Identity Definition and 1000 Threads

THREAD CHANGE *SPINOR IDENTITY*...although it's connected with SuSy in general, it's more basic... I am trying to prove for two spinors the identity: θ^{α}θ^{β}=\frac{1}{2}ε^{αβ}(θθ) I thought that a nice way would be to use the antisymmetry in the exchange of α and β, and propose that...

Homework Statement Maybe this is not possible because i does not represent anything quantile and is merely abstract? I'm not sure and maybe you guys can help! Homework Equations e^{i \pi} + 1 = 0 The Attempt at a Solution e^{i \pi} + 1 = 0 e^{i \pi} = -1 You cannot...

Homework Statement For each of the following statements, select whether the statement is true or false for all n × n matrices A, B , C. (Note that you are being asked whether the statement is true or false for all n × n matrices A, B, C, not just for some A, B, C.) a) (-6 A - 4 B)2 =...

Hi everyone, I have a doubt on Fierz identities. If we define the following quantities: S=1,\; V=\gamma_\mu,\; T=\sigma_{\mu\nu},\; A=\gamma_\mu\gamma_5,\;P=\gamma_5, then we have the identity: $$ (\Gamma_i)_{\alpha\beta}(\Gamma_i)_{\gamma\xi}=\sum_j...

Homework Statement x,y,z are three terms in GP and a,b,c are three terms in AP prove that (xb÷xc)(yc÷ya)(za÷zb)=1Homework Equations The Attempt at a Solution (xb-c)(yc-a)(za-b) since x y z are in GP xb-c÷yc-a=yc-a÷za-b (xb- c)(za-b)=yc-a(yc-a)

1. The pratement, all variables and given/known data If a =1÷(1-b) ,b=1÷(1-c),c=1÷(1-d) prove that a=d Homework Equations The Attempt at a Solution a=1÷(1-b) a-1÷(1-b)=0 {a(1-b)-1)}÷1-b=0 a-ab-1=0 a-ab=1 similarly b-bc=1 c-cd=1 could any of you please give a hint, this was...

Okay so I'm working on this problem $$\int \frac{x^2}{\sqrt{4 - x^2}} \, dx$$ I do a substitution and set $$x={\sqrt{4}}sinu$$ I get to this step fine $$\int 4sin(u)^2$$ I know that u = arcsin(x/2) so I don't see why I can't just substitute in u into sin(u)? I tried this and I got $$\int...

Is it true that: exp(2x)sinh(y)2 + exp(-2x) = exp(2x)cosh(y)2-2sinh(2x) I need this to be correct for an exercise but I don't know how to show it. I tried using something like cosh2+sinh2=1, but it didn't work.

The Reimann curvature tensor has the following symmetry resulting from a Bianchi identityR_{abcd}+R_{acdb}+R_{adbc}=0 The derivative of the electromagnetic field tensor also yields some of Maxwell's equations from a Bianchi identity\partial_\gamma F_{ \alpha \beta } + \partial_\alpha F_{ \beta...

Hi, I just want to double check that 2cos(x)sin(x) = 2sin(x)cos(x) = sin(x)2cos(x) = sin(2x) Thanks, Tim

Anybody think this identity is not super-cool? \begin{eqnarray} \frac {1} {1-x} = (1+x) \prod_{n=1} ^{\infty} [ \frac {(1+x^{2^n})} {(1-x^{2^n})} ]^{2^{-n}} , {\;} for {\;} 0{\le}x<1 \nonumber \end{eqnarray}

Okay the question is, given a plane electromagnetic wave in a vacuum given by E=(Ex,Ey,Ez)exp^{(i(k_{x}x+k_{y}y+k_{z}z-wt)} and B=(Bx,By,Bz)exp^{(i(k_{x}x+k_{y}y+k_{z}z-wt)} , where k = (kx,ky,kz), to show that kXE=wB. So I'm mainly fine with the method. I can see the maxwell's equaion...

Can it be proved? $$\left(\frac{-2\sin A}{1-\cos A}\right)\cos\left(\frac{A}{2}\right)\tan^{-1}\left[\cos \left(\frac{A}{2}\right)\right]=\frac{\pi^2-4A^2}{8}$$

x^(1/n) = the nth root of x (I'd use mathematical notation but I don't really know how I'm new sorry)

Homework Statement A. If the equation Ax=0 has only the trivial solution, then A is row equivalent to the nxn identity[/color] matrix. B. If the columns of A span R^n, the columns are linearly independent. C. If A is an nxn matrix, then the equation Ax=b has at least one solution for each b...

Homework Statement Question #2. Homework Equations The Attempt at a Solution I've drawn a venn diagram for the left-hand side and the right-hand side and I can see that they're not equal but how do I provide a counter-example for this? Wouldn't a counter-example require an infinite number...

