Question Number 224883 by fkwow344 last updated on 09/Oct/25 $$\int\:\mathrm{vol}\left({g}^{\:} \right)=\int_{\:{V}} \:\sqrt{\mathrm{det}\:\boldsymbol{\mathrm{g}}_{\mu\nu} }\:\mathrm{d}{x}^{\mathrm{1}} \wedge\mathrm{d}{x}^{\mathrm{2}} \wedge\mathrm{d}{x}^{\mathrm{3}} \\ $$$$\mathrm{parametric}\:\mathrm{Surface}\: \\ $$$$\overset{\rightarrow} {\mathcal{S}}\left({u},{v},{w}\right);\mathbb{R}^{\mathrm{3}} \rightarrow\mathbb{R}^{\mathrm{3}} \\ $$$$\overset{\rightarrow} {\mathcal{S}}\left({r},\theta,\rho\right)\begin{cases}{{r}\mathrm{sin}\left(\theta\right)\mathrm{cos}\left(\rho\right)}\\{{r}\mathrm{sin}\left(\theta\right)\mathrm{sin}\left(\rho\right)}\\{{r}\mathrm{cos}\left(\theta\right)}\end{cases}\: \\…
Question Number 224861 by fantastic last updated on 08/Oct/25 $$\int\frac{\mathrm{sin}\:{x}}{\:\sqrt{\mathrm{1}+\mathrm{sin}\:{x}}}\:{dx} \\ $$ Answered by Mathswiz last updated on 08/Oct/25 Commented by som(math1967) last updated on…
Question Number 224859 by fantastic last updated on 08/Oct/25 $${A}\:{homogeneous}\:{rod}\:{AB}\:{of}\:{length} \\ $$$${L}=\mathrm{1}.\mathrm{8}{m}\:{and}\:{mass}\:{M}\:{is}\:{pivoted} \\ $$$${at}\:{the}\:{centre}\:{O}\:{in}\:{such}\:{a}\:{way}\:{that} \\ $$$${it}\:{can}\:{rotate}\:{freely}\:{in}\:{the}\:{vertical}\:{plane}. \\ $$$$ \\ $$$${The}\:{rod}\:{is}\:{initially}\:{in}\:{the}\:{horizontal} \\ $$$${position}.{An}\:{insect}\:{S}\:\:{of}\:{the}\: \\ $$$${same}\:{mass}\:{M}\:{falls}\:{vertically} \\…
Question Number 224866 by fkwow344 last updated on 08/Oct/25 $$\mathrm{can}\:\mathrm{you}\:\mathrm{guys}\:\mathrm{explan}\:\mathrm{why} \\ $$$$\mathrm{metric}\:\mathrm{tensor}\:\mathrm{g}_{\mu\nu} =\mathrm{0}\:\:\rightarrow\:\mathrm{Riemann}\:\mathrm{metric}\:\mathrm{tensor}\:{R}_{\alpha\gamma\beta} ^{\delta} =\mathrm{0} \\ $$ Answered by MrAjder last updated on 18/Oct/25 Terms…
Question Number 224852 by fantastic last updated on 07/Oct/25 $$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{3}}+\frac{\mathrm{1}}{\mathrm{4}}+…+\frac{\mathrm{1}}{{n}}}{{ln}\left({n}\right)}=? \\ $$ Answered by vnm last updated on 08/Oct/25 $$\mathrm{lim}=\mathrm{1} \\ $$$$\mathrm{This}\:\mathrm{follows}\:\mathrm{from}\:\mathrm{the}\:\mathrm{double}\:\mathrm{inequality}\: \\ $$$$\mathrm{which}\:\mathrm{is}\:\mathrm{evident}\:\mathrm{if}\:\mathrm{we}\:\mathrm{plot}\:…
