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Question Number 224892    Answers: 1   Comments: 1

lim_(n→0) ∫_0 ^1 e^x^2 sin (nx)dx=?

$$\underset{{n}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:{e}^{{x}^{\mathrm{2}} } \mathrm{sin}\:\left({nx}\right){dx}=? \\ $$

Question Number 224888    Answers: 0   Comments: 2

Question Number 224886    Answers: 1   Comments: 0

Find for acute θ, sin θ and cos θ in terms of 0<k<1, if ((sin θ(1−cos θ))/(cos θ(1−sin θ)))=k.

$${Find}\:{for}\:{acute}\:\theta,\:\mathrm{sin}\:\theta\:{and}\:\mathrm{cos}\:\theta \\ $$$${in}\:{terms}\:{of}\:\mathrm{0}<{k}<\mathrm{1}, \\ $$$${if}\:\:\:\:\frac{\mathrm{sin}\:\theta\left(\mathrm{1}−\mathrm{cos}\:\theta\right)}{\mathrm{cos}\:\theta\left(\mathrm{1}−\mathrm{sin}\:\theta\right)}={k}. \\ $$

Question Number 224883    Answers: 0   Comments: 0

∫ vol(g^ )=∫_( V) (√(det g_(μν) )) dx^1 ∧dx^2 ∧dx^3 parametric Surface S^→ (u,v,w);R^3 →R^3 S^→ (r,θ,ρ) { ((rsin(θ)cos(ρ))),((rsin(θ)sin(ρ))),((rcos(θ))) :} find metric tensor g_(μν) = ((g_(11) ,g_(12) ,g_(13) ),(g_(21) ,g_(22) ,g_(23) ),(g_(31) ,g_(32) ,g_(33) ) ) Describe it in the same as ds^2 =g_(μν) dx^μ dx^ν ds^2 =(dr dθ dρ) ((g_(11) ,g_(12) ,g_(13) ),(g_(21) ,g_(22) ,g_(23) ),(g_(31) ,g_(32) ,g_(33) ) ) ((dr),(dθ),(dρ) ) and find volume V=∫ vol(g)

$$\int\:\mathrm{vol}\left({g}^{\:} \right)=\int_{\:{V}} \:\sqrt{\mathrm{det}\:\boldsymbol{\mathrm{g}}_{\mu\nu} }\:\mathrm{d}{x}^{\mathrm{1}} \wedge\mathrm{d}{x}^{\mathrm{2}} \wedge\mathrm{d}{x}^{\mathrm{3}} \\ $$$$\mathrm{parametric}\:\mathrm{Surface}\: \\ $$$$\overset{\rightarrow} {\mathcal{S}}\left({u},{v},{w}\right);\mathbb{R}^{\mathrm{3}} \rightarrow\mathbb{R}^{\mathrm{3}} \\ $$$$\overset{\rightarrow} {\mathcal{S}}\left({r},\theta,\rho\right)\begin{cases}{{r}\mathrm{sin}\left(\theta\right)\mathrm{cos}\left(\rho\right)}\\{{r}\mathrm{sin}\left(\theta\right)\mathrm{sin}\left(\rho\right)}\\{{r}\mathrm{cos}\left(\theta\right)}\end{cases}\: \\ $$$$\mathrm{find}\:\mathrm{metric}\:\mathrm{tensor}\:\boldsymbol{\mathrm{g}}_{\mu\nu} =\begin{pmatrix}{\mathrm{g}_{\mathrm{11}} }&{\mathrm{g}_{\mathrm{12}} }&{\mathrm{g}_{\mathrm{13}} }\\{\mathrm{g}_{\mathrm{21}} }&{\mathrm{g}_{\mathrm{22}} }&{\mathrm{g}_{\mathrm{23}} }\\{\mathrm{g}_{\mathrm{31}} }&{\mathrm{g}_{\mathrm{32}} }&{\mathrm{g}_{\mathrm{33}} }\end{pmatrix} \\ $$$$\: \\ $$$$\mathrm{Describe}\:\mathrm{it}\:\mathrm{in}\:\mathrm{the}\:\mathrm{same}\:\mathrm{as}\:\mathrm{d}{s}^{\mathrm{2}} =\boldsymbol{\mathrm{g}}_{\mu\nu} \mathrm{d}{x}^{\mu} \mathrm{d}{x}^{\nu} \\ $$$$\mathrm{d}{s}^{\mathrm{2}} =\left(\mathrm{d}{r}\:\:\mathrm{d}\theta\:\:\mathrm{d}\rho\right)\begin{pmatrix}{\mathrm{g}_{\mathrm{11}} }&{\mathrm{g}_{\mathrm{12}} }&{\mathrm{g}_{\mathrm{13}} }\\{\mathrm{g}_{\mathrm{21}} }&{\mathrm{g}_{\mathrm{22}} }&{\mathrm{g}_{\mathrm{23}} }\\{\mathrm{g}_{\mathrm{31}} }&{\mathrm{g}_{\mathrm{32}} }&{\mathrm{g}_{\mathrm{33}} }\end{pmatrix}\begin{pmatrix}{\mathrm{d}{r}}\\{\mathrm{d}\theta}\\{\mathrm{d}\rho}\end{pmatrix} \\ $$$$\mathrm{and}\:\mathrm{find}\:\mathrm{volume}\:{V}=\int\:\:\mathrm{vol}\left(\mathrm{g}\right) \\ $$