Homework Statement Homework Equations The Attempt at a Solution $$A-(A\cap B)=A-B\\ A\cap (A\cap B)^{ C }=A\cap B^{ C }\quad (set\quad difference\quad law)\\ A\cup [A\cap (A\cap B)^{ C }]=A\cup [A\cap B^{ C }]\quad (applied\quad A\cup \quad to\quad both\quad sides)\\ A=A\quad (absorption...

Homework Statement Homework Equations I have to use these set identities: The Attempt at a Solution Pretty sure this is impossible because there's no identity for the Cartesian product.

Homework Statement Homework Equations The Attempt at a Solution $$(A-B)\cup (C-B)=(A\cup C)-B\\ (A\cap B^{ C })\cup (C\cap B^{ C })=(A\cup C)\cap B^{ C }\\ (A\cup C)\cap B^{ C }=(A\cup C)\cap B^{ C }\\$$ I know for algebraic proofs, proofs like these are accepted if they are reversed. But...

Hopefully this will make sense... We have the trig. identities shown below: sin(u)cos(v) = 0.5[sin(u+v) + sin(u-v)] cos(u)sin(v) = 0.5[sin(u+v) - sin(u-v)] How are these different? I realize u and v switched between the sine and cosine functions, but what is the difference between u and...

Homework Statement Show that a relation of the kind ƒ(x,y,z) = 0 then implies the relation (∂x/∂y)_z (∂y/∂z)_x (∂z/∂x)_y = -1 Homework Equations f(x,y) df = (∂f/∂x)_y dx + (∂f/∂y)_x dy The Attempt at a Solution I expressed x = x(y,z) and y = y(x,z) then found dx and...

Use de Moivre's identity to find real values of a and b in the equation below such that the equation is valid. cos^6(x)+sin^6(x)+a(cos^4(x)+sin^4(x))+b=0 Hint: Write cos(x) & sin(x) in terms of e^{ix} & e^{-ix}. Check your values of a and b are valid by substituting in a value of x. State...

I've been looking for this identity: arctan(1/x) = arcot(x) or arccot(1/x) = arctan(x) After just visually inspecting this to be true, I have been unable to find any formal proofs for it. Any references would be great!

Here is the question: I have posted a link there to this thread so the OP can view my work.

(1): $b_1x^3=b_2y^3=b_3z^3$ (2): $\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=1$ prove: $\sqrt[3]{b_1x^2+b_2y^2+b_3z^2}=\sqrt[3] {b_1}+\sqrt[3] {b_2} + \sqrt[3] {b_3}$

Hi Guys, I assume you are familiar with the equations so i do not post them (please write if u want me to post them). One of the steps to prove Kirchhoff's diffraction equation is to use Green's second identity. This identity shows the relation between the solutions in the volume and...

Hi there, This is my very first post, so I'd like to say thanks for reading and hi basically. :biggrin: I'm relatively confident my attempt at the proof is correct, but since the method is quite different from other examples I have seen, it kind of makes me nervous. I was hoping someone...

Homework Statement This is question 1.1 from section 2-1 of New Foundations of Classical Mechanics: Establish the following "vector identities": (a\wedge b) \cdot (c \wedge d) = b\cdot ca \cdot d - b\cdot da \cdot c = b\cdot(c\wedge d)\cdot a Homework Equations The Attempt at...

Homework Statement Prove that there is a constant C such that arctan\sqrt{\frac{1-x}{1+x}} = C - \frac{1}{2}arcsinx for all x in a certain domain. What is the largest domain on which this identity is true? What is the value of the constant C? The Attempt at a Solution Now I know how...

Your help will be greatly appreciated! Thanks!1. The expression \(\sin\pi\) is equal to \(0\), while the expression $\frac{1}{\csc\pi}$ is undefined. Why is $\sin\theta=\frac{1}{\csc\theta}$ still an identity? 2. Prove $\cos(\theta + \frac{\pi}{2})= -\sin\theta$

As a physics student, I understand that α particles are emitted and are the same as helium atoms without electrons. But this raises two questions to me: 1) What happens with the electrons at the atom that emits them? He now has two electrons too much. What happens with those? 2) Why an...

Suppose we have a torsion free connection. Does anyone here know of a slick way to prove that covariant derivatives satisfy the Jacobi identity? I.e. that $$([\nabla_X,[\nabla_Y,\nabla_Z]] + [\nabla_Z,[\nabla_X,\nabla_Y]] +[\nabla_Y,[\nabla_Z,\nabla_X]])V = 0$$ without going into...

My textbook (regarding continuum mechanics) has the following identity that is supposed to be true for all tensors: a\cdotTb = b\cdotTTa But I don't get the same result for both sides when I work it out. For each side, I'm doing the dot product last. For example, I compute Tb first and...