Question Number 224853 by fantastic last updated on 07/Oct/25 $$ \\ $$$$\mathrm{a}\:\mathrm{piece}\:\mathrm{of}\:\mathrm{chalk}\:\mathrm{rests}\:\mathrm{on}\:\mathrm{a} \\ $$$$\mathrm{horizontal}\:\mathrm{board}\:\mathrm{with}\:\mu=\mathrm{0}.\mathrm{1} \\ $$$$\mathrm{Suddenly}\:\mathrm{the}\:\mathrm{board}\:\mathrm{starts}\:\mathrm{to} \\ $$$$\mathrm{move}\:\mathrm{horizontally}\:\mathrm{at}\:\mathrm{a}\:\mathrm{speed}\:\mathrm{of} \\ $$$$\mathrm{2m}\:\mathrm{per}\:\mathrm{second}\:\mathrm{and}\:\mathrm{after}\:\mathrm{a} \\ $$$$\mathrm{time}\:\tau\:\mathrm{it}\:\mathrm{stops}\:\mathrm{abruptly}.\:\mathrm{find}\: \\ $$$$\mathrm{the}\:\mathrm{length}\:\mathrm{of}\:\mathrm{the}\:\mathrm{line}\:\mathrm{drawn} \\…
Question Number 224833 by fantastic last updated on 06/Oct/25 $$\int\mathrm{sec}\:\theta\:{d}\theta \\ $$ Answered by taha3738 last updated on 06/Oct/25 $$\int\:\mathrm{sec}\:\theta\:{d}\theta\:=\:\int\:\frac{\mathrm{sec}\:\theta\:\left(\mathrm{sec}\:\theta\:+\:\mathrm{tan}\:\theta\right)}{\mathrm{sec}\:\theta\:+\:\mathrm{tan}\:\theta}\:{d}\theta \\ $$$$=\:\int\:\:\frac{\mathrm{sec}^{\mathrm{2}} \theta+\:\mathrm{sec}\:\theta\:\mathrm{tan}\:\theta}{\mathrm{sec}\:\theta\:+\:\mathrm{tan}\:\theta\:}\:{d}\theta\:=\:\mathrm{ln}\:\mid\:\mathrm{sec}\:{x}\:+\:\mathrm{tan}\:{x}\:\mid\:+\:{C} \\ $$$${Because}\:\frac{{d}}{{d}\theta}\:\left(\:\mathrm{sec}\:\theta\:+\:\mathrm{tan}\:\theta\:\right)\:=\:\frac{{d}}{{d}\theta}\:\mathrm{sec}\:\theta\:+\:\frac{{d}}{{d}\theta}\:\mathrm{tan}\:\theta…
Question Number 224806 by fkwow344 last updated on 05/Oct/25 $$\mathrm{Use}\:\mathrm{the}\:\mathrm{Gauss}\:\mathrm{Bonnet}\:\mathrm{Theorem}\:\mathrm{to}\:\mathrm{show}\:\mathrm{that}\:\mathrm{the} \\ $$$$\mathrm{number}\:\mathrm{of}\:\mathrm{holes}\:\mathrm{in}\:\mathrm{a}\:\mathrm{straw}\:\mathrm{is}\:\mathrm{1}. \\ $$$$\mathrm{Then}\:\mathrm{associate}\:\mathrm{it}\:\mathrm{and}\: \\ $$$$\mathrm{show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{Genus}\:\mathrm{on}\:\mathrm{the}\:\mathrm{surface}\:\mathrm{is}\:\mathrm{1}. \\ $$ Answered by MrAjder last updated on 18/Oct/25…
Question Number 224814 by fantastic last updated on 05/Oct/25 Answered by mr W last updated on 05/Oct/25 $${v}_{{D}} =\frac{{v}_{{A}} +{v}_{{E}} }{\mathrm{2}}=\frac{\mathrm{4}+\mathrm{0}}{\mathrm{2}}=\mathrm{2}\:{m}/{s} \\ $$$${v}_{{C}} =\frac{{v}_{{B}} +{v}_{{D}}…
Question Number 224807 by fantastic last updated on 05/Oct/25 Commented by fantastic last updated on 05/Oct/25 $${It}\:{took}\:{me}\:\mathrm{10}\:{to}\:\mathrm{15}\:{minutes}\:{to}\:{solve} \\ $$$${this}\:{question}.{It}\:{was}\:{not}\:{too}\:{hard} \\ $$$${but}\:{the}\:{question}\:{is}\:{very}\:{interesting} \\ $$$${Here}\:{is}\:{the}\:{question}: \\ $$$${A}\:{bottle}\:{of}\:{syllendrical}\:{shape}…