Identity Definition and 1000 Threads

Prove identity $$\tan[\tan^{-1} (x) +\tan^{-1} (y)] =(x+y)(x-y)$$ Since $$\tan\left({\tan^{-1} \left({a}\right)}\right)=a$$ And by sum formula of $\tan{(x+y)}$ then $$=\frac{x+y}{1-xy}$$ But then?

Homework Statement I'm trying to derive the vector identity: $$\nabla(\vec{A} \cdot \vec{B})$$Homework Equations $$ \nabla(\vec{A} \cdot \vec{B})=(\vec{B} \cdot \nabla) \vec{A} + ( \vec{A} \cdot \nabla ) \vec{B} + \vec{B} \times (\nabla \times \vec{A})+ \vec{A} \times ( \nabla \times \vec{B})$$...

I want to compute the following when the 4-vectors are already given i.e x^{\mu},y^{\mu} are given and are orthonormal ( x, y are complex vectors); \begin{eqnarray} \left(/\negmedspace\negmedspace x/\negmedspace\negmedspace y\right)^{2} & = & /\negmedspace\negmedspace...

$$(\cot \theta)(\sin \theta)$$ So far I understand that you can make $$(\cot a) \implies (\frac{\cos \theta}{\sin \theta})$$ Then it would come to $$(\frac{\cos \theta}{\sin \theta})(\sin \theta)$$ I'm stuck at when making $$(\sin \theta)$$ into a fraction. The sine in between the asterisks...

Can anyone identity the book please? http://atao.ucsd.edu/258/bandbending.pdf i had some notes from this book. I need more info. with great difficulty I have been able to find the link. any leads will be very useful. thanks

"Two scientists from Japan and Canada were awarded the Nobel Prize in physics for showing that particles known as neutrinos can change identities and also possesses mass, insights that both deepened and challenged our understanding about how the universe works". Friends! Please explain in...

Homework Statement Compute ##\int_0^\infty \frac{\sin x}{x}dx## using that ##\frac{\sin x}{x} = \frac{b_0}{2} +\sum_1^\infty b_n \cos nx \; \; , \; \; 0 < x < \pi## with ##b_n = \frac{1}{\pi} \int_{(n-1)\pi}^{(n+1)\pi} \frac{\sin x}{x}dx##. Homework Equations Perhaps the following...

Hi, I am confused about how I arrive at the contracted epsilon identity. \epsilon_{ijk} \epsilon_{imn} = \delta_{jm} \delta_{kn} - \delta_{jn} \delta_{km} 1. Homework Statement Show that \epsilon_{ijk} \epsilon_{imn} = \delta_{jm} \delta_{kn} - \delta_{jn} \delta_{km} Homework EquationsThe...

Homework Statement Hello everyone, can anyone help me prove this using tensors? Given three arbitrary vectors not on the same line, A, B, C, any other vector D can be expressed in terms of these as: where [A, B, C] is the scalar triple product A · (B × C) Homework Equations I know that...

The problem Show that the left side is equal to right side ## tan (\frac{x}{2}) = \frac{1-cos(x)}{sin(x)} ## The attempt ##\tan(\frac{x}{2}) = \frac{ sin(\frac{x}{2}) }{ cos (\frac{x}{2}) } = \frac{ sin^2(\frac{x}{2}) }{ cos ^2 (\frac{x}{2}) } = \frac{\frac{1-cos(x)}{2}}{\frac{1+cos(x)}{2}} =...

Homework Statement In my statistics notes/lectures my professor will oftentimes use an identity that looks like the following: x_i is a non random variable, the summand is from i=1 to n; This segment comes from notes on linear regression (y_0 = b_0 + b_1*x_i) I actually forgot to mention that...

Homework Statement Prove the following trigonometric identity. The question is sin^4Ө =3/8-3/8cos(2Ө) Homework Equations I think I'm supposed to use the power reducing formulas for trigonometric identities which are sin^2(u)= (1- cos(2u))/2 cos^2(u)=(1+cos(2u))/2 *Let u represent any...

