Identity Definition and 1000 Threads

This is not a homework problem. It was stated in a textbook as trivial but I cannot prove it myself in general. If [A,B]=0 then [A,B^n] = 0 where n is a positive integer. This seems rather intuitive and I can easily see it to be true when I plug in n=2, n=3, n=4, etc. However, I cannot prove it...

I'm trying to proof an identity from Munkres' Topology A \ ( A \ B ) = B By definition A \ B = {x : x in A and x not in B} A \( A \ B) = A \ (A ∩ Bc) = A ∩ (A ∩ Bc)c = A ∩ (Ac ∪ B) = (A ∩ Ac) ∪ (A ∩ B) = ∅ ∪ (A ∩ B) = A ∩ B What did I miss?

Homework Statement Hi all, I'm having trouble working on the following problem. Any assistance will be greatly appreciated. Here, the capital letters stand for Position and Momentum operators while the ##x', p'## stand for eigenvalues.Homework EquationsThe Attempt at a Solution a) and b) It...

Hi there - it has been quite a long time since I took Calculus. I am trying to brush up and understand where to start with this question: Starting with the identity a^x = e^xlna, derive the relationships between (a) e^x and 10^x; (b) ln x and log x. Note: log x = log10 x unless otherwise...

I don't understand from Peskin when can I use Ward Identity? I mean I can see that this identity isn't always valid to use, but when it is? Take for example equation (16.10) page 508 of Peskin's and Schroeder's.

Homework Statement Show that ##\arcsin 2x \sqrt{1-x^2} = 2 \arccos{x}## when 1/√2 < x < 1 Homework Equations All trigonometric and inverse trigonometric identities, special usage of double angle identities here The Attempt at a Solution I can get the answer by puting x=cosy, the term inside...

Hi, For a particle in a box (so that the momentum spectrum is discrete), we can write the identity operator as a sum over all momentum eigenstates of a projection to that eigenstate: $$I=\displaystyle\sum\limits_{p} |p\rangle\langle p|.$$ I was wondering what the corresponding form of the...

What does Tan (Theta - Pie) = ? I know Tan (theta + pie) = tan (theta). They say the answer is tan (theta), but I think it's some kind of typo.

Homework Statement Given the spinors: \Psi_{1}=\frac{1}{\sqrt{2}}\left(\psi-\psi^{c}\right) \Psi_{2}=\frac{1}{\sqrt{2}}\left(\psi+\psi^{c}\right) Where c denotes charge conjugation, show that for a vector boson #A_{\mu}#; A_{\mu}\overline{\Psi_{1}}\gamma^{\mu}\Psi_{2} +...

Homework Statement Prove the following relation. It is assumed that all values of x and y which occur are such that the denominators in the indicated fractions are not equal to 0. $$\frac{x^n-1}{x-1}=x^{n-1}+x^{n-2}+...+x+1$$ Homework EquationsThe Attempt at a Solution Please see attached...

I've been reading Straumann's book "General Relativity & Relativistic Astrophysics". In it, he claims that the twice contracted Bianchi identity: $$\nabla_{\mu}G^{\mu\nu}=0$$ (where ##G^{\mu\nu}=R^{\mu\nu}-\frac{1}{2}g^{\mu\nu}R##) is a consequence of the diffeomorphism (diff) invariance of the...

Has some sense write in the thermodynamics identity the terms TdS and dU at the same side of the equation and with the same sign? what would be this sense?For example PdV=TdS+dU

Homework Statement Find the general solution of the second order DE. y'' + 9y = 0 Homework EquationsThe Attempt at a Solution Problem is straight forward I just don't get why my answer is different than the books. So you get m^2 + 9 = 0 m = 3i and m = -3i so the general solution...

I'm trying to get a more intuitive understanding of Euler's identity, more specifically, what raising e to the power of i means and why additionally raising by an angle in radians rotates the real value into the imaginary plane. I understand you can derive Euler's formula from the cosx, sinx and...

Homework Statement 8 sin3 θ – 6 sin θ + 1 = 0 The answer includes changing this to -2sin3θ+1=0 Homework Equations The double angle identities Sin2θ=sinθcosθ+cosθsinθ The Attempt at a Solution I do not know how to get started with this question

In my EM class, this vector identity for the angular momentum operator (without the ##i##) was stated without proof. Is there anywhere I can look to to actually find a good example/proof on how this works? This is in spherical coordinates, and I can't seem to find this vector identity anywhere...

In Quantum mechanics, when we have momentum operator ##\vec{p}##, and angular momentum operator ##\vec{L}##, then (\vec{p} \times \vec{L})\cdot \vec{p}=\vec{p}\cdot (\vec{L} \times \vec{p}) Why this relation is correct, and not (\vec{p} \times \vec{L})\cdot \vec{p}=\vec{p}\cdot (\vec{p} \times...

