Identity Definition and 1000 Threads

Homework Statement Let f(x,y,z), g(x,y,z), h(x,y,z) be any C^2 scalar functions. Using the standard identities of vector analysis (provided in section 2 below), prove that \nabla \cdot ( f \nabla g \times \nabla h ) = \nabla f \cdot ( \nabla g \times \nabla h) Homework...

One of the basic vector identities is \nabla \cdot (\nabla f \times \nabla g) = 0 Is this true if f and g are C^{1} ? (Or they must be C^{2} functions? Thanks!

Homework Statement Prove the following when p is a positive integer: b^p - a^p = (b-a)(b^{p-1}+b^{p-2}a+b^{p-3}a^2+...+ba^{p-2}+a^{p-1}) Hint: Use the telescoping property for sums. Homework Equations None The Attempt at a Solution I tried reducing it to, (b-a)\sum_{k=1}^p...

Homework Statement Show the following identity (in the sense of distribution): g(\bold x)\delta (\bold x)=g(\bold 0) \delta (\bold x) for a function g. Homework Equations No idea. The Attempt at a Solution I don't have a concrete idea about what a distribution is (It's an...

4/t [cos(wt/2)-1] = -8/t sin(wt/4) ?

Let R be a ring with multiplicative identity 1R. Suppose that R is finite. The elemets xy1, xy2,...xyn are all different. So x y_i=1R for some i. A lemma that is not proven is given. If xyi=1R & yjx=1R, then yi=yj I need to show that yjx=1R. Right now I haven't got much. I took...

After reading an e-mail about a lawyer's identity being stolen and what he did to fight back I thought it was good enough to pass on. See what you think..

Hi, I would like some help in proving the following identity: \sum_{x=0}^{n}x^3 = 6\binom{n+1}{4} + 6\binom{n+1}{3} + \binom{n+1}{2} I tried doing it by induction but that did not go well (perhaps I missed something). Someone told me to use the fact that \binom{x}{0}...

Homework Statement Hi. I need to prove that [b x c, c x a, a x b] = [a, b, c]2 for any three vectors a, b and c. Note that [a, b, c] = a(b x c)Homework Equations I tried using the identify (a x b) x c = (a.c)b - (a.b)c The Attempt at a Solution Using the above identity I got [b x c, c x a...

Hello all, I'm doing a question for the maths module in my physics degree (I'm a second year undergrad) and there's a question I'm doing on basis functions. Homework Statement Verify that functions of the type f_{n}(x) = A cos \frac{2\pi n x}{L} where n = 0,1,2... can be used as a basis...

Say i have a matrix , \begin{bmatrix}{4}&{3}\\{-1}&{7}\end{bmatrix}+5 is it correct if i do it this way , \begin{bmatrix}{4}&{3}\\{-1}&{7}\end{bmatrix}+5\begin{bmatrix}{1}&{0}\\{0}&{1}\end{bmatrix} =\begin{bmatrix}{9}&{3}\\{-1}&{12}\end{bmatrix} is 5 a scalar = 5I where I is an...

Homework Statement prove that: tan(1+cos(x))^2 = 1-cos(x) Homework Equations trig identities, like the pythagorean, sum/difference, double/half angle identities, power reducing identities, etc... The Attempt at a Solution i'm not sure where to start; i tried using the pythagorean...

So I'm given a problem in which I have to prove an identity. It goes: 2csc2x=csc^2xtanx I did the problem myself and could only get to 2csc2x=2\(sin2x)= 2\(2sinxcosx). I had no idea how to get further with the problem so I looked at the answer in the back of my pre-calculus book. It said...

Homework Statement I'm supposed to derive the following: \left({\bf A} \cdot {\bf \sigma} \right) \left({\bf B }\cdot {\bf \sigma} \right) = {\bf A} \cdot {\bf B} I + i \left( {\bf A } \times {\bf B} \right) \cdot {\bf \sigma} using just the two following facts: Any 2x2 matrix can...

Hi, I was looking at an EM problem today and realized I wasn't sure why (kxH)\dotk = 0 I tried writing it out explicitly and got (w 1,2,3 representing directions) A1(A2*B3-A3*B2) - A2(A1*B3-A3*B1) + A3(A1*B2-A2*B1) and I can't see why this should equal zero. This is troubling...

Hi, I'm reading a book at the moment in which the author states the identity: \frac{1}{x-i\epsilon}=\frac{x}{x^2+\epsilon^2}+\frac{i\epsilon}{x^2+\epsilon^2} Which is fine, but then he goes on to state that this is equal to: P\frac{1}{x}+i\pi\delta(x) Where P is the principal...

Homework Statement Find two functions f, g \in C[0,1] (i.e. continuous functions on [0,1]) which do not satisfy 2 ||f||^2_{sup} + 2 ||g||^2_{sup} = ||f+g||^2_{sup} + ||f-g||^2_{sup} (where || \cdot ||_{sup} is the supremum or infinity norm) Homework Equations Parallelogram identity...