Homework Statement Prove that ##A\cap(B\Delta C)=(A\cap B)\Delta(A\cap C)## Homework EquationsThe Attempt at a Solution [/B] L.H.S.=##A\cap(B\Delta C)## =##A\cap[(B - C) \cup (C - B)]## =##A\cap[(B \cap \bar{C}) \cup (C \cap \bar{B})]## =##[A\cap (B \cap \bar{C})] \cup [A\cap (C \cap...

Homework Statement ## y^({9}) + y''' = 6 ## Homework EquationsThe Attempt at a Solution ## y^({9}) + y''' = 6 ## ## r^9 + r^3 = r^{3}(r^{6}+1)=0 ## ## r = 0, m = 3 ## ## r^6 + 1 = 0 = e^{(i(\pi + 2k\pi)} ## ## r = -1 = e^{i(\frac {\pi +2k\pi} {6})} ## ## k = 0 , r = e^{i(\frac {\pi}...

How does e-Δ2/δ2 ≈ 1-Δ2/δ2 When Δ<<δ ? I'm sure it's a basic summation I'm unaware of.

$$\cos\left({4x}\right) =8\sin^4\left({x}\right) -8\sin^2\left({x}\right) +1$$ I thought this would break down nice from the quadratic but it didn't.

Hi - just started complex analysis for the 1st time. I have been a little confused as to the chicken and egg-ness of Cauchy-Riemann conditions... 1) Wiki says: "Then f = u + iv is complex-differentiable at that point if and only if the partial derivatives of u and v satisfy the Cauchy–Riemann...

Show $ |sinh(z)|^2 = sin^2x + sinh^2y $ Since I posted this, I found new info - cos(iy) = cosh(y) and sin(iy) = i sinh(y) which made the above easy; don't want to bother anyone so will mark this solved.

Homework Statement Prove the following identity \nabla (\vec{F}\cdot \vec{G}) = (\vec{F}\cdot \nabla)\vec{G} + (\vec{G}\cdot \nabla)\vec{F} + \vec{F} \times (\nabla \times \vec{G}) + \vec{G}\times (\nabla \times \vec{F}) Homework Equations vector triple product \vec{a} \times (\vec{b}...

I am reading Paolo Aluffi's book, Algebra: Chapter 0. I am currently focused on Chapter III, Section 2: The Category Ring. I need some help in getting started on Problem 2.15 in this section. Problem 2.15 at the end of Chapter III, Section 2 reads as follows: I would welcome some help in...

I cannot seem to prove the following identity (1+sin(x))/(1-sin(x))=2tan^2(x)+1+2tan(x)sec(x) Can you assist?

Homework Statement Let ##P## be a permutation matrix. Show that for some ##N>0## P^N := \underbrace{PP...P}_{N \ \text{times}} = I 2. Relevant definitions A permutation matrix is a ##n\times n## matrix containing only zeros and ones such that there is exactly one ##1## per row and per column...

So, essentially, all I wonder is: What is the The Matrix Exponent of the Identity Matrix, I? Silly question perhaps, but here follows my problem. Per definition, the Matrix Exponent of the matrix A is, e^{A} = I + A + \frac{A^2}{2} + \ldots = I + \sum_{k=1}^{\infty} \frac{A^k}{k!} =...

I am doig trigonometric identities and i got this one, (all will be in the picture the solution and my work) i used the double angle for this but i am afraid i didn't get the exact idea, just guessing, good guessing, so i want to know how is the proper way to reach the solution

Homework Statement I have had a brain malfunction and I need help to understand something simple. It would be great if someone could show the process of attaining the end form. How does; ##a\cos{(x)}+b\sin{(x)} = c\sin{(x+\phi)}## where a,b are arbitrary constants, c results from whatever...

Homework Statement I have to prove the Fierz rearrengement identity for Weyl Fermions. Eq 2.20 in Martin's supersymmetry primer: \chi_\alpha(\xi\eta)=-\xi_\alpha(\eta\chi)-\eta_\alpha(\chi\xi) Homework Equations We have that the antisimetric tensor raises and lowers indices. The Attempt at...

Premise 1: Physics don't believe in sense "organs" of the human "robot" (more commonly said "common sense deceives us"). Premise 2: Physics believes in logic or mathematics. Background thrust: Quantum mechanics. Premise 3: Everything which "revolves" around the nucleus might not have...

