Identity Definition and 1000 Threads

1. Homework Statement Given ##\nabla## a torsionless connection, the Ricci identity for co-vectors is $$\nabla_a \nabla_b \lambda_c - \nabla_b \nabla_a \lambda_c = -R^d_{\,\,cab}\lambda_d.$$ Prove ##R^a_{[bcd]} = 0## by considering the co-vector field ##\lambda_c = \nabla_c f## Homework...

What is bianchi identity? Can anyone explain it to me as simple as possible? Is it something that allows us to convert riemannian tensor to ricci curvature tensor?

In our particle physics lecture this term comes up often, it doesn't look right to me but the lecturer uses it so it must be: ##{\partial }^{2}A^{\mu} = - {\partial }_{\mu}{\partial }^{\mu}A^{\mu}+ {\partial }_{\mu}{\partial }^{2}A^{\mu}## I understand if you have: ##F^{\mu v} = {\partial...

I kind of understand what to do with this when there are no numbers in front of the expressions, I also kind of understand that you can rewrite 4Sine(4x) as 4Sine(2x+2x) hat do I do with the 4 and 8? In an algebra problem you could divide the 4 into the -8, then simplify that expression, am I...

Homework Statement Let f be a function with a power series representation on a disk, say D(0,1). In each case, use the given information to identify the function. Is it unique? (a) f(1/n)=4 for n=1,2,\dots (b) f(i/n)=-\frac{1}{n^2} for n=1,2,\dots A side question: Is corollary 1 from my...

Homework Statement Prove that: \cos^6{(x)} + \sin^6{(x)} = \frac{5}{8} + \frac{3}{8} \cos{(4x)} Homework Equations I am not sure. I used factoring a sum of cubes. The Attempt at a Solution I tried \cos^6{(x)} + \sin^6{(x)} = \cos^4{(x)} - \cos^2{(x)} \sin^2{(x)} + \sin^4{(x)} . But I...

Homework Statement Apply corollary to show that 2 sinz*sinw = cos(z-w) - cos(z+w) for any z,w ∈ ℂ Homework Equations 2 sinz*sinw = cos(z-w) - cos(z+w) for any z,w ∈ ℂ Corollary: Let f and g be analytic functions defined on a domain D ⊂ ℂ. Let E ⊂ D be a subset that has at least one limit...

I have J - matrix x and y - vector d [ J(x) y(x)] / dx I can multiply the matrix and vector together and then differentiate but I think for my application it would be better to find an identity like {d [ J(x) y(x)] / dx } = J(x) d y(x) / dx + d J (x) / dx y(x) I am not sure if this identity...

1) Question: Show that (sin3A-sinA)/(cosA+cos3A)=tanA 2) Relevant equations: tan A=sinA/cos A 1+tan^2A=sec^A cot A=1/tanA cot A=cos A/sinA sin^2A+cos^2A=1 secA=1/cos A cosecA=1/sinA 1+cosec^2A= cot^2A sin2A=2sinAcosA cos2A=1-2sin^2A=cos^2A-sin^2A=2cos^A-1 tan2A=(2tanA)/1-tan^2A 3)Attempt...

Homework Statement Given a triangle ABC prove that c sin \frac{A-B}{2} = (a-b)cos \frac{C}{2} Homework EquationsThe Attempt at a Solution It looks rather similar to a formula mentioned in my book's lead-in to this exercise: \frac{a-b}{a+b}=tan\frac{A-B}{2}tan\frac{C}{2} Which can be...

Hi, I need help proving the following trig identity, (2sinx)\overline{secxtan(2x)}=2cos^2x-csc^2x+cot^2x I have tried starting from the left hand side, the right hand side, and doing both together, but nothing seems to work. One of the ways I tried: LHS...

Hi, I need help proving the following trig identity: \frac{\cot^2(x)-\cot(x)+1}{1-2\tan(x)+\tan^2(x)}=\frac{1+\cot^2(x)}{1+\tan^2(x)} Me and my friend have spent several hours determined to figure this out, starting from the left hand side, the right hand side, and doing both together, but...

