Author: Tinku Tara

Question Number 226581 by klipto last updated on 06/Dec/25 $$\boldsymbol{\mathrm{find}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{polar}}\:\boldsymbol{\mathrm{of}} \\ $$$$\left(\mathrm{1}+\boldsymbol{\mathrm{i}}\right)\left(\mathrm{1}+\boldsymbol{\mathrm{i}}\sqrt{\mathrm{3}}\right) \\ $$ Answered by Frix last updated on 06/Dec/25 $$=\sqrt{\mathrm{2}}\mathrm{e}^{\mathrm{i}\frac{\pi}{\mathrm{4}}} ×\mathrm{2e}^{\mathrm{i}\frac{\pi}{\mathrm{3}}} =\mathrm{2}\sqrt{\mathrm{2}}\mathrm{e}^{\mathrm{i}\frac{\mathrm{7}\pi}{\mathrm{12}}} \\…

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}\:…