Prove the following identity ($n = 1,2,3,...$): \[\sqrt{n - \sqrt{n+\sqrt{n-\sqrt{n +...}}}} = \sqrt{(n-1)-\sqrt{(n-1)-\sqrt{(n-1)-...}}}\]

Prove the identity \[\sum_{j=1}^{n-1}\csc^2\left ( \frac{j\pi}{n} \right ) = \frac{n^2-1}{3 }.\]

Can some websites such as search engines know of our identity? Can they guess or learn our name, surname or personality characters etc? If so, is it better not to use them? Google is offering me results from websites which are used in searches i.e site: searches. This seems very strange to me...

Prove that $$\sum_{n=0}^\infty \frac{n^2}{n!}=2e.$$

Hi, I'm struggling to understand something. Does domain restriction work the same way for composition of inverse functions as it does for other composite functions? I would assume it does, but the end result seems counter-intuitive. For example: If I have the function f(x) = 1/(1+x), with...

I seem to remember this Algebra identity being covered in one of my classes years ago, but it has cropped back up in studying the relativistic doppler effect for light. Can anyone please show me the intermediate steps to show that: (1+x)/(sqrt(1-x^2) = sqrt((1+x)/(1-x)) or similarly...

I am searching for a shortcut in the calculation of a proof. The question is as follows: 2.12 Prove that: $$|z_1|+|z_2| = |\frac{z_1+z_2}{2}-u|+|\frac{z_1+z_2}{2}+u|$$ where $z_1,z_2$ are two complex numbers and $u=\sqrt{z_1z_2}$. I thought of showing that the squares of both sides of the...

I performed a laboratory experiment using a Dumas bulb to find the molar mass of an unknown, clear liquid in order to identify it. The Dumas bulb was submerged in a beaker filled with water (with the tip out of the water) and the water was boiled to evaporate the sample. I eventually got a...

Homework Statement I've been given the Bianchi identity in the form ##\nabla _{\kappa} R^{\mu}_{\nu\rho\sigma} + \nabla _{\rho} R^{\mu}_{\nu\sigma \kappa} + \nabla _{\sigma} R^{\mu}_{\nu\kappa\rho} =0##Homework EquationsThe Attempt at a Solution In order to get from this to the Einstein...

Homework Statement Homework Equations Thermodynamic Identity The Attempt at a Solution While I was able to work out the problem with the help of the hint, I couldn't completely understand the implication of said hint. The hint suggests that the equations for Chemical Potential in a process...

Homework Statement Homework Equations ##dS = \frac{1}{T} (dU - PdV)## assuming dN = 0 The Attempt at a Solution I have actually managed to solve all 4 parts correctly, except for the fact that I solved Part d) with the Sackur-Tetrode equation rather than the thermodynamic identity. I...

$\frac{1 -\cos A}{1 + \cos A} = (\cot A - \csc A)^2$

Suppose that L: ##S^1## ---> ##R## is a lift of the identity map of ##S^1##, where e is the covering map from ##R## to ##S^1##, where ##R## is the real numbers and ##S^1## is the circle. Then the equation e * L = ##Id_{S^1}## (where * is composition) means that 2*pi*L is a continuous choice of...

Can someone please tell me how sin(180 - x) = sin x? Here my attempt: sin (180 - x) = sin 180 . cos x - cos 180 . sin x Next? I have no idea...

Homework Statement sin2x + cos2x = 1 but would sin23x + cos23x = 1?Homework Equations none. The Attempt at a Solution [/B] I'm pretty sure sin23x + cos23x can't equal 1 otherwise the identity would probably be written as sin2cx + cos2cx = 1 and I've never seen it written like this. I was...

Homework Statement Prove that the conjugate of ##g(x) = f(Ax + b)## is ## g^*(y) = f^*(A^{-T}y) - b^TA^{-T}y ## where A is nonsingular nXm matrix in R, and b is in ##R^n##. Homework Equations This is from chapter 3 of Boyd's Convex Optimization. 1. The conjugate function is defined as ##...

Hi there! I am reading textbook "Supergravity" by Freedman and Van Proeyen and got stuck on a simple exercise (Ex 2.4). Usually I would proceed further marking it as a typo but I've checked the errata list on the website and didn't find this exercise there Exercise 2.4 Show that ##...

$\sin\left({A}\right)+\sin\left({A+\frac{2\pi}{3}}\right)+\sin\left({A+\frac{4\pi}{3}}\right)=0$...

$\sin^8\left({A}\right)-\cos^8\left({A}\right)=(\sin^2\left({A}\right)-\cos^2\left({A}\right)(1-2\sin^2\left({A}\right)\cos^2\left({A}\right))$ $L.H.S=(\sin^2\left({A}\right)-\cos^2\left({A}\right)(1-2\sin^2\left({A}\right)\cos^2\left({A}\right))$ $...

