Question-226467

Question Number 226467 by Lara2440 last updated on 30/Nov/25

Answered by Lara2440 last updated on 30/Nov/25

let differantable Smooth curve φ;R^2 →R^3 φ(u,v)= { ((−sin(u)−3sin(v))),((cos(u)+3cos(v))),((4v)) :} , −2π≤u≤2π , −2π≤v≤2π Find Normal curvature , Principal curvature , Principal dirction each point p on a Regular Surface M⊂R^3 , the Shape Operator is a linear map S;M_p →M_p Shape Operator S(v)=−D_v N^�

$$\:\mathrm{let}\:\mathrm{differantable}\:\mathrm{Smooth}\:\mathrm{curve}\:\phi;\mathbb{R}^{\mathrm{2}} \rightarrow\mathbb{R}^{\mathrm{3}} \\ $$$$\:\phi\left({u},{v}\right)=\begin{cases}{−\mathrm{sin}\left({u}\right)−\mathrm{3sin}\left({v}\right)}\\{\mathrm{cos}\left({u}\right)+\mathrm{3cos}\left({v}\right)}\\{\mathrm{4}{v}}\end{cases}\:\:\:,\:−\mathrm{2}\pi\leq{u}\leq\mathrm{2}\pi\:,\:−\mathrm{2}\pi\leq{v}\leq\mathrm{2}\pi \\ $$$$\mathrm{Find}\:\mathrm{Normal}\:\mathrm{curvature}\:,\:\mathrm{Principal}\:\mathrm{curvature}\:,\:\mathrm{Principal}\:\mathrm{dirction} \\ $$$$\: \\ $$$$\mathrm{each}\:\mathrm{point}\:\boldsymbol{\mathrm{p}}\:\mathrm{on}\:\mathrm{a}\:\mathrm{Regular}\:\mathrm{Surface}\:{M}\subset\mathbb{R}^{\mathrm{3}} , \\ $$$$\mathrm{the}\:\mathrm{Shape}\:\mathrm{Operator}\:\mathrm{is}\:\mathrm{a}\:\mathrm{linear}\:\mathrm{map} \\ $$$${S};{M}_{\boldsymbol{\mathrm{p}}} \rightarrow{M}_{\boldsymbol{\mathrm{p}}} \\ $$$$\mathrm{Shape}\:\mathrm{Operator}\:{S}\left(\boldsymbol{\mathrm{v}}\right)=−{D}_{\boldsymbol{\mathrm{v}}} \hat {\boldsymbol{\mathrm{N}}} \\ $$

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