Question Number 226562 by thetpainghtun_111 last updated on 04/Dec/25 $$\mathrm{If}\:\mathrm{x}\:+\:\mathrm{y}\:{i}\:=\:\frac{\mathrm{a}\:+\:{i}}{\mathrm{a}\:−\:{i}}\:,\:\mathrm{prove}\:\mathrm{that}\:\mathrm{ay}\:−\:\mathrm{1}\:=\:\mathrm{x}. \\ $$$$\:\left(\mathrm{x}+\mathrm{y}{i}\right)\left(\mathrm{a}−{i}\right)=\mathrm{a}+{i} \\ $$$$\:\:\:\mathrm{ax}\:−\:\mathrm{x}{i}\:+\:\mathrm{ay}{i}\:−\:\mathrm{y}{i}^{\mathrm{2}} \:=\:\mathrm{a}\:+\:{i} \\ $$$$\:\:\left(\mathrm{ax}\:+\:\mathrm{y}\right)\:+\:\left(\mathrm{ay}\:−\:\mathrm{x}\right){i}\:=\:\mathrm{a}\:+\:{i} \\ $$$$\:\:\mathrm{ay}\:−\:\mathrm{x}\:=\:\mathrm{1} \\ $$$$\:\:\mathrm{x}\:=\:\mathrm{ay}\:−\mathrm{1} \\ $$ Terms of…
Question Number 226542 by Lara2440 last updated on 03/Dec/25 Commented by Lara2440 last updated on 04/Dec/25 $$\: \\ $$$$\mathrm{Smooth}\:\mathrm{Manifold}\:{M},{N}\:\mathrm{and}\:\mathrm{differentiable}\:\mathrm{Smooth}\:\mathrm{function}\: \\ $$$$\overset{\rightarrow} {\phi}\left({u},{v}\right);{M}\rightarrow{N} \\ $$$$\: \\…
Question Number 226507 by Lara2440 last updated on 01/Dec/25 Answered by Lara2440 last updated on 01/Dec/25 $$\mathrm{Smooth}\:\mathrm{manifold}\:{M},{N}\:\mathrm{and} \\ $$$$\mathrm{Differentiable}\:\mathrm{smooth}\:\mathrm{function}\:\:\phi;{M}\rightarrow{N} \\ $$$$\: \\ $$$$\phi\left({u},{v}\right)=\begin{cases}{−\mathrm{sin}\left({u}\right)−\mathrm{3sin}\left({v}\right)}\\{\:\:\:\:\mathrm{cos}\left({u}\right)+\mathrm{3cos}\left({v}\right)\:\:\:\:\:\:,\:{u}\in\left[−\pi,\pi\right]\:,\:{v}\in\left[−\mathrm{2}\pi,\mathrm{2}\pi\right]}\\{\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{5}{v}}\end{cases} \\ $$$$\:…
Question Number 226467 by Lara2440 last updated on 30/Nov/25 Answered by Lara2440 last updated on 30/Nov/25 $$\:\mathrm{let}\:\mathrm{differantable}\:\mathrm{Smooth}\:\mathrm{curve}\:\phi;\mathbb{R}^{\mathrm{2}} \rightarrow\mathbb{R}^{\mathrm{3}} \\ $$$$\:\phi\left({u},{v}\right)=\begin{cases}{−\mathrm{sin}\left({u}\right)−\mathrm{3sin}\left({v}\right)}\\{\mathrm{cos}\left({u}\right)+\mathrm{3cos}\left({v}\right)}\\{\mathrm{4}{v}}\end{cases}\:\:\:,\:−\mathrm{2}\pi\leq{u}\leq\mathrm{2}\pi\:,\:−\mathrm{2}\pi\leq{v}\leq\mathrm{2}\pi \\ $$$$\mathrm{Find}\:\mathrm{Normal}\:\mathrm{curvature}\:,\:\mathrm{Principal}\:\mathrm{curvature}\:,\:\mathrm{Principal}\:\mathrm{dirction} \\ $$$$\: \\…
