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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

Question Number 224763    Answers: 1   Comments: 3

Question Number 224760    Answers: 1   Comments: 0

log _5 (5^(1/x) +125)=log _5 6+1+(1/(2x)) x=??

$$\mathrm{log}\:_{\mathrm{5}} \left(\mathrm{5}^{\frac{\mathrm{1}}{{x}}} +\mathrm{125}\right)=\mathrm{log}\:_{\mathrm{5}} \mathrm{6}+\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}{x}} \\ $$$${x}=?? \\ $$

Question Number 224753    Answers: 1   Comments: 2

Question Number 224739    Answers: 1   Comments: 0

z = r (cos θ + i sin θ), find (z/z^− ) +(z^− /z).

$$\mathrm{z}\:=\:\mathrm{r}\:\left(\mathrm{cos}\:\theta\:+\:\mathrm{i}\:\mathrm{sin}\:\theta\right),\:\mathrm{find}\:\frac{\mathrm{z}}{\overset{−} {\mathrm{z}}}\:+\frac{\overset{−} {\mathrm{z}}}{\mathrm{z}}. \\ $$

Question Number 224735    Answers: 1   Comments: 0

Let u=((y^2 −x^2 )/(x^2 y^2 )), v=((z^2 −y^2 )/(y^2 z^2 )) for x≠0,y≠0z≠0. Let w=f(u,v), where f is a real valued function defined on R^2 having continuous first order partial derivatives. the value of x^3 (∂w/∂x)+y^3 (∂w/∂y)+z^3 (∂w/∂z) at point (1,2,3) is

$${Let}\:{u}=\frac{{y}^{\mathrm{2}} −{x}^{\mathrm{2}} }{{x}^{\mathrm{2}} {y}^{\mathrm{2}} },\:{v}=\frac{{z}^{\mathrm{2}} −{y}^{\mathrm{2}} }{{y}^{\mathrm{2}} {z}^{\mathrm{2}} }\:{for}\:{x}\neq\mathrm{0},{y}\neq\mathrm{0}{z}\neq\mathrm{0}. \\ $$$${Let}\:{w}={f}\left({u},{v}\right),\:{where}\:{f}\:{is}\:{a}\:{real} \\ $$$${valued}\:{function}\:{defined}\:{on}\:{R}^{\mathrm{2}} \\ $$$${having}\:{continuous}\:{first}\:{order} \\ $$$${partial}\:{derivatives}. \\ $$$${the}\:{value}\:{of} \\ $$$${x}^{\mathrm{3}} \:\frac{\partial{w}}{\partial{x}}+{y}^{\mathrm{3}} \:\frac{\partial{w}}{\partial{y}}+{z}^{\mathrm{3}} \:\frac{\partial{w}}{\partial{z}}\:{at}\:{point}\:\left(\mathrm{1},\mathrm{2},\mathrm{3}\right)\:{is} \\ $$

Question Number 224733    Answers: 1   Comments: 0

Let f be a continuously differentiable function such that ∫_0 ^(2x^2 ) f(t)dt=e^(cos x^2 ) for all x∈(0,∞) the value of f ′(π)=?

$${Let}\:{f}\:{be}\:{a}\:{continuously}\:{differentiable}\:{function} \\ $$$${such}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{2}{x}^{\mathrm{2}} } {f}\left({t}\right){dt}={e}^{\mathrm{cos}\:{x}^{\mathrm{2}} } \:{for}\:{all}\:{x}\in\left(\mathrm{0},\infty\right) \\ $$$${the}\:{value}\:{of}\:{f}\:'\left(\pi\right)=? \\ $$

Question Number 224732    Answers: 0   Comments: 0

The value of n for which the divergence of the function F=(r/ determinant ((r))^n ), r=xi^ +yj^ +zk^ , determinant ((r))≠0, vanishes is a)1 b)−1 c)3 d)−3 p=38

$${The}\:{value}\:{of}\:{n}\:{for}\:{which}\:{the}\:{divergence} \\ $$$${of}\:{the}\:{function} \\ $$$$\mathrm{F}=\frac{\mathrm{r}}{\begin{vmatrix}{\mathrm{r}}\end{vmatrix}^{{n}} },\:\mathrm{r}=\mathrm{x}\hat {\mathrm{i}}+{y}\hat {\mathrm{j}}+{z}\hat {\mathrm{k}},\begin{vmatrix}{\mathrm{r}}\end{vmatrix}\neq\mathrm{0}, \\ $$$${vanishes}\:{is} \\ $$$$\left.{a}\right)\mathrm{1} \\ $$$$\left.{b}\right)−\mathrm{1} \\ $$$$\left.{c}\right)\mathrm{3} \\ $$$$\left.{d}\right)−\mathrm{3} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{p}=\mathrm{38} \\ $$

Question Number 224728    Answers: 2   Comments: 2

Question Number 224723    Answers: 0   Comments: 9

So here is what it shows.

$${So}\:{here}\:{is}\:{what}\:{it}\:{shows}. \\ $$

Question Number 224714    Answers: 0   Comments: 1

∫(1/( (√(tan θ)))) dθ

$$\int\frac{\mathrm{1}}{\:\sqrt{\mathrm{tan}\:\theta}}\:{d}\theta \\ $$

Question Number 224713    Answers: 1   Comments: 0

lim_(x→0) (((sin x)/x))^((x−3sin x)/x) .?

$$\:\: \underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{\mathrm{sin}\:\mathrm{x}}{\mathrm{x}}\right)^{\frac{\mathrm{x}−\mathrm{3sin}\:\mathrm{x}}{\mathrm{x}}} .? \\ $$$$\: \\ $$

Question Number 224709    Answers: 1   Comments: 0

Question Number 224702    Answers: 3   Comments: 1

Question Number 224692    Answers: 0   Comments: 10

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Question Number 224691    Answers: 1   Comments: 0

3k+4=n^2 . k,n ∈N Find all n numbers .

$$\mathrm{3}{k}+\mathrm{4}={n}^{\mathrm{2}} .\:{k},{n}\:\in\mathbb{N} \\ $$$${Find}\:{all}\:{n}\:{numbers}\:. \\ $$

Question Number 224688    Answers: 2   Comments: 0

A gun, kept on a straight horizontal road, is used to hit a car travelling along the same road away from it with a uniform speed of 72 km/h. The car is at a distance of 500 m from the gun, when the gun is fired at an angle of 45° with the horizontal. Find the distance of the car from the gun, when the shell hits it. g = 10m/s²

A gun, kept on a straight horizontal road, is used to hit a car travelling along the same road away from it with a uniform speed of 72 km/h. The car is at a distance of 500 m from the gun, when the gun is fired at an angle of 45° with the horizontal. Find the distance of the car from the gun, when the shell hits it. g = 10m/s²

Question Number 224678    Answers: 1   Comments: 8

Guys I just turned 15 today :)

$$\left.\mathrm{Guys}\:\mathrm{I}\:\mathrm{just}\:\mathrm{turned}\:\mathrm{15}\:\mathrm{today}\::\right) \\ $$

Question Number 224672    Answers: 1   Comments: 0

Question Number 224665    Answers: 0   Comments: 5

Question Number 224642    Answers: 1   Comments: 0

Question Number 224641    Answers: 1   Comments: 0

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