How prove $\cos\frac{8\pi}{35}+\cos\frac{12\pi}{35}+\cos\frac{18\pi}{35}=\frac{1}{2}\cdot\left(\cos\frac{\pi}{5}+\sqrt7\cdot\sin\frac{\pi}{5}\right)$?

Homework Statement For a single molecule, derive the internal energy U = 3/2kBT In terms of the partition function Z, F = -kBTlnZ Where Z = V(aT)3/2 Homework Equations Thermodynamic identity: δF = -SδT - pδV p = kBT/V S = kB[ln(Z) + 3/2]The Attempt at a Solution U = F + TS δU = δF +...

I've worked with Euler's Identity for physics applications quite a few times, but ran into a "proof" of a contradiction in it, which I can't seem to find a flaw in (the only time I've ever had to do any proofs was in high school). I've derived Euler's equation in two different ways in past...

Prove the following identity: \frac{1-tan^2(\theta)}{1-cot^2(\theta)}=1-sec^2(\theta) By tan/sec identity, \frac {2-sec^2(\theta)}{2-csc^2(\theta)} separated the variables \frac {2}{2-csc^2}- \frac {sec^2}{2-csc^2} There's absolutely nothing useful I can do with either of those terms, so...

Mod note: Fixed the LaTeX. ##a=sinθ+sinϕ## ##b=tanθ+tanϕ## ##c=secθ+secϕ##Show that, ##8bc=a[4b^2 + (b^2-c^2)^2]## I tried to solve this for hours and have gotten no-where. Here's what I've got so far : ##a= 2\sin(\frac{\theta+\phi}{2})\cos(\frac{\theta-\phi}{2}) ## ## b =...

Homework Statement For the space Lw2(a,b), we can write the basis in a discrete fashion as {en|n∈ℤ} or in a continuous fashion as |x> (as we would in quantum mechanics for the position representation), such that we may write the identity operator as either I=∑n|en><en| or I=∫abdxw(x)|x><x|...

Prove the following \sum_{k=1}^n \frac{H_k}{k} = \frac{H_n^2+H^{(2)}_n}{2} where we define H^{(k)}_n = \sum_{j=1}^n \frac{1}{j^k} \,\,\, ; \,\,\, H^2_n = \left( \sum_{j=1}^n \frac{1}{j}\right)^2

Hey fellahs, got another whopper that's killing me. 1 - \cos2\theta + \cos8\theta - \cos10\theta=? My objective here is to complete the identity, and my worksheet lists the correct solution as: 4\sin\theta\cos4\theta\sin5\theta And once again I've had trouble beating this one. This is what...

Why ##\sin^2 2x + \cos^2 2x = 1##? Will ##\sin^2 3x + \cos^2 3x## or ##\sin^2 4x + \cos^2 4x## and so on, be = 1? How to proof this?

Hi I'm taking a math course at university that covers introductory group theory. The textbook's definition of the identity element of a group defines it as two sided; that is, they say that a group ##G## must have an element ##e## such that for all ##a \in G##, ##e \cdot a = a = a \cdot e## ...

Homework Statement I am trying to prove $$\vec{\nabla}(\vec{a}.\vec{b}) = (\vec{a}.\vec{\nabla})\vec{b} + (\vec{b}.\vec{\nabla})\vec{a} + \vec{b}\times\vec{\nabla}\times\vec{a} + \vec{a}\times\vec{\nabla}\times\vec{b}.$$ I can go from RHS to LHS by writng...

Homework Statement IF G, H and G+H are invertible matrices and have the same dimensions Prove that G(G^-1 + H^-1)H(G+H)^-1 = I 3. Attempt G(G^-1 +H^-1)(G+H)H^-1 = G(G^-1G +G^-1H + H^-1G + H^-1H)H^-1 = (GG^-1GH^-1 +GG^-1HH^-1 +GH^-1GH^-1 +GH^-1HH^-1) = GH^-1+I +GH^-1GH^-1 +GH^-1 =2GH^-1+...

