Question Number 224789 by fkwow344 last updated on 04/Oct/25 $$\mathrm{prove}\:\mathrm{Sphere}\:\mathcal{S};\mathbb{R}^{\mathrm{3}} \rightarrow\mathbb{R} \\ $$$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} ={R}^{\mathrm{2}} \:,\:\mathrm{Euler}\:\mathrm{characteristic}\:\boldsymbol{\chi}=\mathrm{2} \\ $$$$\mathrm{by}\:\mathrm{gauss}-\mathrm{Bonnet}\:\mathrm{theorem} \\ $$$$\mathrm{2}\pi\boldsymbol{\chi}\left(\boldsymbol{\Omega}\right)=\int_{\:\boldsymbol{\Omega}} \:\mathrm{d}{A}\:{K} \\ $$$$\mathrm{Gauss}\:\mathrm{curvature}\:\mathrm{defined}\:\mathrm{as}\:{K}=\frac{\mathrm{det}\:\Pi}{\mathrm{det}\:\mathrm{I}}=\frac{{LN}−{M}^{\mathrm{2}} }{{EG}−{F}^{\mathrm{2}}…
Question Number 224773 by Ismoiljon_008 last updated on 03/Oct/25 Answered by mr W last updated on 03/Oct/25 Commented by mr W last updated on 03/Oct/25…
Question Number 224760 by fantastic last updated on 02/Oct/25 $$\mathrm{log}\:_{\mathrm{5}} \left(\mathrm{5}^{\frac{\mathrm{1}}{{x}}} +\mathrm{125}\right)=\mathrm{log}\:_{\mathrm{5}} \mathrm{6}+\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}{x}} \\ $$$${x}=?? \\ $$ Answered by som(math1967) last updated on 02/Oct/25 $$\:{log}_{\mathrm{5}}…
Question Number 224732 by fantastic last updated on 30/Sep/25 $${The}\:{value}\:{of}\:{n}\:{for}\:{which}\:{the}\:{divergence} \\ $$$${of}\:{the}\:{function} \\ $$$$\mathrm{F}=\frac{\mathrm{r}}{\begin{vmatrix}{\mathrm{r}}\end{vmatrix}^{{n}} },\:\mathrm{r}=\mathrm{x}\hat {\mathrm{i}}+{y}\hat {\mathrm{j}}+{z}\hat {\mathrm{k}},\begin{vmatrix}{\mathrm{r}}\end{vmatrix}\neq\mathrm{0}, \\ $$$${vanishes}\:{is} \\ $$$$\left.{a}\right)\mathrm{1} \\ $$$$\left.{b}\right)−\mathrm{1} \\…
Question Number 224735 by fantastic last updated on 30/Sep/25 $${Let}\:{u}=\frac{{y}^{\mathrm{2}} −{x}^{\mathrm{2}} }{{x}^{\mathrm{2}} {y}^{\mathrm{2}} },\:{v}=\frac{{z}^{\mathrm{2}} −{y}^{\mathrm{2}} }{{y}^{\mathrm{2}} {z}^{\mathrm{2}} }\:{for}\:{x}\neq\mathrm{0},{y}\neq\mathrm{0}{z}\neq\mathrm{0}. \\ $$$${Let}\:{w}={f}\left({u},{v}\right),\:{where}\:{f}\:{is}\:{a}\:{real} \\ $$$${valued}\:{function}\:{defined}\:{on}\:{R}^{\mathrm{2}} \\ $$$${having}\:{continuous}\:{first}\:{order} \\…
Question Number 224733 by fantastic last updated on 30/Sep/25 $${Let}\:{f}\:{be}\:{a}\:{continuously}\:{differentiable}\:{function} \\ $$$${such}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{2}{x}^{\mathrm{2}} } {f}\left({t}\right){dt}={e}^{\mathrm{cos}\:{x}^{\mathrm{2}} } \:{for}\:{all}\:{x}\in\left(\mathrm{0},\infty\right) \\ $$$${the}\:{value}\:{of}\:{f}\:'\left(\pi\right)=? \\ $$ Answered by…
Question Number 224739 by thetpainghtun_111 last updated on 30/Sep/25 $$\mathrm{z}\:=\:\mathrm{r}\:\left(\mathrm{cos}\:\theta\:+\:\mathrm{i}\:\mathrm{sin}\:\theta\right),\:\mathrm{find}\:\frac{\mathrm{z}}{\overset{−} {\mathrm{z}}}\:+\frac{\overset{−} {\mathrm{z}}}{\mathrm{z}}. \\ $$ Answered by Frix last updated on 30/Sep/25 $${z}={r}\mathrm{e}^{\mathrm{i}\theta} \:\Leftrightarrow\:\bar {{z}}={r}\mathrm{e}^{−\mathrm{i}\theta} \\…
Question Number 224723 by TonyCWX last updated on 29/Sep/25 $${So}\:{here}\:{is}\:{what}\:{it}\:{shows}. \\ $$ Commented by TonyCWX last updated on 29/Sep/25 Commented by Tinku Tara last updated…
Question Number 224714 by fantastic last updated on 28/Sep/25 $$\int\frac{\mathrm{1}}{\:\sqrt{\mathrm{tan}\:\theta}}\:{d}\theta \\ $$ Commented by Ghisom_ last updated on 29/Sep/25 $$\int\frac{{d}\theta}{\:\sqrt{\mathrm{tan}\:\theta}}= \\ $$$$\:\:\:\:\:\left[{t}=\sqrt{\mathrm{tan}\:\theta}\:\rightarrow\:{d}\theta=\mathrm{2cos}^{\mathrm{2}} \:\theta\:\sqrt{\mathrm{tan}\:\theta}\right] \\ $$$$=\mathrm{2}\int\frac{{dt}}{{t}^{\mathrm{4}}…
Question Number 224692 by TonyCWX last updated on 27/Sep/25 $$ \\ $$ Commented by TonyCWX last updated on 27/Sep/25 Commented by TonyCWX last updated on…