This surface (dreamed up by Steiner while in Rome) looks smooth but is non-orientable, with the origin having three different normal vectors when arriving along different values for the of parameters. The parametrization for this surface is given by o(u, v)=(x, y, z) = (sin 2u sin v, sin 2u cos v, cos² u sin 2v). (a) Show o parametrizes the surface given by the quartic equation x²y² + y²z²+z²x² = 2xyz. (b) Find at least three different values of (u, v) with o(u, v) = (0,0,0) such that none of the three normal vectors are the same (and, in fact, they are all orthogonal to each other).
This surface (dreamed up by Steiner while in Rome) looks smooth but is non-orientable, with the origin having three different normal vectors when arriving along different values for the of parameters. The parametrization for this surface is given by o(u, v)=(x, y, z) = (sin 2u sin v, sin 2u cos v, cos² u sin 2v). (a) Show o parametrizes the surface given by the quartic equation x²y² + y²z²+z²x² = 2xyz. (b) Find at least three different values of (u, v) with o(u, v) = (0,0,0) such that none of the three normal vectors are the same (and, in fact, they are all orthogonal to each other).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:This surface (dreamed up by Steiner while in Rome) looks smooth but is non-orientable,
with the origin having three different normal vectors when arriving along different values
for the of parameters. The parametrization for this surface is given by
o(u,v) = (x, y, z) = (sin 2u sin v, sin 2u cos v, cos u sin 2v).
(a) Show o parametrizes the surface given by the quartic equation
r'y + y'z? +z°x² = 2.xyz.
(b) Find at least three different values of (u,v) with o(u,v) (0,0,0) such that
none of the three normal vectors are the same (and, in fact, they are all
orthogonal to each other).
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