb2 — k2, the xz-plane bisecting the anglebetween them. When x = dbk, k2z2 + a2y2 = b2k2
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectcalculu, bookyear1912