Prove that \lfloor nx \rfloor = \sum_{k=0}^{n-1}\lfloor x+k/n \rfloor. Note \lfloor x\rfloor means the greatest integer less than or equal to x. I proved the cases where n=2 and n=3 by writing x=\lfloor x\rfloor + \{x\}, where \{x\} is the fractional part of x, and then using cases where 0\leq...

Homework Statement Hi, I have been trying to prove one of the corollaries of the Bezout's Identity in the general form.Unfortunately,I can't figure it out by myself.I hope someone could solve the problem. If A1,...,Ar are all factors of m and (Ai,Aj) = 1 for all i =/= j,then A1A2...Ar is a...

I was reading Schwartz's qft book. I saw the proof of ward identity taking pair annihilation as an example. he claimed he didn't assume that photon is massless in this derivation. but i have confusion with this statement. gauge invariance is a fact related to massless particles. now he has...

Homework Statement [/B] This is a seemingly simple problem. All I have to do is multiply two matrices: [ 1 0 ] [ 0 1 ] (A) and [ 2 ] [ 3 ] (B) The Attempt at a Solution [/B] Because the matrix A has the same number of columns as matrix B has rows, and because matrix A is an identity matrix...

So, I would like to prove that \gamma^{\mu_{1}...\mu_{r}}=(-)^{r(r-1)/2}\gamma^{\mu_{r}...\mu_{1}} where the matrix gamma is a totally antisymmetric matrix defined as \gamma^{\mu_{1}...\mu_{r}}=\gamma^{[\mu_{1}}\gamma^{\mu_{2}}...\gamma^{\mu_{r}]} What I have done is to prove that...

For EM wave, magnetic and electrical components are in phase, meaning when E = 0, then B = 0. Thus, I understand if it is written: f(x,t) = A(cos(kx - wt) + icos(kx - wt)) Then why plane wave is always described: f(x,t) = Aei(kx-wt) = A(cos(kx-wt) + isin(kx - wt)) Implying that Real and...

Kindly help me with this problem. I'm stuck! $\frac{\csc\theta+1}{\cot\theta}=\frac{\cot\theta}{\csc\theta-1}$ this is how far I can get to $\sec\theta+\tan\theta=\frac{1}{\sec\theta-\tan\theta}$

I can't quite work out this derivation I ran into which is essentially...Asin^2(wt) + Bcos^2(wt) = A = B. Is this correct? I know that sin^2(wt) + cos^2(wt) = 1, but I can't reason out how the factoring works here? Any help?

The context of the following identity is in the Classical Normal Linear Regression Model, ie, $\boldsymbol{y} = \boldsymbol{X}\boldsymbol{\beta}+ \boldsymbol{u}$ where $\boldsymbol{u}$ is a $n \times 1$ matrix and $u_i \sim iid.N(0, \sigma^2)$ for $i = 1, 2, \cdots, n$ Show that...

Dear All, Please see the attachment for the text i will be referring to. what does "I = τ1 · · · τm, where each τj is a transposition acting on {1, . . . , n}. Clearly, m != 1, thus m ≥ 2. Suppose, for the moment, that τm does not fix n". what does fixing n mean in a...

Hello everyone, I'm currently writing my dissertation based around the question of whether photo editing within the spheres of fashion and beauty media and advertising rids the subjects of their identity - I have looked at numerous books about personal identity, such as Paul Ricoeur's Oneself...

Homework Statement Let S be the subset of group G that contains identity element 1 such that left co sets aS with a in G, partition G .Probe that S is a subgroup of G. Homework Equations {hS : h belongs to G } is a partition of G. The Attempt at a Solution For h in S if I show that hS is S...

