Question Number 192397 by Mastermind last updated on 16/May/23 $$\mathrm{Let}\:\mathrm{f}:\mathrm{D}\left(\mathrm{f}\right)\subseteq\mathbb{R}^{\mathrm{n}} \rightarrow\mathbb{R}^{\mathrm{m}} \\ $$$$\mathrm{let}\:'\mathrm{a}'\:\mathrm{be}\:\mathrm{an}\:\mathrm{interior}\:\mathrm{point}\:\mathrm{of}\:\mathrm{Dom}\left(\mathrm{f}\right) \\ $$$$\mathrm{and}\:\mathrm{let}\:'\mathrm{u}'\:\mathrm{be}\:\mathrm{any}\:\mathrm{vector}\:\mathrm{in}\:\mathbb{R}^{\mathrm{n}} ,\:\mathrm{when} \\ $$$$\mathrm{is}\:\mathrm{a}\:\mathrm{vector}\:\mathrm{v}\in\mathbb{R}^{\mathrm{m}} \:\mathrm{called}\:\mathrm{the}\:\mathrm{directional} \\ $$$$\mathrm{derivative}\:\mathrm{of}\:\mathrm{f}\:\mathrm{at}\:'\mathrm{a}'\:\mathrm{along}\:\mathrm{the}\:\mathrm{line} \\ $$$$\mathrm{determine}\:\mathrm{by}\:\mathrm{u}\:? \\ $$$$…
Question Number 192396 by moh777 last updated on 16/May/23 Answered by mehdee42 last updated on 16/May/23 $$\mathrm{4}{x}−{x}^{\mathrm{2}} =\mathrm{3}\Rightarrow{x}=\mathrm{1},\mathrm{3} \\ $$$${v}_{\mathrm{1}} =\pi\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{4}{x}−{x}^{\mathrm{2}} \right)^{\mathrm{2}} {dx}=\pi\int_{\mathrm{0}}…
Question Number 192398 by Mastermind last updated on 16/May/23 $$\mathrm{Let}\:\mathrm{f}:\mathbb{R}^{\mathrm{3}} \rightarrow\mathbb{R}\:\mathrm{be}\:\mathrm{define}\:\mathrm{by}\: \\ $$$$\mathrm{f}\left(\mathrm{x},\:\mathrm{y},\:\mathrm{z}\right)\:=\:\mathrm{2x}^{\mathrm{2}} −\mathrm{y}+\mathrm{6xy}−\mathrm{z}^{\mathrm{3}} +\mathrm{3z}. \\ $$$$\mathrm{calculate}\:\mathrm{the}\:\mathrm{directional}\:\mathrm{deriva}− \\ $$$$\mathrm{tive}\:\mathrm{of}\:\mathrm{the}\:\mathrm{vector}\:\mathrm{u}=\left(\mathrm{2},\:\mathrm{1},\:−\mathrm{3}\right) \\ $$$$ \\ $$$$\mathrm{help}! \\ $$…
Question Number 61318 by Tawa1 last updated on 31/May/19 Answered by tanmay last updated on 01/Jun/19 $$\overset{\rightarrow} {{V}}_{{train}\:{w}.{r}.{t}\:{ground}} =\left(\frac{\mathrm{64000}}{\mathrm{3600}}\right){m}/{s}\:\overset{\rightarrow} {{i}}=\left(\frac{\mathrm{160}}{\mathrm{9}}\right){m}/{s}\:\overset{\rightarrow} {{i}} \\ $$$$\overset{\rightarrow} {{V}}_{{rain}\:{w}.{r}.{t}\:{ground}} =\mathrm{5}{m}/{s}\:\left(−\overset{\rightarrow}…
Question Number 192341 by Mastermind last updated on 15/May/23 $$\left.\mathrm{1}\right)\:\mathrm{Find}\:\mathrm{the}\:\mathrm{sign}\:\mathrm{of}\:\mathrm{odd}\:\mathrm{or}\:\mathrm{even}\:\left(\mathrm{or}\:\mathrm{pality}\right) \\ $$$$\mathrm{of}\:\mathrm{permutation}\:\theta=\left(\mathrm{1}\:\mathrm{2}\:\mathrm{3}\:\mathrm{4}\:\mathrm{5}\:\mathrm{6}\:\mathrm{7}\:\mathrm{8}\right) \\ $$$$ \\ $$$$\left.\mathrm{2}\right)\:\mathrm{prove}\:\mathrm{that}\:\mathrm{any}\:\mathrm{permutation} \\ $$$$\theta:\mathrm{S}\rightarrow\mathrm{S}\:\mathrm{where}\:\mathrm{S}\:\mathrm{is}\:\mathrm{a}\:\mathrm{finite}\:\mathrm{set}\:\mathrm{can}\:\mathrm{be} \\ $$$$\mathrm{written}\:\mathrm{as}\:\mathrm{a}\:\mathrm{product}\:\mathrm{of}\:\mathrm{disjoint} \\ $$$$\mathrm{cycle} \\ $$$$ \\…
