Identity Definition and 1000 Threads

Homework Statement Prove the following vector identity: Any vector a dotted with its time derivative is equal to the vector's scalar magnitude times the vector's derivative's scalar magnitude. Homework Equations (a)dot(d(a)/dt)=||a|| x ||da/dt|| The Attempt at a Solution I...

Homework Statement I need to prove the identity: (a×b)\cdot(c×d)= (a\cdotc)(b\cdotd)-(a\cdotd)(b\cdotc) using the properties of the vector and triple products: Homework Equations a×(b×c)=b(a\cdotc)-c(a\cdotb) a\cdot(b×c)=c\cdot(a×b)=b\cdot(c×a) The Attempt at a Solution I...

Hi there, working on a physical problem I found two functions that should be equivalent, and indeed they seem to be after a numerical check. The functions are shown in the attached PDF. I can not figure a way to prove their equivalence analytically (the double integral especially gives me...

Can anyone help me in proving the following identity: (\gamma ^{\mu} )^T = \gamma ^0 \gamma ^{\mu} \gamma ^0 I understand that one can proceed by proving it say in standard representation and then proving that it's invariant under unitary transformations. this last thing is the one...

Hey guys, this is for my classical E&M class but it's more of a math problem. Homework Statement Show: ∇(\vec{A} . \vec{B}) = \vec{B} \times (∇ \times \vec{A}) + (\vec{B} \times ∇)\vec{A} + \vec{A} \times (∇ \times \vec{B}) + (\vec{A} \times ∇)\vec{B} Homework Equations I tried...

Hello everybody. Here's the problem: $$\text{Let } R \text{ be a ring with identity. Let }a \in R \text{ and suppose that exists an unique } a' \in R \text{ such that }a a' =1. \text{ Prove that } a'a=1.$$ My solution: Since we have an identity, it has an inverse (itself), which means we can...

Homework Statement If G is a group, a is in G, and a*b=b for some b in G (* is a certain operation), prove that a is the identity element of G Homework Equations The Attempt at a Solution I feel like you should assume a is not the identity element and eventually show that a= the...

Hello everyone, I came across this identity while browsing Wikipedia, and I decided to try to prove it for myself. ( It was discovered by S Ramanujan) \int_0^\infty \cfrac{1+{x}^2/({b+1})^2}{1+{x}^2/({a})^2} \times\cfrac{1+{x}^2/({b+2})^2}{1+{x}^2/({a+1})^2}\times\cdots\;\;dx =...

It seems that this term comes up in solving the cubic equation. While there is the identity for the half-angle, there doesn't seem to be one for third-angle.

Homework Statement I do not understand equal signs 2 and 3 the following Angular momentum operator identity: Homework Equations \hat{J}^2 = \hat{J}_1^2+\hat{J}_2^2 +\hat{J}_3^2 = \left(\hat{J}_1 +i\hat{J}_2 \right)\left(\hat{J}_1 -i\hat{J}_2 \right) +\hat{J}_3^2 + i...

Homework Statement Simplify the following: (1/cos2θ) - (1/cot2θ)Homework Equations Various trig identities The Attempt at a Solution I tried to make cos2θ into 1-sin2θ and cot2θ into csc2θ-1 but still couldn't find any obvious solution. Help?

Show that: 4(\sin^4x+\cos^4x) \equiv \cos4x + 3. Really stuck with this, no idea how to go ahead with it. The book gives a hint: \sin ^4 x = (\sin ^2 x)^2 and use \cos 2x = 1 - 2\sin ^2 x But I don't even understand the hint, where did they get \cos 2x = 1 - 2\sin ^2 x from?

Prove: \frac{CosθSinθ}{1 + Tanθ} = Cos2θ =========================== I multiply out the denominator to get: CosθSinθ = Cos2θ + CosθSinθ I cannot seem to prove it. Starting to think it's a trick question.. :/

1 + 1 = 2 sec2x ______ ________ 1+sinx 1-sinx PLEASE SOMEONE HELP! UGH! In case you can't tell what that says, it's 1/1+sinx + 1/1-sinx = 2sec2x​

I was thinking about identities, and seem to have arrived at a contradiction. I'm sure I'm missing something. A(n) (two-sided) identity for a binary operation must be unique. I will reproduce the familiar proof: Proof: Suppose a is an arbitrary element of a set S, e and e' are both...