Question Number 224879    Answers: 0   Comments: 0

prove lim_(n→∞) (1/(ln(p_n ))) Π_k (1/(1−(1/p_k )))=e^Υ_0 , Υ_0 =0.57721566490153286060..

$$\mathrm{prove} \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{1}}{\mathrm{ln}\left({p}_{{n}} \right)}\:\underset{{k}} {\prod}\:\frac{\mathrm{1}}{\mathrm{1}−\frac{\mathrm{1}}{{p}_{{k}} }}={e}^{\Upsilon_{\mathrm{0}} } \:,\: \\ $$$$\Upsilon_{\mathrm{0}} =\mathrm{0}.\mathrm{57721566490153286060}.. \\ $$

Question Number 224873    Answers: 0   Comments: 6

Question Number 224872    Answers: 0   Comments: 0

Question Number 224866    Answers: 0   Comments: 0

can you guys explan why metric tensor g_(μν) =0 → Riemann metric tensor R_(αγβ) ^δ =0

$$\mathrm{can}\:\mathrm{you}\:\mathrm{guys}\:\mathrm{explan}\:\mathrm{why} \\ $$$$\mathrm{metric}\:\mathrm{tensor}\:\mathrm{g}_{\mu\nu} =\mathrm{0}\:\:\rightarrow\:\mathrm{Riemann}\:\mathrm{metric}\:\mathrm{tensor}\:{R}_{\alpha\gamma\beta} ^{\delta} =\mathrm{0} \\ $$

Question Number 224861    Answers: 2   Comments: 0

∫((sin x)/( (√(1+sin x)))) dx

$$\int\frac{\mathrm{sin}\:{x}}{\:\sqrt{\mathrm{1}+\mathrm{sin}\:{x}}}\:{dx} \\ $$

Question Number 224859    Answers: 1   Comments: 1

A homogeneous rod AB of length L=1.8m 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 with speed V on point C, midway between the points O and B. Immediaty after falling, the insect moves towards the end B such that the rod rotates with a constant angular velocity ω. a)determine the angular velocity ω in terms of V and L. b)If the imsect reaches the end B when the rod turned through an angle 90^0 , determine V