EDIT: I figured out my mistake...no option to delete silly post. Oh well. 1. Homework Statement The problem is: use iterated integrals in polar form to find the area of one leaf of the rose-shaped curve r = cos(3*theta). My setup agrees exactly with the solutions manual...but then something...

Does ∇⋅A = A ⋅∇? If not then, what does the latter actually equal?

Homework Statement Prove that ##(x^2+y^2+z^2)\nabla^2[\delta(x)\delta(y)\delta(z)]=6\delta(x)\delta(y)\delta(z)## Homework Equations ##\delta''(x)/2=\delta(x)/x^2## The Attempt at a Solution I have obtained this: ##6\delta(x)\delta(y)\delta(z) +...

Hi all, I make some exercises in particle physics but I'm stuck in two problems related to Gamma matrices identities, First: the Fermion propagator ## \frac {i } { /\!\!\!p - m} = i \frac { /\!\!\!p + m } { p^2 - m^2} ## So how ##/ \!\!\!\!p ^2 = p^2 ## ? Where ## /\!\!\!p = \gamma_\mu p^\mu...

I'm looking at the informal arguements in deriving the EFE equation. The step that by the bianchi identity the divergence of the einstein tensor is automatically zero. So the bianchi identity is ##\bigtriangledown^{u}R_{pu}-\frac{1}{2}\bigtriangledown_{p}R=0##...

In peskin p. 160 forth paragraph they say to verefy Ward identity in equation 5.74. I don't succeed, they say some algebra is needed. I conjecture that this some algebra is what i miss. Any help will be appreciated - thanks a lot.

I am in an independent study working through probabilistic graph theory and I am stuck on part of a theorem from chapter 4 of The Probabilistic Method by Joel Spencer and Noga Alon (specifically theorem 4.2.1). In this context, $p$ is a prime number. The part where I am confused comes from a...

Homework Statement Need to prove that: (v⋅∇)v=(1/2)∇(v⋅v)+(∇×v)×v Homework Equations Vector triple product (a×b)×c=-(c⋅b)a+(c⋅a)b The Attempt at a Solution I know I could prove that simply by applying definitions directly to both sides. I haven't done that because that is tedious, and I...

Hi, the question (from math methods for physicists) is: If A is orthogonal and det(A)=+1, show that (det A)aij = Cij(A). I know that if det(A)=+1, then we are looking at a rotation. (Side question - I have seen that det(A) =-1 can be a reflection, but is 'mostly not reflections'; what does...

Homework Statement What is the likely identity of a weak acid if a 0.50 mol/L solution of the acid has a pH of 3.18? Homework Equations HA + H2O <=> H3O + A K_{a} = \frac{[P]}{[R]} [H3O] = 10^{-pH} The Attempt at a Solution [/B] I set up an ICE table with the concentration given...

Please be patient as I struggle with latex here ... Part 1 of the problem says to start with: $ \frac{\partial\bar{r}}{\partial{q}_{1}} ={h}_{1} \hat{q}_{1} $ and then to find an expression for $ {h}_{1} $ that agrees with $ {g}_{ij}=\sum_{l}...

Learned this identity a year ago randomly studying for adv biomechanics and was wondering if there were real-world applications for this outside of mathematicians appreciating the formula.

Homework Statement Let us consider three scalar fields u(x), v(x), and w(x). Show that they have a relationship such that f(u, v, w) = 0 if and only if (∇u) × (∇v) · (∇w) = 0. Homework EquationsThe Attempt at a Solution I could think nothing but...

Homework Statement Given that [A_i,J_j]=i\hbar\epsilon_{ijk}Ak where A_i is not invariant under rotation Show that [J^2,Ai]=-2i\hbar\epsilon_{ijk}J_jAk-2\hbar^2A_i Homework Equations [AB,C]=A[B,C]+[A,C]B [A,B]=-[B,A]The Attempt at a Solution [J^2,Ai]=[J_x^2,Ai]+[J_y^2,Ai]+[J_z^2,Ai]...

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...