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Question Number 221417 Answers: 0 Comments: 0

Let S be delimited by the equations x=0; y=0 ; z=0 and x+y+z=0 Find the flux of vector field V(x,y,z)=(x,y,x^2 +y^2 ) through S

$${Let}\:\:{S}\:{be}\:{delimited}\:{by}\:{the}\:{equations}\: \\ $$$${x}=\mathrm{0};\:{y}=\mathrm{0}\:;\:{z}=\mathrm{0}\:{and}\:{x}+{y}+{z}=\mathrm{0} \\ $$$${Find}\:{the}\:{flux}\:{of}\:{vector}\:{field}\: \\ $$$${V}\left({x},{y},{z}\right)=\left({x},{y},{x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right)\:{through}\:{S} \\ $$

Question Number 221416 Answers: 0 Comments: 0

ex3. prove f^((n)) (α)=((n!)/(2πi)) ∮_( ∂S) ((f(z))/((z−α)^(n+1) )) dz ex4. Let z_0 be any point interior to a positively oriented simple closed contour C show that a. ∮_C (dz/(z−z_0 ))=2πi b. ∮_( C) (dz/((z−z_0 )^(n+1) ))=0 , n∈R^+ ex 5. Let C be any simple closed contour, described in the positive sense in the z plane and write g(z)= ∮_( C) ((s^3 +2s)/((s−z)^3 )) ds show that g(z)=6πi when z is inside C and that g(z)=0 when z is outside

$$\mathrm{ex3}. \\ $$$$\mathrm{prove} \\ $$$${f}^{\left({n}\right)} \left(\alpha\right)=\frac{{n}!}{\mathrm{2}\pi\boldsymbol{{i}}}\:\oint_{\:\partial{S}} \:\frac{{f}\left({z}\right)}{\left({z}−\alpha\right)^{{n}+\mathrm{1}} }\:\mathrm{d}{z} \\ $$$$\mathrm{ex4}. \\ $$$$\mathrm{Let}\:{z}_{\mathrm{0}} \:\mathrm{be}\:\mathrm{any}\:\mathrm{point}\:\mathrm{interior}\:\mathrm{to}\:\mathrm{a}\:\mathrm{positively} \\ $$$$\mathrm{oriented}\:\mathrm{simple}\:\mathrm{closed}\:\mathrm{contour}\:\mathcal{C} \\ $$$$\mathrm{show}\:\mathrm{that} \\ $$$${a}.\:\oint_{{C}} \:\frac{\mathrm{d}{z}}{{z}−{z}_{\mathrm{0}} }=\mathrm{2}\pi\boldsymbol{{i}} \\ $$$$\mathrm{b}.\:\oint_{\:{C}} \frac{\mathrm{d}{z}}{\left({z}−{z}_{\mathrm{0}} \right)^{{n}+\mathrm{1}} }=\mathrm{0}\:,\:{n}\in\mathbb{R}^{+} \\ $$$$\mathrm{ex}\:\mathrm{5}. \\ $$$$\mathrm{Let}\:\mathcal{C}\:\mathrm{be}\:\mathrm{any}\:\mathrm{simple}\:\mathrm{closed}\:\mathrm{contour}, \\ $$$$\mathrm{described}\:\mathrm{in}\:\mathrm{the}\:\mathrm{positive}\:\mathrm{sense}\:\mathrm{in}\:\mathrm{the}\:{z}\:\mathrm{plane} \\ $$$$\mathrm{and}\:\mathrm{write}\: \\ $$$$\mathrm{g}\left({z}\right)=\:\oint_{\:\mathcal{C}} \:\frac{{s}^{\mathrm{3}} +\mathrm{2}{s}}{\left({s}−{z}\right)^{\mathrm{3}} }\:\mathrm{d}{s} \\ $$$$\mathrm{show}\:\mathrm{that}\:\mathrm{g}\left({z}\right)=\mathrm{6}\pi\boldsymbol{{i}}\:\mathrm{when}\:{z}\:\mathrm{is}\:\mathrm{inside}\:\mathcal{C}\:\mathrm{and} \\ $$$$\mathrm{that}\:\mathrm{g}\left({z}\right)=\mathrm{0}\:\mathrm{when}\:{z}\:\mathrm{is}\:\mathrm{outside} \\ $$

