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Question Number 225652 by fantastic last updated on 05/Nov/25 $$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {e}^{{i}\pi{x}} \:{dx} \\ $$ Answered by mahdipoor last updated on 05/Nov/25 $$\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{ae}^{\mathrm{i}\pi\mathrm{x}} \right)=\left(\mathrm{i}\pi\mathrm{a}\right)\mathrm{e}^{\mathrm{i}\pi\mathrm{x}} =\mathrm{e}^{\mathrm{i}\pi\mathrm{x}}…
Question Number 225625 by fantastic last updated on 04/Nov/25 $${quick}\:{short}\:{Q} \\ $$$$\mathrm{2}\:{things}\:{are}\:{mixed}\: \\ $$$$\left.\mathrm{1}\right){both}\:{same}\:{volume} \\ $$$$\left.\mathrm{2}\right){both}\:{same}\:{mass} \\ $$$${density}\Rightarrow{d}_{{v}} \:{and}\:{d}_{{m}} \\ $$$${d}_{{v}} \:?\:{d}_{{m}} \left[=,<\:{or}>\right] \\ $$…
Question Number 225610 by Lara2440 last updated on 06/Nov/25 $$\mathrm{prove} \\ $$$$\mathrm{Gauss}\:\mathrm{curvature}\:{K}\:\mathrm{is}\:\mathrm{intrinsic}\:\mathrm{by}\:\mathrm{showing} \\ $$$${K}=\frac{\begin{vmatrix}{−\frac{\mathrm{1}}{\mathrm{2}}{E}_{{vv}} +{F}_{{uv}} −{G}_{{uu}} }&{\frac{\mathrm{1}}{\mathrm{2}}{E}_{{u}} }&{{F}_{{u}} −\frac{\mathrm{1}}{\mathrm{2}}{E}_{{v}} }\\{\:\:\:\:\:\:\:\:\:\:\:\:{F}_{{v}} −\frac{\mathrm{1}}{\mathrm{2}}{G}_{{u}} }&{\:\:\:{E}}&{\:\:\:\:\:\:{F}}\\{\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{1}}{\mathrm{2}}{G}_{{v}} }&{\:\:\:{F}}&{\:\:\:\:\:{G}}\end{vmatrix}−\begin{vmatrix}{\:\:\:\:\mathrm{0}}&{\frac{\mathrm{1}}{\mathrm{2}}{E}_{{v}} }&{\frac{\mathrm{1}}{\mathrm{2}}{G}_{{u}} }\\{\frac{\mathrm{1}}{\mathrm{2}}{E}_{{v}}…
Question Number 225581 by Lara2440 last updated on 03/Nov/25 $$\mathrm{prove} \\ $$$$\mathrm{Gauss}\:\mathrm{curvature}\:{K}\:\mathrm{intrinsic} \\ $$$$\mathrm{it}'\mathrm{s}\:\mathrm{the}\:\mathrm{same}\:\mathrm{thing}\:\mathrm{as}\:\mathrm{saying}; \\ $$$$\mathrm{Show}\:\mathrm{that}\:\mathrm{Gauss}\:\mathrm{curvature}\:{K}\:\mathrm{can}\:\mathrm{only}\:\mathrm{consist}\:\mathrm{of} \\ $$$$\mathrm{First}\:\mathrm{Fundamental}\:\mathrm{Form}\:\mathrm{and}\:\mathrm{it}'\mathrm{s}\:\mathrm{Derivatives}. \\ $$ Terms of Service Privacy Policy…
Question Number 225506 by fantastic last updated on 31/Oct/25 Commented by fantastic last updated on 31/Oct/25 $${youre}\:{right}\:{sir} \\ $$ Commented by MirHasibulHossain last updated on…
Question Number 225488 by Lara2440 last updated on 30/Oct/25 $$\mathrm{Diffenrantial}\:\mathrm{Geometry}…..=\wedge= \\ $$$$\mathrm{Christoffel}\:\mathrm{symbol}\: \\ $$$$\Gamma_{\sigma\mu\nu} \:\mathrm{first}\:\mathrm{kind}\:\mathrm{and}\:\Gamma_{\mu\nu} ^{\:\sigma} \:\mathrm{second}\:\mathrm{kind}… \\ $$$$\mathrm{and}\:\mathrm{Chritoffel}\:\mathrm{symbol}\:\mathrm{satisfy} \\ $$$$\Gamma_{\mu\nu} ^{\:\sigma} =\frac{\mathrm{1}}{\mathrm{2}}{g}^{\sigma{l}} \Gamma_{{l}\mu\nu} \\…
Question Number 225479 by Lara2440 last updated on 28/Oct/25 $$\mathrm{Let}\:\mathcal{S};\mathbb{R}^{\mathrm{2}} \rightarrow\mathbb{R}^{\mathrm{3}} \:\mathrm{Sphere}\: \\ $$$${Q}\mathrm{1}.\:\mathrm{Find}\:\mathrm{metric}\:\mathrm{tensor}\:\mathrm{g}_{\mu\nu} \: \\ $$$${Q}\mathrm{2}.\:\mathrm{Find}\:\mathrm{Riemann}\:\mathrm{metric}\:\mathrm{tensor}\:{R}_{{jkl}} ^{{i}} \\ $$$${Q}.\mathrm{3}\:\:\mathrm{Find}\:\mathrm{Ricci}\:\mathrm{tensor}\:\mathrm{R}_{\alpha\beta} \\ $$$${Q}.\mathrm{4}\:\:\mathrm{Find}\:\mathrm{Ricci}\:\mathrm{Scalar}\:\mathcal{R} \\ $$$$\mathrm{Christoffel}\:\mathrm{symbol}\:\mathrm{first}\:\mathrm{kind}\: \\…
Question Number 225461 by Jyrgen last updated on 26/Oct/25 $${Which}\:{one}\:{is}\:{the}\:{oddest}\:{prime}\:{number}? \\ $$ Answered by Frix last updated on 26/Oct/25 $$\mathrm{Obviously}\:\mathrm{2} \\ $$$$\left[\mathrm{All}\:\mathrm{other}\:\mathrm{primes}\:\mathrm{are}\:\mathrm{odd}\:\Rightarrow\:\mathrm{it}'\mathrm{s}\:\mathrm{odd}\:\mathrm{for}\:\mathrm{a}\right. \\ $$$$\left.\mathrm{prime}\:\mathrm{number}\:\mathrm{to}\:\mathrm{be}\:\mathrm{not}\:\mathrm{odd}.\right] \\…
Question Number 225397 by fantastic last updated on 24/Oct/25 Commented by fantastic last updated on 24/Oct/25 $${looks}\:{easy}\:{right}? \\ $$ Answered by mr W last updated…