$${A}\:{homogeneous}\:{rod}\:{AB}\:{of}\:{length} \\ $$$${L}=\mathrm{1}.\mathrm{8}{m}\:{and}\:{mass}\:{M}\:{is}\:{pivoted} \\ $$$${at}\:{the}\:{centre}\:{O}\:{in}\:{such}\:{a}\:{way}\:{that} \\ $$$${it}\:{can}\:{rotate}\:{freely}\:{in}\:{the}\:{vertical}\:{plane}. \\ $$$$ \\ $$$${The}\:{rod}\:{is}\:{initially}\:{in}\:{the}\:{horizontal} \\ $$$${position}.{An}\:{insect}\:{S}\:\:{of}\:{the}\: \\ $$$${same}\:{mass}\:{M}\:{falls}\:{vertically} \\ $$$${with}\:{speed}\:{V}\:{on}\:{point}\:{C},\:{midway} \\ $$$${between}\:{the}\:{points}\:{O}\:{and}\:{B}. \\ $$$${Immediaty}\:{after}\:{falling},\:{the} \\ $$$${insect}\:{moves}\:{towards}\:{the}\:{end}\:{B} \\ $$$${such}\:{that}\:{the}\:{rod}\:{rotates} \\ $$$${with}\:{a}\:{constant}\:{angular}\:{velocity}\:\omega. \\ $$$$\left.{a}\right){determine}\:{the}\:{angular}\:{velocity}\:\omega \\ $$$${in}\:{terms}\:{of}\:{V}\:{and}\:{L}. \\ $$$$\left.{b}\right){If}\:{the}\:{imsect}\:{reaches}\:{the}\:{end}\:{B} \\ $$$${when}\:{the}\:{rod}\:{turned}\:{through} \\ $$$${an}\:{angle}\:\mathrm{90}^{\mathrm{0}} ,\:{determine}\:{V} \\ $$

Question Number 224853    Answers: 1   Comments: 15

a piece of chalk rests on a horizontal board with μ=0.1 Suddenly the board starts to move horizontally at a speed of 2m per second and after a time τ it stops abruptly. find the length of the line drawn by the chalk on the board for folowing cases τ=5sec τ=1sec g=10m/s^2

$$ \\ $$$$\mathrm{a}\:\mathrm{piece}\:\mathrm{of}\:\mathrm{chalk}\:\mathrm{rests}\:\mathrm{on}\:\mathrm{a} \\ $$$$\mathrm{horizontal}\:\mathrm{board}\:\mathrm{with}\:\mu=\mathrm{0}.\mathrm{1} \\ $$$$\mathrm{Suddenly}\:\mathrm{the}\:\mathrm{board}\:\mathrm{starts}\:\mathrm{to} \\ $$$$\mathrm{move}\:\mathrm{horizontally}\:\mathrm{at}\:\mathrm{a}\:\mathrm{speed}\:\mathrm{of} \\ $$$$\mathrm{2m}\:\mathrm{per}\:\mathrm{second}\:\mathrm{and}\:\mathrm{after}\:\mathrm{a} \\ $$$$\mathrm{time}\:\tau\:\mathrm{it}\:\mathrm{stops}\:\mathrm{abruptly}.\:\mathrm{find}\: \\ $$$$\mathrm{the}\:\mathrm{length}\:\mathrm{of}\:\mathrm{the}\:\mathrm{line}\:\mathrm{drawn} \\ $$$$\mathrm{by}\:\mathrm{the}\:\mathrm{chalk}\:\mathrm{on}\:\mathrm{the}\:\mathrm{board}\:\mathrm{for} \\ $$$$\mathrm{folowing}\:\mathrm{cases} \\ $$$$\tau=\mathrm{5}{sec} \\ $$$$\tau=\mathrm{1}{sec} \\ $$$${g}=\mathrm{10}{m}/{s}^{\mathrm{2}} \\ $$

Question Number 224852    Answers: 2   Comments: 0

lim_(n→∞) ((1+(1/2)+(1/3)+(1/4)+...+(1/n))/(ln(n)))=?