Question Number 192340 by Mastermind last updated on 15/May/23 $$\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{order}\:\mathrm{of}\:\mathrm{any}\:\mathrm{permuta}− \\ $$$$\mathrm{tion}\:\theta\:\mathrm{is}\:\mathrm{the}\:\mathrm{least}\:\mathrm{common}\:\mathrm{multiple} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{length}\:\mathrm{of}\:\mathrm{its}\:\mathrm{disjoint}\:\mathrm{cycles}. \\ $$$$ \\ $$$$\:\mathrm{hi} \\ $$ Answered by aleks041103 last updated…
Question Number 192342 by Mastermind last updated on 15/May/23 $$\left.\mathrm{1}\right)\:\mathrm{Compute}\:\mathrm{in}\:\mathrm{S}_{\mathrm{a}} \:,\:\mathrm{a}^{−\mathrm{1}} \mathrm{ba}\:\:\mathrm{where}\: \\ $$$$\mathrm{a}=\left(\mathrm{1}\:\mathrm{2}\right)\left(\mathrm{1}\:\mathrm{3}\:\mathrm{5}\right),\:\mathrm{b}=\left(\mathrm{1}\:\mathrm{5}\:\mathrm{7}\:\mathrm{1}\right) \\ $$$$ \\ $$$$\left.\mathrm{2}\right)\:\mathrm{Given}\:\mathrm{permutation}\:\alpha\:=\:\left(\mathrm{1}\:\mathrm{2}\right)\left(\mathrm{3}\:\mathrm{4}\right), \\ $$$$\beta\:=\:\left(\mathrm{1}\:\mathrm{3}\right)\left(\mathrm{5}\:\mathrm{6}\right).\:\mathrm{Find}\:\mathrm{a}\:\mathrm{permutation} \\ $$$$\mathrm{x}\in\mathrm{S}_{\mathrm{6}} \:\exists\alpha\mathrm{x}\:=\:\beta. \\ $$$$…
Question Number 192336 by Mastermind last updated on 14/May/23 Answered by Rajpurohith last updated on 27/May/23 $${Since}\:\boldsymbol{{S}}\:{is}\:{bounded}\:,{inf}\left(\boldsymbol{{S}}\right)\:{and}\:{sup}\left(\boldsymbol{{S}}\right)\:{exist}\:{and}\:\lambda\in\mathbb{R}. \\ $$$$\left({a}\right)\forall{s}\in\boldsymbol{{S}}\:,\:{inf}\left({S}\right)\leqslant{s} \\ $$$$\Rightarrow\:\:\forall{s}\in\boldsymbol{{S}}\:,\:{inf}\left({S}\right)+\lambda\leqslant{s}+\lambda \\ $$$$\Rightarrow{inf}\left(\boldsymbol{{S}}\right)+\lambda\:{is}\:{a}\:{lower}\:{bound}\:{of}\:\boldsymbol{{S}}+\lambda. \\ $$$${if}\:\:{inf}\left(\boldsymbol{{S}}\right)+\lambda<{t}\:\left({be}\:{a}\:{lower}\:{bound}\:{of}\:\boldsymbol{{S}}+\lambda\right)…
Question Number 192339 by Mastermind last updated on 15/May/23 $$\mathrm{Express}\:\mathrm{as}\:\mathrm{the}\:\mathrm{product}\:\mathrm{of}\:\mathrm{disjoint}\: \\ $$$$\mathrm{cycle}\:\mathrm{the}\:\mathrm{permutation} \\ $$$$\left.\mathrm{a}\right)\:\theta\left(\mathrm{1}\right)=\mathrm{4}\:\:\theta\left(\mathrm{2}\right)=\mathrm{6}\:\:\theta\left(\mathrm{1}\right)=\mathrm{5}\:\:\theta\left(\mathrm{4}\right)=\mathrm{1} \\ $$$$\theta\left(\mathrm{5}\right)=\mathrm{3}\:\:\theta\left(\mathrm{6}\right)=\mathrm{2} \\ $$$$ \\ $$$$\left.\mathrm{b}\right)\:\left(\mathrm{1}\:\mathrm{6}\:\mathrm{3}\right)\left(\mathrm{1}\:\mathrm{3}\:\mathrm{5}\:\mathrm{7}\right)\left(\mathrm{6}\:\mathrm{7}\right)\left(\mathrm{1}\:\mathrm{2}\:\mathrm{3}\:\mathrm{4}\:\mathrm{5}\right) \\ $$$$ \\ $$$$\left.\mathrm{c}\right)\:\left(\mathrm{1}\:\mathrm{2}\:\mathrm{3}\:\mathrm{4}\:\mathrm{5}\right)\left(\mathrm{6}\:\mathrm{7}\right)\left(\mathrm{1}\:\mathrm{3}\:\mathrm{5}\:\mathrm{7}\right) \\…
Question Number 126780 by TITA last updated on 24/Dec/20 Commented by TITA last updated on 24/Dec/20 $$\mathrm{pleaae}\:\mathrm{help} \\ $$ Answered by physicstutes last updated on…