Question Number 221415 Answers: 1 Comments: 0

∫ dz [−(1/π)((2/z))^ν ∙Σ_(k=0) ^(ν−1) ((Γ(ν−k))/(k!))((z/2))^(2k) +(2/π)ln((1/2)z)J_ν (z)−(1/π)((z/2))^ν Σ_(k=0) ^∞ (((−1)^k (ψ^((0)) (k+ν+1)+ψ^((0)) (k+1)))/(k!(k+ν)!))((z/2))^(2k) ]

$$\int\:\:\mathrm{d}{z}\:\left[−\frac{\mathrm{1}}{\pi}\left(\frac{\mathrm{2}}{{z}}\right)^{\nu} \centerdot\underset{{k}=\mathrm{0}} {\overset{\nu−\mathrm{1}} {\sum}}\:\frac{\Gamma\left(\nu−{k}\right)}{{k}!}\left(\frac{{z}}{\mathrm{2}}\right)^{\mathrm{2}{k}} +\frac{\mathrm{2}}{\pi}\mathrm{ln}\left(\frac{\mathrm{1}}{\mathrm{2}}{z}\right){J}_{\nu} \left({z}\right)−\frac{\mathrm{1}}{\pi}\left(\frac{{z}}{\mathrm{2}}\right)^{\nu} \underset{{k}=\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{\left(−\mathrm{1}\right)^{{k}} \left(\psi^{\left(\mathrm{0}\right)} \left({k}+\nu+\mathrm{1}\right)+\psi^{\left(\mathrm{0}\right)} \left({k}+\mathrm{1}\right)\right)}{{k}!\left({k}+\nu\right)!}\left(\frac{{z}}{\mathrm{2}}\right)^{\mathrm{2}{k}} \right] \\ $$

Question Number 221413 Answers: 7 Comments: 0

Question Number 221411 Answers: 1 Comments: 0

Question Number 221407 Answers: 0 Comments: 0

A and B are two angles such that 0^0 <B<A<90^0 then prove geometrycaly that cos (A+B)=cos Acos B−sin Asin B

$${A}\:{and}\:{B}\:{are}\:{two}\:{angles}\:{such}\:{that}\:\mathrm{0}^{\mathrm{0}} <{B}<{A}<\mathrm{90}^{\mathrm{0}} \:{then}\:{prove}\:{geometrycaly}\:{that}\: \\ $$$$\mathrm{cos}\:\left({A}+{B}\right)=\mathrm{cos}\:{A}\mathrm{cos}\:{B}−\mathrm{sin}\:{A}\mathrm{sin}\:{B} \\ $$

Question Number 221406 Answers: 1 Comments: 1

Question Number 221405 Answers: 0 Comments: 0

Question Number 221404 Answers: 0 Comments: 0

Question Number 221400 Answers: 1 Comments: 0

∫∫∫_(0≤x≤y≤z≤1) [(y − x)^2 (z − y)^2 (z − x)^2 ] dxdydz

$$ \\ $$$$\:\:\:\:\:\:\:\int\int\int_{\mathrm{0}\leqslant{x}\leqslant{y}\leqslant{z}\leqslant\mathrm{1}} \:\left[\left({y}\:−\:{x}\right)^{\mathrm{2}} \left({z}\:−\:{y}\right)^{\mathrm{2}} \left({z}\:−\:{x}\right)^{\mathrm{2}} \right]\:{dxdydz}\:\:\:\:\:\:\: \\ $$$$ \\ $$

Question Number 221399 Answers: 0 Comments: 0

let a,b ≥ 0 and a + b + ab = 3 show that; ((38)/(55)) ≤ (1/(a^2 + 2)) + (1/(b^2 +2)) + (1/(a^2 + b^2 + 1)) ≤ 1 , ((463)/(812)) ≤ (1/(a^3 + 2)) + (1/(b^3 + 2)) + (1/(a^3 + b^3 + 1)) ≤ 1 , ((193)/(308)) ≤ (1/(a^2 + 2)) + (1/(b^3 + 2)) + (1/(a^3 + b^2 + 1)) ≤1 , and ((463)/(812)) ≤ (1/(a^2 + 2)) + (1/(b^3 + 2)) + (1/(a^2 + b^3 + 1)) ≤ ((11)/(10))