Homework Statement Show that for all integers n,m where 0 ≤ m ≤ n The sum from k=m to n of {(nCk)*(kCm)} = (nCm)*2^(n-m) The Attempt at a Solution So for the proof, I have to use a real example, such as choosing committees, binary sequences, giving fruit to kids, etc. I have been...

Cos^2 (wt+a) = 1+Cos(2wt+2a) ?

Hi I've got this problem which has really been bothering me. How are you supposed to prove that: (Sin[A] Sin[2 A] + Sin[3 A] Sin[6 A])/(Sin[A] Cos[2 A] + Sin[3 A] Cos[6 A]) is identicle to tan[5A]. I am almost sure that I've got to use the factor formulae, but I've had no luck. Maybe...

Homework Statement For ease of writing, a covariant tensor \bf G.. will be written as \bf G and a,b,c,d are vectors. Let \bf S and \bf G be two non-zero symmetric covariant tensors in a four-dimensional vector space. Furthermore, let S and G satisfy the identity: [\bf G \otimes \bf...

Homework Statement Verify the Identity: cos(x)-[cos(x)/1-tan(x)] = [(sin(x)cos(x)]/[sin(x)-cos(x)] b]2. Homework Equations [/b] reciprocal Identities, quotient Identities, Pythagorean Identities [b]3. The Attempt at a Solution cos(x)-[cos(x)/1-tan(x)] =...

Homework Statement let I_n be as an identity matrix where a_ij = 1 when i=j I just want to ask that is it true that all identity matrix has an inverse (determinant is not 0) for n>=2? The Attempt at a Solution

Hi all, I'm studying my mathematics lesson, and there is an example I can't understand: Consider the matrix A=(0 In) (-In 0) with In the identity nxn We want to compute detA : We introduce the permutation p=(1 2 ... n n+1 ... 2n)...

When you study physics, you never really delve into the reasons behind some of mathematical identities, i was curious about this one as it occurs in Bloch's Theorem (correct me if I go wrong)...

Homework Statement Show |sin z|^2 = \frac{1}{2}[cosh(2y)-cos(2x)]Homework Equations cosh2y = cosh^2y+sinh^2y cos2x = cos^2x-sin^2xThe Attempt at a Solution Here is what I have so far |sinz|^2=|sin(x+iy)|^2=|sin(x)cosh(y)+icos(x)sinh(y)|^2 =sin^2(x)cosh^2(y)+cos^2(x)sinh^2(y)...

I'm trying to do excercise 4.8 in "Riemannian manifolds" by John Lee. (It's about showing that the geodesics of \mathbb R^n are straight lines). The result I'm getting is that the definition of a geodesic implies the well-known identity 0=0, which isn't very useful. I must have made a mistake...

Hi, I was asked to prove this identity, I found the determinants for both the left and the right side, and now I basically have to prove that (d/dy)(f(dg/dz))=(df/dy)(dg/dz), the d's are actual partials though. Can anyone give me an idea on how to prove this? Thanks.

While seeing a signals example problem, I encountered this: cos^2 (wt+θ) = [1+cos(2wt+2θ)] What identity is this?

In Eq 11.72 in the QFT text by Peskin, the following equality is stated: i\int\frac{d^{d}k_{E}}{(2\pi)^{d}}\log(k_{E}^{2}+m^{2})=-i\frac{\partial}{\partial\alpha}\int\frac{d^{d}k_{E}}{(2\pi)^{d}}\frac{1}{(k_{E}^{2}+m^{2})^{\alpha}}|_{\alpha=0} This suggests that...

Hi, I am new with linear algebra, and I'm having a hard time wrapping my mind around the 0 vector and the additive identity v + 0 = v, where 0 is the 0 vector. If I had a 2x2 matrix, and v + w = C + (C^T)*D ... (where (C^T) is the transpose, v & w are vectors, and C & D are matrices)...

Could someone please explain to me in simple words (i.e., without referring to forms on the frame bundle, etc) why the Bianchi identity is the relativistic generalisation of the momentum-conservation law? Here comes my hypothesis, but I am not 100% convinced that it is correct. In Newtonian...

Homework Statement Let z=x+iy prove that Exp[z1]*Exp[z2]=Exp[z1+z2] Homework Equations Binomial thm (x+y)^n=Sum[Bin[n,k]*x^n-k*y^k,{k,1,n}] The Attempt at a Solution I have no idea about this question... Please give me some help.

Matrix A= 2x2, R1= -1, -1, R2= -7, 3 Matrix b= 2x2, R1= 1,0, R2= 0, 1 A*?=b ____________ To solve, I put ? on the one side of the equation as ?=A^(-1)b. My answer is then just the inverse of A, because what is multiplied by the identity matrix is itself. It is shown to be incorrect...