Question Number 226400 by Lara2440 last updated on 27/Nov/25 $$\mathrm{Prove}\:\mathrm{klein}\:\mathrm{bottle}\:\mathrm{is}\:\mathrm{Immersion} \\ $$$$\mathrm{but}\:\mathrm{klein}\:\mathrm{bottle}\:\mathrm{can}'\mathrm{t}\:\mathrm{Imbedding}\:\mathrm{in}\:\mathbb{R}^{\mathrm{3}} \:\mathrm{Space}\: \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 226401 by fantastic2 last updated on 27/Nov/25 $${most}\:{hated}\:{trigonometric} \\ $$$${problem}: \\ $$$$\mathrm{sin}\left(\:\frac{\pi}{\mathrm{7}}\right)\mathrm{sin}\:\left(\frac{\mathrm{2}\pi}{\mathrm{7}}\right)\mathrm{sin}\:\left(\frac{\mathrm{3}\pi}{\mathrm{7}}\right)=? \\ $$ Commented by AgniMath last updated on 27/Nov/25 $${bhannat}\:{maths}\:{op}. \\…
Question Number 226366 by klipto last updated on 26/Nov/25 $$\boldsymbol{\mathrm{compute}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{double}}\:\boldsymbol{\mathrm{integral}} \\ $$$$\int_{\boldsymbol{\mathrm{y}}=\mathrm{0}} ^{\mathrm{1}} \int_{\boldsymbol{\mathrm{x}}=\mathrm{0}} ^{\mathrm{2}} \boldsymbol{\mathrm{x}}^{\mathrm{2}} \boldsymbol{\mathrm{dxdy}}\:\boldsymbol{\mathrm{and}}\:\:\int_{\boldsymbol{\mathrm{y}}=\mathrm{0}} ^{\mathrm{1}} \int_{\boldsymbol{\mathrm{x}}=\mathrm{0}} ^{\mathrm{2}} \boldsymbol{\mathrm{y}}^{\mathrm{2}} \boldsymbol{\mathrm{dxdy}} \\ $$$$ \\…
Question Number 226384 by Lara2440 last updated on 26/Nov/25 Answered by Lara2440 last updated on 29/Nov/25 $$\mathrm{help} \\ $$ Terms of Service Privacy Policy Contact:…
Question Number 226386 by fantastic2 last updated on 26/Nov/25 Commented by fantastic2 last updated on 27/Nov/25 $${l}=\mathrm{5}\pi{R} \\ $$$${a}\:{force}\:{F}\:{is}\:{given}\:{horizontally} \\ $$$${for}\:\mathrm{1}{sec}\:{such}\:{that}\:{the}\:{speed}\:{of}\:{the}\:{ball} \\ $$$${becomes}\:\mathrm{0}\:{just}\:{before}\:{touching}\:{the}\:{cylinder} \\ $$$${find}\:{the}\:{time}\:{taken}\:{to}\:{touch}\:{and}\:{the}\:{force}…
Question Number 226351 by Lara2440 last updated on 26/Nov/25 $$\mathrm{Prove}\:\mathrm{M}\ddot {\mathrm{o}bious}\:\mathrm{String}\:\mathrm{is}\:\mathrm{Not}\:\mathrm{a}\:\mathrm{Orientated}\:\mathrm{Surface}. \\ $$$$\sigma\left({u},\theta\right)=\begin{cases}{\left(\mathrm{1}−{u}\centerdot\mathrm{sin}\left(\frac{\mathrm{1}}{\mathrm{2}}\theta\right)\right)\mathrm{cos}\left(\theta\right)}\\{\left(\mathrm{1}−{u}\centerdot\mathrm{sin}\left(\frac{\mathrm{1}}{\mathrm{2}}\theta\right)\right)\mathrm{sin}\left(\theta\right)}\\{{u}\centerdot\mathrm{cos}\left(\frac{\mathrm{1}}{\mathrm{2}}\theta\right)}\end{cases}\:\:,\:−\frac{\mathrm{1}}{\mathrm{2}}\leq{u}\leq\frac{\mathrm{1}}{\mathrm{2}}\:,\:\mathrm{0}\leq\theta\leq\mathrm{2}\pi \\ $$ Answered by Lara2440 last updated on 26/Nov/25 $$\: \\ $$$$\mathrm{To}\:\mathrm{show}\:\mathrm{that}\:\mathrm{this}\:\mathrm{Surface}\:\mathrm{is}\:\mathrm{Oriented},…