$$ \\ $$$$\:\:\:\:\:\boldsymbol{\mathrm{let}}\:\boldsymbol{{a}},\boldsymbol{{b}}\:\geqslant\:\mathrm{0}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{{a}}\:+\:\boldsymbol{{b}}\:+\:\boldsymbol{{ab}}\:=\:\mathrm{3} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{show}}\:\boldsymbol{\mathrm{that}}; \\ $$$$\:\:\frac{\mathrm{38}}{\mathrm{55}}\:\leqslant\:\frac{\mathrm{1}}{\boldsymbol{{a}}^{\mathrm{2}} \:+\:\mathrm{2}}\:+\:\frac{\mathrm{1}}{\boldsymbol{{b}}^{\mathrm{2}} \:+\mathrm{2}}\:+\:\frac{\mathrm{1}}{\boldsymbol{{a}}^{\mathrm{2}} \:+\:\boldsymbol{{b}}^{\mathrm{2}} \:+\:\mathrm{1}}\:\leqslant\:\mathrm{1}\:\:,\:\:\:\:\:\:\: \\ $$$$\:\:\frac{\mathrm{463}}{\mathrm{812}}\:\leqslant\:\frac{\mathrm{1}}{\boldsymbol{{a}}^{\mathrm{3}} \:+\:\mathrm{2}}\:+\:\frac{\mathrm{1}}{\boldsymbol{{b}}^{\mathrm{3}} \:+\:\mathrm{2}}\:+\:\frac{\mathrm{1}}{\boldsymbol{{a}}^{\mathrm{3}} \:+\:\boldsymbol{{b}}^{\mathrm{3}} \:+\:\mathrm{1}}\:\leqslant\:\mathrm{1}\:\:\:,\:\:\:\:\:\:\:\: \\ $$$$\:\:\frac{\mathrm{193}}{\mathrm{308}}\:\leqslant\:\frac{\mathrm{1}}{\boldsymbol{{a}}^{\mathrm{2}} \:+\:\mathrm{2}}\:+\:\frac{\mathrm{1}}{\boldsymbol{{b}}^{\mathrm{3}} \:+\:\mathrm{2}}\:+\:\frac{\mathrm{1}}{\boldsymbol{{a}}^{\mathrm{3}} \:+\:\boldsymbol{{b}}^{\mathrm{2}} \:+\:\mathrm{1}}\:\:\:\leqslant\mathrm{1}\:\:,\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{and}} \\ $$$$\:\:\frac{\mathrm{463}}{\mathrm{812}}\:\leqslant\:\frac{\mathrm{1}}{\boldsymbol{{a}}^{\mathrm{2}} \:+\:\mathrm{2}}\:+\:\frac{\mathrm{1}}{\boldsymbol{{b}}^{\mathrm{3}} \:+\:\mathrm{2}}\:+\:\frac{\mathrm{1}}{\boldsymbol{{a}}^{\mathrm{2}} \:+\:\boldsymbol{{b}}^{\mathrm{3}} \:+\:\mathrm{1}}\:\leqslant\:\frac{\mathrm{11}}{\mathrm{10}}\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\: \\ $$

Question Number 221397 Answers: 0 Comments: 1

Question Number 221393 Answers: 0 Comments: 0

a, b, c are complex number and ∣a∣ = ∣b∣=∣c∣= 1 and (a^2 /(bc))+(b^2 /(ac)) +(c^2 /(ab)) = −1 where ∣.∣ is modules function then ∣a+b+c∣ can be (A) 0 (B) 1 (C) (3/2) (D) 2

$$\:\:\:\:\:{a},\:{b},\:{c}\:{are}\:{complex}\:{number}\:{and}\: \\ $$$$\:\:\:\mid{a}\mid\:=\:\mid{b}\mid=\mid{c}\mid=\:\mathrm{1}\:{and}\:\:\frac{{a}^{\mathrm{2}} }{{bc}}+\frac{{b}^{\mathrm{2}} }{{ac}}\:+\frac{{c}^{\mathrm{2}} }{{ab}}\:=\:−\mathrm{1} \\ $$$$\:\:\:\:{where}\:\mid.\mid\:{is}\:\:{modules}\:{function} \\ $$$${then}\:\mid{a}+{b}+{c}\mid\:{can}\:{be}\: \\ $$$$\left({A}\right)\:\mathrm{0}\:\:\:\:\:\:\:\left({B}\right)\:\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\left({C}\right)\:\frac{\mathrm{3}}{\mathrm{2}}\:\:\:\:\:\:\:\:\:\:\left({D}\right)\:\mathrm{2} \\ $$

Question Number 221392 Answers: 1 Comments: 0

lim_(x→3) (√(x−3))=? 1) 0 2) 3 3) Does not exist 4) Undefined

$$\underset{{x}\rightarrow\mathrm{3}} {\mathrm{lim}}\sqrt{{x}−\mathrm{3}}=? \\ $$$$\left.\mathrm{1}\right)\:\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:\mathrm{3} \\ $$$$\left.\mathrm{3}\right)\:{Does}\:{not}\:{exist} \\ $$$$\left.\mathrm{4}\right)\:{Undefined} \\ $$