Hi friends, I am not able to understand how the below shown identity becomes (-1) power v cosθ. cos(vπ − θ ) = cos vπ cos θ + sin vπ sin θ = (−1)power v cos θ ==> (-1)vcos θ Please help me understand this basic problem. Thanks, Nesta

I have a simple technical problem. I'm following a paper [Shore, G. Ann Phys. 137, 262-305 (1981)], and I am unable to show a very simple identity for the non-abelian fluctuation operator (eq 4.37): D_\mu\left[-D^2\delta_{\mu\nu}+D_\mu D_\nu-2F_{\mu\nu}\right]\,\phi=-(D_\mu F_{\mu\nu})\,\phi ...

Hello, in a paper I have the identity \int_{-\infty}^{\infty} d x \sqrt{(x-i\epsilon)^2-1}= I_+ - I_- + i \int_{-1}^{1}(\ldots) where I_+ = \int_1^{\infty}(\ldots), I_=\int_{-\infty}^{-1}(\ldots) and \epsilon is a small positive number that will be taken to zero at the end. My...

Homework Statement Prove that the additive identity in a vector space is unique Homework Equations Additive identity There is an element 0 in V such that v + 0 = v for all v in V The Attempt at a Solution Assume that the additive identity is NOT unique, then there exists y...

interval from a to b \int f(x) dx = interval from b to a (-)\int f(x) dx Is this correct? Swapping the interval endpoints changes the sign of the integral? It seems like they should be equal. Thanks for the help. By the way, I saw this property here...

sin^2(x)-sin^2(2x)=cos^2(2x)-cos^2(x) I need help with proving this trig identity. Every thing I've tried just makes the problem more confusing. How would you guys go about this?

URGENT:trigonometric identity question Homework Statement tan2x+cos2x+sin2x=sec2x *the 2 stands for squared since I don't know how to make the squared symbol appear on a compter Homework Equations http://www.analyzemath.com/Trigonometry_2/Trigonometric_identities.html stuff from...

I know I haven't entered the formulae with the proper syntax, but I'm extremely exhausted at the time of posting, so please just read it and give advice, forgiving me this once for not using proper form (it's basically in latex code format). Homework Statement Show...

Homework Statement I'm a bit confused as to the following vector calculus identity: [∇ (∇.A)]_i = (δ/δx_i )( δA_j/δx_j) Shouldn’t it be = (δ/δx_i )( δA_i/δx_i) why is it ‘j’ if we are taking it over ‘i’ ? Thanks.

Homework Statement I'm supposed to verify this: \frac{cos(2x)-cos(4x)}{sin(2x)+sin(4x)}=tanx The attempt at a solution I reworked it every way I could think of, but it just won't work. I got desperate so I plugged it into some site and it said it was not a real identity, so I now I'm...

There is the identity \dfrac{1}{1-x+i0} = PV \dfrac{1}{x} - i \pi \delta(1-x) PV corresponds to Cauchy principal value. But how can I handle a term like \dfrac{1}{(x-i0)(1-x+i0)} and how can I use the identity above? I tried several things such as writing \dfrac{1}{(x-i0)(1-x+i0)} =...

Homework Statement This is a problem from a textbook, Riley Hobson and Bence 'Mathematical Methods for Physics and Engineering'. It asks to check the validity of a vector identity. If a, b and c are general vectors satisfying a x c = b x c, does this imply c . a - c . b = c|a-b| 2. The...

Homework Statement sec^2(x) tan^2(x) + sec^2(x) = sec^4(x) Homework Equations sin^2 + cos^2 = 1 1+tan^2 = sec^2 1+cot^2 = csc^2 The Attempt at a Solution First, I changed everything to sin and cos to try and make it clearer. 1/cos^2 * sin^2/cos^2 + 1/cos^2 = sec^4...

I would like to demonstrate an identity with the INDICIAL NOTATION. I have attached my attempt. Please let me know where I made mistakes. Any suggestion? I am trying to understand tensors all by myself because they are the keys in continuum mechanics Thanks

Hello. Homework Statement I would like to solve the following: \[\int\limits_{ - \infty }^{ + \infty } {{\rm{d}}x\,f\left( x \right)\frac{{\rm{d}}}{{{\rm{d}}x}}\delta \left[ {a\left( {x - x_0 } \right)} \right]} \] The solution I found in a paper is: \[\int\limits_{ -...

Homework Statement Hi all , again i am stuck onto this question :( , tried over 3 sheets alone on it lol.btw. thanks for your replies ;) . Prove the Identity (cosecA+cotA)^2 similar to 1+cosA/1-cosA Homework Equations hmm let's see.. sin2+cos2=1 , sec2= 1+tan2 cosec2= 1+cot2...