Question-226624

Question Number 226624 by Lara2440 last updated on 07/Dec/25

Commented by Lara2440 last updated on 08/Dec/25

Pseudo-sphere is surface of revolving of the tractrix paremetric by v^� (t)→(α(t−tanh(t)),α∙sech(t)) α>0 , 0≤t<∞ Pseudo sphere′s precise mapping r^→ ;R^2 →R^3 is r^→ (u,v)→(r∙sech(u)cos(v),r∙sech(u)sin(v),u−tanh(u)) r>0 , −π≤u≤π , −∞≤v≤∞ 1. show that volume and surface are finite 2. show that gauss curvature K always K<0 3.can pseudo-sphere piecewise smoothly immersion in R^3 ?? 4.can pseudo sphere be Embedded in R^3 Space ??

$$\: \\ $$$$\mathrm{Pseudo}-\mathrm{sphere}\:\mathrm{is}\:\mathrm{surface}\:\mathrm{of}\:\mathrm{revolving}\:\mathrm{of}\:\mathrm{the}\:\mathrm{tractrix} \\ $$$$\mathrm{paremetric}\:\mathrm{by}\:\:\hat {\boldsymbol{{v}}}\left({t}\right)\rightarrow\left(\alpha\left({t}−\mathrm{tanh}\left({t}\right)\right),\alpha\centerdot\mathrm{sech}\left({t}\right)\right) \\ $$$$\alpha>\mathrm{0}\:,\:\mathrm{0}\leq{t}<\infty \\ $$$$\mathrm{Pseudo}\:\mathrm{sphere}'\mathrm{s}\:\mathrm{precise}\:\mathrm{mapping}\:\overset{\rightarrow} {\boldsymbol{\mathrm{r}}};\mathbb{R}^{\mathrm{2}} \rightarrow\mathbb{R}^{\mathrm{3}} \:\mathrm{is}\:\: \\ $$$$\overset{\rightarrow} {\boldsymbol{\mathrm{r}}}\left({u},{v}\right)\rightarrow\left({r}\centerdot\mathrm{sech}\left({u}\right)\mathrm{cos}\left({v}\right),{r}\centerdot\mathrm{sech}\left({u}\right)\mathrm{sin}\left({v}\right),{u}−\mathrm{tanh}\left({u}\right)\right) \\ $$$${r}>\mathrm{0}\:,\:−\pi\leq{u}\leq\pi\:,\:−\infty\leq{v}\leq\infty \\ $$$$\: \\ $$$$\mathrm{1}.\:\mathrm{show}\:\mathrm{that}\:\mathrm{volume}\:\mathrm{and}\:\mathrm{surface}\:\mathrm{are}\:\mathrm{finite} \\ $$$$\mathrm{2}.\:\mathrm{show}\:\mathrm{that}\:\mathrm{gauss}\:\mathrm{curvature}\:{K}\:\mathrm{always}\:{K}<\mathrm{0} \\ $$$$\mathrm{3}.\mathrm{can}\:\mathrm{pseudo}-\mathrm{sphere}\:\mathrm{piecewise}\:\mathrm{smoothly}\:\mathrm{immersion}\:\mathrm{in}\:\mathbb{R}^{\mathrm{3}} \:?? \\ $$$$\mathrm{4}.\mathrm{can}\:\mathrm{pseudo}\:\mathrm{sphere}\:\mathrm{be}\:\mathrm{Embedded}\:\mathrm{in}\:\mathbb{R}^{\mathrm{3}} \:\mathrm{Space}\:?? \\ $$

Answered by MrAjder last updated on 09/Dec/25