$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{3}}+\frac{\mathrm{1}}{\mathrm{4}}+...+\frac{\mathrm{1}}{{n}}}{{ln}\left({n}\right)}=? \\ $$

Question Number 224848    Answers: 0   Comments: 1

Question Number 224847    Answers: 0   Comments: 0

Question Number 224839    Answers: 0   Comments: 0

∫_( 0) ^( 1) ((x tan^(− 1) (x) ln(1 − x))/(1 + x^2 )) dx

$$\int_{\:\mathrm{0}} ^{\:\mathrm{1}} \:\frac{\mathrm{x}\:\mathrm{tan}^{−\:\mathrm{1}} \left(\mathrm{x}\right)\:\mathrm{ln}\left(\mathrm{1}\:\:\:−\:\:\:\mathrm{x}\right)}{\mathrm{1}\:\:\:+\:\:\:\mathrm{x}^{\mathrm{2}} }\:\mathrm{dx} \\ $$

Question Number 224833    Answers: 3   Comments: 0

∫sec θ dθ

$$\int\mathrm{sec}\:\theta\:{d}\theta \\ $$

Question Number 224819    Answers: 2   Comments: 0

Question Number 224814    Answers: 1   Comments: 0

Question Number 224807    Answers: 1   Comments: 1

Question Number 224806    Answers: 0   Comments: 0

Use the Gauss Bonnet Theorem to show that the number of holes in a straw is 1. Then associate it and show that the Genus on the surface is 1.

$$\mathrm{Use}\:\mathrm{the}\:\mathrm{Gauss}\:\mathrm{Bonnet}\:\mathrm{Theorem}\:\mathrm{to}\:\mathrm{show}\:\mathrm{that}\:\mathrm{the} \\ $$$$\mathrm{number}\:\mathrm{of}\:\mathrm{holes}\:\mathrm{in}\:\mathrm{a}\:\mathrm{straw}\:\mathrm{is}\:\mathrm{1}. \\ $$$$\mathrm{Then}\:\mathrm{associate}\:\mathrm{it}\:\mathrm{and}\: \\ $$$$\mathrm{show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{Genus}\:\mathrm{on}\:\mathrm{the}\:\mathrm{surface}\:\mathrm{is}\:\mathrm{1}. \\ $$

Question Number 224798    Answers: 0   Comments: 0

∫_( 0) ^( 1) ((arctan^2 (x) ln(1 − x))/x^2 ) dx

$$\int_{\:\mathrm{0}} ^{\:\mathrm{1}} \:\frac{\mathrm{arctan}^{\mathrm{2}} \left(\mathrm{x}\right)\:\mathrm{ln}\left(\mathrm{1}\:\:\:−\:\:\:\mathrm{x}\right)}{\mathrm{x}^{\mathrm{2}} }\:\mathrm{dx} \\ $$

Question Number 224790    Answers: 3   Comments: 4

Question Number 224789    Answers: 0   Comments: 0

prove Sphere S;R^3 →R x^2 +y^2 +z^2 =R^2 , Euler characteristic 𝛘=2 by gauss-Bonnet theorem 2π𝛘(𝛀)=∫_( 𝛀) dA K Gauss curvature defined as K=((det Π)/(det I))=((LN−M^2 )/(EG−F^2 )) such that I=(dζ^1 dζ^2 ) ((E,F),(F,G) ) ((dζ^1 ),(dζ^2 ) )=Σ_(jk) (∂f/∂ζ^j )∙(∂f/∂ζ^k ) dζ^j dζ^k E=x_u ∗x_u ,F=x_u ∗x_v , G=x_v ∗x_v Π=(dζ^1 dζ^2 ) ((L,M),(M,N) ) ((dζ^1 ),(dζ^2 ) )=Σ_(ıȷ) n^ ∗(∂^2 f/(∂ζ^ı ∂ζ^ȷ )) dζ^ı dζ^ȷ n^ =((x_u ×x_v )/(∣∣x_u ×x_v ∣∣)) L=x_(uu) ∗n^ , M=x_(uv) ∗n^ , N=x_(vv) ∗n^