I am learning about statistical design of experiments, and in the process of mathematically rigorizing the concepts behind fractional factorial designs of resolution III, I derived an interesting equation: $$k = \sum_{i=1}^{3}{\lceil{\log_2{k}}\rceil \choose i},$$ for which the solutions $k$...

Homework Statement A and B are two unit vectors in the x-y plane. A = <cos(a), sin(a)> B = <cos(b), sin(b)> I need to derive the trig identity: sin(a-b) = sin(a) cos(b) - sin(b) cos (a) I'm told to do it using the properties of the cross product A x B Homework Equations A x B =...

Homework Statement Just like my title says, we are to prove the trig identity sin^2x+cos^2x=1 using the Euler identity. Homework Equations Euler - e^(ix) = cosx + isinx trig identity - sin^2x + cos^2x = 1 The Attempt at a Solution I tried solving the Euler for sinx and cosx...

Hi, I'm hoping to clear up a few uncertainties in my mind about proving that the identity element and inverses of elements in a group are unique. Suppose we have a group \left(G, \ast\right). From the group axioms, we know that at least one element b exists in G, such that a \ast b = b \ast...

Hi there. I was following a deduction on continuum mechanics for the invariant nature of the first two laws of thermodynamics. The thing is that this deduction works with an identity, and there is something I'm missing to get it. I have the vector product: ##\vec \omega \times grad \theta##...

Am I the only one who sees the resemblance between these two identities? Schouten: <p q> <r s> +<p r> <s q>+ <p s > <q r> =0 Jacobi: [A,[B,C]]+[C,[A,B]]+[B,[C,A]]=0 In Schouten the p occours in each term in the three terms, so we can regard it as dumby variable, and somehow get a...

Homework Statement Prove that [tan(a) + 1][cot(a+pi/4) + 1] = 2 Homework Equations [tan(a) + 1][cot(a+pi/4) + 1] = 2 The Attempt at a Solution This was very hard, I tried my best at expanding. [tan(a) + 1][cot(a+pi/4) + 1] = tan(a)cot(a+pi/4) + tan(a) + cot(a+pi/4) + 1...

Homework Statement Prove that if G is a cyclic group with more than two elements, then there always exists an isomorphism: ψ: G--> G that is not the identity mapping. Homework Equations The Attempt at a Solution So if G is a cyclic group of prime order with n>2, then by Euler's...

prove: $3=\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+5\sqrt{1+6\sqrt{1+7\sqrt{1+8\sqrt{1+9--}}}}}}}}$

In 'Supersymmetry in Particle Physics, An Elementary Introduction', the author Ian Aitchison used for several times the following identity: λa(ζ · ρ) + ζa(ρ · λ) + ρa(λ · ζ) = 0. I know that this identity is called Schouten's Identity, which is correct when all the variables are common...

Does anyone know how to prove this identity? I don't quote understand why the associated Legendre function is allowed to have arguments where |x|>1. h_n(kr)P_n^m(\cos\theta)=\frac{(-i)^{n+1}}{\pi}\int_{-\infty}^\infty e^{ikzt}K_m(k\rho\gamma(t))P_n^m(t)\,dt where \gamma(t)=\begin{cases}...

Hello guys, I am working on Ch22 "Continuous symmetries and conserved currents" of Srednicki QFT book. I am trying to understand how to prove the Ward-Takahashi identity using path integral method, done in page 136 of Srednicki. I understood everything up to Equation 22.22, which is 0 =...

Let S={p,q,r} and S=(S,+,*) a ring with identity. Let p be the identity for + and q the identity for *. Use the equation r*(r+q)=r*r+r*q to deduce that r*r=q. Attempt of a solution r*r=r*(r+q)- r*q =r*r+r*q - r*q But I'm not finding a clever way to deduce what is required. Any type of help...

Double integral of (52-x^2-y^2)^.5 2<_ x <_ 4 2<_ y <_ 6 I get up to this simplicity that results in a zero! 1-cos^2(@) - sin^2(@) = 0 This identity seems to be useless. HELP PLEASE.