Parametric-Surface-r-u-v-R-2-R-3-r-u-v-a-sin-u-cos-v-a-sin-u-sin-v-a-cos-u-a-gt-0-0-u-pi-0-v-2pi-1-Find-Principal-Direction-2-Find-Principal-Curvature-3-Find-Gaus

Question Number 226573 by Lara2440 last updated on 06/Dec/25

$$\mathrm{Parametric}\:\mathrm{Surface}\:\hat {\boldsymbol{\mathrm{r}}}\left({u},{v}\right);\mathbb{R}^{\mathrm{2}} \rightarrow\mathbb{R}^{\mathrm{3}} \\ $$$$\hat {\boldsymbol{\mathrm{r}}}\left({u},{v}\right)=\begin{cases}{{a}\centerdot\mathrm{sin}\left({u}\right)\mathrm{cos}\left({v}\right)}\\{{a}\centerdot\mathrm{sin}\left({u}\right)\mathrm{sin}\left({v}\right)}\\{{a}\centerdot\mathrm{cos}\left({u}\right)}\end{cases}\:\:\:{a}>\mathrm{0}\:,\:\mathrm{0}\leq{u}\leq\pi\:,\:\mathrm{0}\leq{v}\leq\mathrm{2}\pi \\ $$$$\mathrm{1}.\:\mathrm{Find}\:\mathrm{Principal}\:\mathrm{Direction} \\ $$$$\mathrm{2}.\:\mathrm{Find}\:\mathrm{Principal}\:\mathrm{Curvature} \\ $$$$\mathrm{3}.\:\mathrm{Find}\:\mathrm{Gauss}\:\mathrm{Curvature} \\ $$$$\mathrm{4}.\:\mathrm{Find}\:\mathrm{Euler}\:\mathrm{Characteristic}\: \\ $$$$\mathrm{Hint}\: \\ $$$$\mathrm{Shape}\:\mathrm{Operator}\:\mathcal{S}={r}_{,\mu\nu} ^{\lambda} \ast\hat {\mathcal{N}}_{\lambda} \: \\ $$

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