$$\mathrm{prove}\:\mathrm{Sphere}\:\mathcal{S};\mathbb{R}^{\mathrm{3}} \rightarrow\mathbb{R} \\ $$$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} ={R}^{\mathrm{2}} \:,\:\mathrm{Euler}\:\mathrm{characteristic}\:\boldsymbol{\chi}=\mathrm{2} \\ $$$$\mathrm{by}\:\mathrm{gauss}-\mathrm{Bonnet}\:\mathrm{theorem} \\ $$$$\mathrm{2}\pi\boldsymbol{\chi}\left(\boldsymbol{\Omega}\right)=\int_{\:\boldsymbol{\Omega}} \:\mathrm{d}{A}\:{K} \\ $$$$\mathrm{Gauss}\:\mathrm{curvature}\:\mathrm{defined}\:\mathrm{as}\:{K}=\frac{\mathrm{det}\:\Pi}{\mathrm{det}\:\mathrm{I}}=\frac{{LN}−{M}^{\mathrm{2}} }{{EG}−{F}^{\mathrm{2}} } \\ $$$$\mathrm{such}\:\mathrm{that}\: \\ $$$$\mathrm{I}=\left(\mathrm{d}\zeta^{\mathrm{1}} \:\mathrm{d}\zeta^{\mathrm{2}} \right)\begin{pmatrix}{{E}}&{{F}}\\{{F}}&{{G}}\end{pmatrix}\begin{pmatrix}{\mathrm{d}\zeta^{\mathrm{1}} }\\{\mathrm{d}\zeta^{\mathrm{2}} }\end{pmatrix}=\underset{{jk}} {\sum}\:\frac{\partial\boldsymbol{\mathrm{f}}}{\partial\zeta^{{j}} }\centerdot\frac{\partial\boldsymbol{\mathrm{f}}}{\partial\zeta^{{k}} }\:\mathrm{d}\zeta^{{j}} \mathrm{d}\zeta^{{k}} \\ $$$$\mathrm{E}={x}_{{u}} \ast{x}_{{u}} \:,\mathrm{F}={x}_{{u}} \ast{x}_{{v}} \:,\:{G}={x}_{{v}} \ast{x}_{{v}} \: \\ $$$$\Pi=\left(\mathrm{d}\zeta^{\mathrm{1}} \:\mathrm{d}\zeta^{\mathrm{2}} \right)\begin{pmatrix}{{L}}&{{M}}\\{{M}}&{{N}}\end{pmatrix}\begin{pmatrix}{\mathrm{d}\zeta^{\mathrm{1}} }\\{\mathrm{d}\zeta^{\mathrm{2}} }\end{pmatrix}=\underset{\imath\jmath} {\sum}\:\hat {\boldsymbol{\mathrm{n}}}\ast\frac{\partial^{\mathrm{2}} \boldsymbol{\mathrm{f}}}{\partial\zeta^{\imath} \partial\zeta^{\jmath} }\:\mathrm{d}\zeta^{\imath} \mathrm{d}\zeta^{\jmath} \\ $$$$\hat {\boldsymbol{\mathrm{n}}}=\frac{{x}_{{u}} ×{x}_{{v}} }{\mid\mid{x}_{{u}} ×{x}_{{v}} \mid\mid} \\ $$$${L}={x}_{{uu}} \ast\hat {\boldsymbol{\mathrm{n}}},\:{M}={x}_{{uv}} \ast\hat {\boldsymbol{\mathrm{n}}}\:,\:{N}={x}_{{vv}} \ast\hat {\boldsymbol{\mathrm{n}}} \\ $$

Question Number 224770    Answers: 0   Comments: 1

(2x^3 +x−3)^3 =3−x^2

$$\:\:\:\left(\mathrm{2}{x}^{\mathrm{3}} +{x}−\mathrm{3}\right)^{\mathrm{3}} =\mathrm{3}−{x}^{\mathrm{2}} \\ $$

Question Number 224771    Answers: 0   Comments: 0

K=∫_0 ^( ∞) ((sinx)/(coshx)) (e^(−2x) −e^(−4x) )dx=?

$$ \\ $$$$\:\:{K}=\int_{\mathrm{0}} ^{\:\infty} \frac{{sinx}}{{coshx}}\:\left({e}^{−\mathrm{2}{x}} −{e}^{−\mathrm{4}{x}} \right){dx}=?\:\:\:\:\: \\ $$$$\:\: \\ $$

Question Number 224773    Answers: 1   Comments: 0

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