Question Number 221391 Answers: 0 Comments: 0

∫_0 ^( ∞) J_ν ^((1)) (t)Y_ν (t)sin(t)dt−∫_0 ^( ∞) J_ν (t)Y_ν ^((1)) (t)sin(t)dt=?? J_ν (t) is ν th Bessel function first Kind Y_ν (t) is ν th Bessel function second Kind sin(t) is sine function

$$\int_{\mathrm{0}} ^{\:\infty} {J}_{\nu} ^{\left(\mathrm{1}\right)} \left({t}\right){Y}_{\nu} \left({t}\right)\mathrm{sin}\left({t}\right)\mathrm{d}{t}−\int_{\mathrm{0}} ^{\:\infty} {J}_{\nu} \left({t}\right){Y}_{\nu} ^{\left(\mathrm{1}\right)} \left({t}\right)\mathrm{sin}\left({t}\right)\mathrm{d}{t}=?? \\ $$$${J}_{\nu} \left({t}\right)\:\mathrm{is}\:\nu\:\mathrm{th}\:\mathrm{Bessel}\:\mathrm{function}\:\mathrm{first}\:\mathrm{Kind} \\ $$$${Y}_{\nu} \left({t}\right)\:\mathrm{is}\:\nu\:\mathrm{th}\:\mathrm{Bessel}\:\mathrm{function}\:\mathrm{second}\:\mathrm{Kind} \\ $$$$\mathrm{sin}\left({t}\right)\:\mathrm{is}\:\mathrm{sine}\:\mathrm{function} \\ $$

Question Number 221388 Answers: 0 Comments: 0

Question Number 221387 Answers: 1 Comments: 0

Σ_(k = 1) ^∞ (2Σ_(n = 1) ^∞ (1/(n^2 + kn)))^2 = ?

$$ \\ $$$$\:\:\:\:\:\:\:\underset{{k}\:=\:\mathrm{1}} {\overset{\infty} {\sum}}\left(\mathrm{2}\underset{{n}\:=\:\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}^{\mathrm{2}} \:+\:{kn}}\right)^{\mathrm{2}} \:=\:? \\ $$$$ \\ $$

Question Number 221382 Answers: 0 Comments: 1

Σ_(k=0) ^n ((n),(k) )^(−1)

$$\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\begin{pmatrix}{{n}}\\{{k}}\end{pmatrix}^{−\mathrm{1}} \\ $$

Question Number 221380 Answers: 1 Comments: 0

Problem 3.11 Find the momentum space wave function 𝚿(p,t) for a particle in the ground state of the harmoic oscillator. What is the probability (to two signficant digits)that a measurement of on a particle in this state would yield value outside the classical range(for the samenergy) Hint Look in a math table under Normal Distribution Error Function for the numerical partor use Mathematica

$$ \\ $$$$\mathrm{Problem}\:\mathrm{3}.\mathrm{11}\:\mathrm{Find}\:\mathrm{the}\:\mathrm{momentum}\:\mathrm{space}\:\mathrm{wave}\: \\ $$$$\mathrm{function}\:\boldsymbol{\Psi}\left({p},{t}\right)\:\mathrm{for}\:\mathrm{a}\:\mathrm{particle}\:\mathrm{in}\:\mathrm{the}\:\mathrm{ground}\:\mathrm{state}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{harmoic}\:\mathrm{oscillator}.\:\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{probability} \\ $$$$\left(\mathrm{to}\:\mathrm{two}\:\mathrm{signficant}\:\mathrm{digits}\right)\mathrm{that}\:\mathrm{a}\:\mathrm{measurement}\:\mathrm{of}\:\mathrm{on}\:\mathrm{a}\:\mathrm{particle}\: \\ $$$$\mathrm{in}\:\mathrm{this}\:\mathrm{state}\:\mathrm{would}\:\mathrm{yield}\:\mathrm{value}\:\mathrm{outside}\:\mathrm{the}\: \\ $$$$\mathrm{classical}\:\mathrm{range}\left(\mathrm{for}\:\mathrm{the}\:\mathrm{samenergy}\right) \\ $$$$\mathrm{Hint}\:\mathrm{Look}\:\mathrm{in}\:\mathrm{a}\:\mathrm{math}\:\mathrm{table}\:\mathrm{under}\:\mathrm{Normal}\:\mathrm{Distribution} \\ $$$$\mathrm{Error}\:\mathrm{Function}\:\mathrm{for}\:\mathrm{the}\:\mathrm{numerical}\:\mathrm{partor}\:\mathrm{use}\:\mathrm{Mathematica} \\ $$

Question Number 221377 Answers: 2 Comments: 0

Find: Ω =lim_(n→∞) (2 ((10))^(1/n) − 1)^n = ?

$$\mathrm{Find}:\:\:\:\Omega\:=\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\:\left(\mathrm{2}\:\sqrt[{\boldsymbol{\mathrm{n}}}]{\mathrm{10}}\:−\:\mathrm{1}\right)^{\boldsymbol{\mathrm{n}}} \:=\:? \\ $$

Question Number 221373 Answers: 0 Comments: 1