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((−1))^(1/i)

$$\sqrt[{{i}}]{−\mathrm{1}} \\ $$

Question Number 225676 Answers: 0 Comments: 0

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If a(x)=1 and W_(n=−1) ^(Qw_(fr.) ((1/(x−1))×x)) a′′(ust^n x)=0; What value of Tk(x^2 )? (This is nonstandartmath exercise)

$${If}\:{a}\left({x}\right)=\mathrm{1}\:{and}\:\underset{{n}=−\mathrm{1}} {\overset{{Qw}_{{fr}.} \left(\frac{\mathrm{1}}{{x}−\mathrm{1}}×{x}\right)} {{W}}a}''\left({ust}^{{n}} {x}\right)=\mathrm{0}; \\ $$$${What}\:{value}\:{of}\:{Tk}\left({x}^{\mathrm{2}} \right)? \\ $$$$\left({This}\:{is}\:{nonstandartmath}\:{exercise}\right) \\ $$

Question Number 225661 Answers: 1 Comments: 5

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Question Number 225652 Answers: 2 Comments: 0

∫_0 ^(π/2) e^(iπx) dx

$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {e}^{{i}\pi{x}} \:{dx} \\ $$

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Question Number 225625 Answers: 0 Comments: 4

quick short Q 2 things are mixed 1)both same volume 2)both same mass density⇒d_v and d_m d_v ? d_m [=,< or>]

$${quick}\:{short}\:{Q} \\ $$$$\mathrm{2}\:{things}\:{are}\:{mixed}\: \\ $$$$\left.\mathrm{1}\right){both}\:{same}\:{volume} \\ $$$$\left.\mathrm{2}\right){both}\:{same}\:{mass} \\ $$$${density}\Rightarrow{d}_{{v}} \:{and}\:{d}_{{m}} \\ $$$${d}_{{v}} \:?\:{d}_{{m}} \left[=,<\:{or}>\right] \\ $$

Question Number 225613 Answers: 7 Comments: 0

Question Number 225610 Answers: 0 Comments: 0

prove Gauss curvature K is intrinsic by showing K=(( determinant (((−(1/2)E_(vv) +F_(uv) −G_(uu) ),((1/2)E_u ),(F_u −(1/2)E_v )),(( F_v −(1/2)G_u ),( E),( F)),(( (1/2)G_v ),( F),( G)))− determinant ((( 0),((1/2)E_v ),((1/2)G_u )),(((1/2)E_v ),( E),( F)),(((1/2)G_v ),( F),( G))))/((EG−F^( 2) )^2 )) E,F,G is First Fundametal form of metric tensor.

$$\mathrm{prove} \\ $$$$\mathrm{Gauss}\:\mathrm{curvature}\:{K}\:\mathrm{is}\:\mathrm{intrinsic}\:\mathrm{by}\:\mathrm{showing} \\ $$$${K}=\frac{\begin{vmatrix}{−\frac{\mathrm{1}}{\mathrm{2}}{E}_{{vv}} +{F}_{{uv}} −{G}_{{uu}} }&{\frac{\mathrm{1}}{\mathrm{2}}{E}_{{u}} }&{{F}_{{u}} −\frac{\mathrm{1}}{\mathrm{2}}{E}_{{v}} }\\{\:\:\:\:\:\:\:\:\:\:\:\:{F}_{{v}} −\frac{\mathrm{1}}{\mathrm{2}}{G}_{{u}} }&{\:\:\:{E}}&{\:\:\:\:\:\:{F}}\\{\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{1}}{\mathrm{2}}{G}_{{v}} }&{\:\:\:{F}}&{\:\:\:\:\:{G}}\end{vmatrix}−\begin{vmatrix}{\:\:\:\:\mathrm{0}}&{\frac{\mathrm{1}}{\mathrm{2}}{E}_{{v}} }&{\frac{\mathrm{1}}{\mathrm{2}}{G}_{{u}} }\\{\frac{\mathrm{1}}{\mathrm{2}}{E}_{{v}} }&{\:\:\:\:{E}}&{\:\:\:\:{F}}\\{\frac{\mathrm{1}}{\mathrm{2}}{G}_{{v}} }&{\:\:\:\:\:{F}}&{\:\:\:{G}}\end{vmatrix}}{\left({EG}−{F}^{\:\mathrm{2}} \right)^{\mathrm{2}} } \\ $$$${E},{F},{G}\:\mathrm{is}\:\mathrm{First}\:\mathrm{Fundametal}\:\mathrm{form}\:\mathrm{of}\:\mathrm{metric}\:\mathrm{tensor}. \\ $$

Question Number 225604 Answers: 3 Comments: 2

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