$\frac{\cot\left({A}\right)\cos\left({A}\right)}{\cot\left({A}\right)+\cos\left({A}\right)}=\frac{\cot\left({A}\right)-\cos\left({A}\right)}{\cot\left({A}\right)\cos\left({A}\right)}$ $L.H.S=\frac{\cot\left({A}\right)\cos\left({A}\right)}{\cot\left({A}\right)+\cos\left({A}\right)}$...

Suppose $f$ is a polynomial in $n$ variables, of degree $ \le n − 1$, ($n = 2, 3, 4 ...$ ).Prove the identity: \[\sum (-1)^{\epsilon_1+\epsilon_2+\epsilon_3+ ...+\epsilon_n}f(\epsilon_1,\epsilon_2,\epsilon_3,...,\epsilon_n) = 0\;\;\;\;\; (1)\] where $\epsilon_i$ is either $0$ or $1$, and the...

$\frac{1-\tan\left({A}\right)}{1+\tan\left({A}\right)}=\frac{\cot\left({A}\right)-1}{\cot\left({A}\right)+1}$ $L..H.S=\frac{1-\frac{\sin\left({A}\right)}{\cos\left({A}\right)}}{1+\frac{\sin\left({A}\right)}{\cos\left({A}\right)}}$...

Homework Statement ##(\hat A \times \hat B)^*=-\hat B^* \times \hat A^*## Note that ##*## signifies the dagger symbol. Homework Equations ##(\hat A \times \hat B)=-(\hat B \times \hat A)+ \epsilon_{ijk} [a_j,b_k]## The Attempt at a Solution Using as example ##R## and ##P## operators: ##(\hat...

Prove $Cos^6A+Sin^6A=1-3 \hspace{0.2cm}Sin^2 A\hspace{0.02cm}Cos^2A$ So far, $Cos^6A+Sin^6A=1-3 \hspace{0.2cm}Sin^2 A\hspace{0.02cm}Cos^2A$ $L.H.S=(Cos^2A)^3+(Sin^2A)^3$ $=(Cos^2A+Sin^2A)(Cos^4A-Cos^2ASin^2A+Sin^4A)$...

Hello everyone. Iam working on a course in multivariable control theory and I stumbled over the Identity Matrix. I understand what the identity matrix is, though the use of it is a mistery... I was reading about going from state space to transfer functions and I found this expressions...

Homework Statement exp(z)=-4+3i, find z in x+iy form Homework Equations See attached image. The Attempt at a Solution See attached image. exp(z)=exp(x+iy)=exp(x)*exp(iy)=exp(x)*[cos(y)+isin(y)] ... y=inv(tan(-3/4)=-.6432 ... mag(-4+3i)=5, x= ln (5)..exp(ln(5))=5 ...

Homework Statement Let ##{w_1,w_2} ## be a basis for ##\Omega## the period lattice. Use ##\zeta (z+ w_{i})=\zeta(z)+ n_i## , ##i=1,2## ; ## m \in N## for the weierstrass zeta function to show that ##\sigma ( z + mw_i )=(-1)^m \exp^{(mn_i(z+mwi/2))}\sigma(z)## Homework Equations [/B] To...

I am familiar with the derivation of the resolution of the identity proof in Dirac notation. Where ## | \psi \rangle ## can be represented as a linear combination of basis vectors ## | n \rangle ## such that: ## | \psi \rangle = \sum_{n} c_n | n \rangle = \sum_{n} | n \rangle c_n ## Assuming an...

Hi, I have the following homework question: Let Xt be the continuous-time simple random walk on a circle as in Example 2, Section 7.2. Show that there exists a c,β > 0, independent of N such that for all initial probability distributions ν and all t > 0 ∥νe^tA−π∥_TV ≤ ce^(−βt/N2) Here's what...

Consider an amplitude for some subprocess involving an off shell external state photon with polarisation ##\epsilon_{\mu}## and momentum ##q_{\mu}##, stripped of the polarisation vectors so that e.g ##T = \epsilon_{\mu} \epsilon_{\nu}^* T^{\mu \nu}## (##\epsilon_{\nu}^*## is polarisation vector...

Homework Statement "Suppose that ##F(s) = L[f(t)]## exists for ##s > a ≥ 0##. (a) Show that if c is a positive constant, then ##L[f(ct)]=\frac{1}{c}F(\frac{s}{c})## Homework Equations ##L[f(t)]=\int_0^\infty f(t)e^{-st}dt## The Attempt at a Solution ##L[f(ct)]=\int_0^\infty f(ct)e^{-st}dt##...

Homework Statement Attached Homework EquationsThe Attempt at a Solution So the question says 'some point'. So just a single point of space-time to be isotropic is enough for this identity hold? I don't quite understand by what is meant by 'these vectors give preferred directions'. Can...