Rudin Theorem 1.21. How does he get "The identity"? In Theorem 1.21, Rudin says: The identity b^n-a^n=(b-a)(b^{n-1}+b^{n-2}a+...+a^{n-1}) yields etc etc. What is this "identity", and do we need to prove it first? If not, what assumption is Rudin making?

Homework Statement We are given two sets of functions: sin(x) and cos(x); S(x) and C(x). In the former, x is measured in radians, in the latter x is measured in degrees. It is possible to convert between the two using the following relations: sin(x) = S(mx), cos(x) = C(mx) where m=180/pi...

Homework Statement I was doing this practice exam and I had to calculate the eigenvalues en vectors. The matrix had two eigenvalues, I calculated one eigenvector. But when I was performing row operations for the second eigenvector, the matrix with the second eigenvalue substitued became an...

Here is the question: I have posted a link there to this topic so the OP can see my work,

Hi all, I've been playing around with spin 1/2 Lagrangians, and found the very interesting Fierz identities. In particular for the S x S product, (\bar{\chi}\psi)(\bar{\psi}\chi)=\frac{1}{4}(\bar{\chi} \chi)(\bar{\psi} \psi)+\frac{1}{4}(\bar{\chi}\gamma^{\mu}\chi)(\bar{\psi}\gamma_{\mu}...

What is the physical meaning of Jacobi identity for Poisson brackets? When does it come in handy? It goes as follows: {f,{g,h}}+{g,{h,f}}+{h,{f,g}}=0 Thanks.

Homework Statement Good day. May I know, for Dirac Delta Function, Is δ(x+y)=δ(x-y)? The Attempt at a Solution Since δ(x)=δ(-x), I would say δ(x+y)=δ(x-y). Am I correct?

Prove that: $\displaystyle\sum_{k=0}^n \frac{\cos(k x)}{\cos^kx} = \frac{1+(-1)^n}{2\cos^nx} + \dfrac{2\sin\big(\lfloor\frac{n+1}{2}\rfloor x\big) \cos\big(\lfloor\frac{n+2}{2}\rfloor x\big)} {\sin x\cos^n x} \qquad\qquad (\frac{2x}{\pi}\not\in \mathbb Z)$ *note: $\lfloor x\rfloor$ is floor...

I am looking for a counting interpretation to make the following identity evident: \sum_{k=0}^{n-j}(-1)^k\binom{j-1+k}{j-1}\binom{n}{j+k} = 1 The form of it looks like inclusion-exclusion. The sum is 1, more or less independent of j. So that makes me think it would be something like "how...

I am back again, with more tensor questions. I am getting better at this, but it is still a tough challenge of pattern recognition. Problem Statement Prove the following identity is true, using indicial notation: \nabla\times(\nabla \vec{v})^T = \nabla(\nabla\times\vec{v}) Attempt at...

1. Which of the following is a group? To find the identity element, which in these problems is an ordered pair (e1, e2) of real numbers, solve the equation (a,b)*(e1, e2)=(a,b) for e1 and e2. 2. (a,b)*(c,d)=(ac-bd,ad+bc), on the set ℝxℝ with the origin deleted. 3. The question...

Homework Statement Prove that \gamma^{a}\gamma^{b}\gamma^{c}\gamma^{d}\gamma^{e}\gamma_{a} = 2\left(\gamma^{e}\gamma^{b}\gamma^{c}\gamma^{d}+\gamma^{d}\gamma^{c} \gamma^{b}\gamma^{e}\right) Each of the \gamma^{i}s are as used in the Dirac equation. Homework Equations...

Hi. I'm trying to understand a derivation of the Bianchi idenity which starts from the torsion tensor in a torsion free space; $$ 0 = T(X,Y) = \nabla_X Y - \nabla_Y X - [X,Y]$$ according to the author, covariant differentiation of this identity with respect to a vector Z yields $$$ 0 =...

Homework Statement Show that.. e^{L}a e^{-L}=a+[L,a]+\frac{1}{2!}[L,[L,a]]+\frac{1}{3!}[L[L[L,a]]]+... Where L and a are operators. The Attempt at a Solution Right now I am writing the exponentials as their Taylor expansions, and then expanding the RHS of the above equation to see...

I understand the combinatorial proof and the common sense behind why it works but lately I am trying to play around with proofs since I am still new to them. So I understand part of this...

(I know the title contains a typo, but I can't edit it!) I'm trying to understand and/or prove this identity: blogbx = x I've inserted numbers, and it does work, but I just don't seem to understand why. I mean, some identities are obvious, like: logb bx = x since bx = bx but I can't make...

I've been watching a TV series recently called Person of Interest, it's really cool you should check it out. I'm curious though about people who somehow manage to completely wipe their identity and literally become a nobody and then acquire a new identity. I think legally the most thorough...