Identity Definition and 1000 Threads

I am attempting to work through a paper that involves some slightly unfamiliar vector calculus, as well as many omitted steps. It begins with the potential energy due to an electric field, familiarly expressed as: U_{el} = \frac{\epsilon_r\epsilon_0}{2} \iiint_VE^2dV =...

Hey! :o We know that: $$(x,x)=0 \Rightarrow x=0$$ When we have $\displaystyle{(x,y)=0}$, do we conclude that $\displaystyle{x=0 \text{ AND } y=0}$. Or is this wrong? (Wondering)

If the group G/[G,G] is abelian then how do we show that xyx^{-1}y^{-1}=1? Thanx

Hi everybody! First post!(atleast in years and years). The stationary KdV equation given by $$ 6u(x)u_{x} - u_{xxx} = 0 $$. It has a solution given by $$ \bar{u}(x)=-2\sech^{2}(x) + \frac{2}{3} $$ This solution obeys the indentity $$ \int_{0}^{z}\left(\bar{u}(y) -...

I have no idea how to go about proving this trig identiy. I mean, I've been taught that it's a safe bet to convert everything to sines and cosines, but other than that, I've no clue. Am I even on the right path?

Homework Statement Show that: curl(r \times curlF)+(r.\nabla)curlF+2curlF=0, where r is a vector and F is a vector field. (Or letting G=curlF=\nabla \times F i.e. \nabla \times (r \times G) + (r.\nabla)G+2G=0) The Attempt at a Solution I used an identity to change it to reduce (?) it to...

So, this question says "prove each identity. State any restrictions on the variables". 5a) \frac{sinx}{tanx} = cosx I did the first part of the question correctly (proving it), but I don't understand how you determine the restrictions on the variables. In the textbook, it says that it cannot...

hey, question is attached thanks in advance!

In a Hilbert-space whose dimensionality is either finite or countably infinite, we have the discrete resolution of identity \sum_n |n\rangle \langle n| = 1 In many cases, for example to obtain the wavefunctions of the discrete states, one employs the continuous form of the resolution...

Homework Statement From Mary Boas' "Mathematical Methods in the Physical Sciences" 3rd ed. Ch 3 Sec 4 problem 24 Where A and B are vectors. What is the value of (AXB)^2+(A dot B)^2=? Comment: This is a special case of Lagrange's Identity. Homework Equations Cross product and dot...

Hi everyone. I'm studying Heavy Quark Effective Theory and I have some problems in proving an equality. I'm am basically following Wise's book "Heavy Quark Physics" where, in section 4.1, he claims the following identity: $$ \bar Q_v\sigma^{\mu\nu}v_\mu Q_v=0 $$ Does any of you have an...

I'm slightly confused at the proof of this theorem, hopefully someone can help. Identity theorem: Suppose X and Y are Riemann surfaces, and f_1,f_2:X \to Y are holomorphic mappings which coincide on a set A \subseteq X having a limit point a \in X. Then f_1 and f_2 are identically equal. The...

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 4 *...

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...

Show that $$\sum_{k=1}^{N} (-1)^{k} \sin(kx) = - \frac{1}{2} \tan \left(\frac{x}{2} \right) + \frac{(-1)^{N} \sin \Big((N+\frac{1}{2})x \Big)}{2\cos (\frac{x}{2})} \ .$$ Now assuming $ \displaystyle \int_{a}^{b} f(x) \tan \left(\frac{x}{2} \right) \ dx $ converges, argue that $$ \int_{a}^{b}...

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 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 in R^n...

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,