Question Number 226507 by Lara2440 last updated on 01/Dec/25
Answered by Lara2440 last updated on 01/Dec/25
![Smooth manifold M,N and Differentiable smooth function φ;M→N φ(u,v)= { ((−sin(u)−3sin(v))),(( cos(u)+3cos(v) , u∈[−π,π] , v∈[−2π,2π])),(( 5v)) :} at Each point p on a regular surface M⊂R^3 the Shape Operator is linear map S;M_p →M_p and Shape operator defiened as S(v)=−D_v N^ S(x_u )=−N_u ^ S(x_v )=−N_v ^ 1) Find Normal Curvature 2) Find Principal Curvature 3) Find Principal Direction 4) Find Gaussian Curvature](https://www.tinkutara.com/question/Q226508.png)
$$\mathrm{Smooth}\:\mathrm{manifold}\:{M},{N}\:\mathrm{and} \\ $$$$\mathrm{Differentiable}\:\mathrm{smooth}\:\mathrm{function}\:\:\phi;{M}\rightarrow{N} \\ $$$$\: \\ $$$$\phi\left({u},{v}\right)=\begin{cases}{−\mathrm{sin}\left({u}\right)−\mathrm{3sin}\left({v}\right)}\\{\:\:\:\:\mathrm{cos}\left({u}\right)+\mathrm{3cos}\left({v}\right)\:\:\:\:\:\:,\:{u}\in\left[−\pi,\pi\right]\:,\:{v}\in\left[−\mathrm{2}\pi,\mathrm{2}\pi\right]}\\{\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{5}{v}}\end{cases} \\ $$$$\: \\ $$$$\mathrm{at}\:\mathrm{Each}\:\mathrm{point}\:\boldsymbol{\mathrm{p}}\:\mathrm{on}\:\mathrm{a}\:\mathrm{regular}\:\mathrm{surface}\:{M}\subset\mathbb{R}^{\mathrm{3}} \\ $$$$\mathrm{the}\:\mathrm{Shape}\:\mathrm{Operator}\:\mathrm{is}\:\mathrm{linear}\:\mathrm{map} \\ $$$${S};{M}_{\boldsymbol{\mathrm{p}}} \rightarrow{M}_{\boldsymbol{\mathrm{p}}} \: \\ $$$$\mathrm{and}\:\mathrm{Shape}\:\mathrm{operator}\:\mathrm{defiened}\:\mathrm{as}\:\boldsymbol{\mathrm{S}}\left(\boldsymbol{\mathrm{v}}\right)=−{D}_{\boldsymbol{\mathrm{v}}} \hat {\boldsymbol{\mathrm{N}}} \\ $$$$\boldsymbol{\mathrm{S}}\left({x}_{{u}} \right)=−\hat {\boldsymbol{\mathrm{N}}}_{{u}} \\ $$$$\boldsymbol{\mathrm{S}}\left({x}_{{v}} \right)=−\hat {\boldsymbol{\mathrm{N}}}_{{v}} \\ $$$$\left.\mathrm{1}\right)\:\mathrm{Find}\:\mathrm{Normal}\:\mathrm{Curvature} \\ $$$$\left.\mathrm{2}\right)\:\mathrm{Find}\:\mathrm{Principal}\:\mathrm{Curvature} \\ $$$$\left.\mathrm{3}\right)\:\mathrm{Find}\:\mathrm{Principal}\:\mathrm{Direction} \\ $$$$\left.\mathrm{4}\right)\:\mathrm{Find}\:\mathrm{Gaussian}\:\mathrm{Curvature} \\ $$