I need to verify the given identity. I've tried every which way i can think of, but can't figure this one out. I am self-studying this book "College Trigonometry 5th Edition by Aufmann. This is exercise set 3.3, problem 63. cos^2(x) - 2sin^2(x)cos^2(x) - sin^2(x) + 2sin^4(x) = cos^2(2x)...

\frac{2 i}{(2 i + 1)!} = \frac{1}{(2 i)!} - \frac{1}{(2 i + 1)!} Could anybody please show what it is that needs to be done on LHS to get to RHS in this identity.

i am completely lost as to how to go from p^ \frac{1}{m}∇p to \frac{m}{m+1} ∇p^\frac{m+1}{m}

Homework Statement Prove the identity. Homework Equations http://postimage.org/image/vjhwki1ax/ The Attempt at a Solution http://s13.postimage.org/jkhubi4lz/DSC03534.jpg

I got trouble to understand the cyclic sum identity (the first Bianchi identity) of the Riemann curvature tensor: {R^\alpha}_{[ \beta \gamma \delta ]}=0 or equivalently, {R^\alpha}_{\beta \gamma \delta}+{R^\alpha}_{\gamma \delta \beta}+{R^\alpha}_{\delta \beta \gamma}=0. I can understand the...

Homework Statement I'm trying to understand a proof of the LC-KD identity involving determinants (see attachment), from the book Introduction to Tensor Calculus and Continuum Mechanics by Herinbockel. What is the author saying in the last line of text? How can we sum the deltas in the upper...

I look for trig identitiy to simplify this expression: \frac{\sin(nx/2)}{\sin(x/2)} is there one specficic to use, or is there other ones that will help to simplifiy? I have been trying but can't make it simplier! Thanks!

Homework Statement To prove that \sum over m=1 to 15 of sin(4m-2) = 1/4sin2, where all angles are in degress Homework Equations The Attempt at a Solution Tried to solve it using identity sinx+siny=2sin((x+y)/2)cos((x-y)/2)..but all attempts failed..help

Hi, I'm looking for intuition and/or logic as to why we would want or need morphisms according to axioms in category theory, to imply that in the category of rings, they must preserve the identity (unless codomain is "0"). Thank you very much.

Homework Statement sin4x=(4sinxcosx)(1-2sin^2x) Homework Equations Trig identities. The Attempt at a Solution sin4x=(4sinxcosx)(1-2sin^2x) (4sinxcosx)(cos2x) stuck right here...

Homework Statement Prove sin(x-iy) = sin(x) cosh(y) - i cos(x) sinh(y) Homework Equations The Attempt at a Solution I tried to prove it by developing sinh into it's exponential form, but I get stuck. sinh(x-iy) = [ ei(x-iy) - e-i(x-iy) ] /2i = [ eixey - e-ix e-y ] /2i...

Homework Statement Show that : 1 + cos(2∏/5)= 2 cos(∏/5) Homework Equations cos(2x) = cos^2(x)-sin^2(x) cos^2(x)+sin^2(x) = 1 The Attempt at a Solution I have tried using the two formulas above but i couldn't show the required result.

Homework Statement In a worked example I have of an integration it states the integral of (cosx)^2 = the integral of (1 + cos2x)/2 How is this equality reached? Is this a known identity, (cosx)^2 = (1 + cos2x)/2 ? Thank you. Homework Equations The Attempt at a Solution

Homework Statement I'm trying to follow some solution to an exercise in physics and apparently e^{-im \frac{3\pi}{2}}=i^m where m \in \mathbb{Z}. I don't realize why this is true. Homework Equations Euler's formula.The Attempt at a Solution I applied Euler's formula but this is still a...

I am trying to understand Derek Parfit view on persistence of personal identity. You don’t have to take in weird examples such as branching. Can you also show some of the flaws of his view. Can you advice?

Homework Statement Let G be a Lie group, e be its identity, and \mathfrak g its Lie algebra. Let i be group inversion map. Show that d i_e = -\operatorname{id} . The Attempt at a Solution So this isn't terribly difficult if we have the exponentiation functor, since in that case e^{-\xi}...

This isn't a homework problem, but it won't let me post on the other page. A well known vector identity is that rot(rot(E)) = grad(div(E)) - div(grad(E)). I've actually used this before without encountering any problems, so I don't know if I'm just having a brain fart or something, but...

Homework Statement Here is the question given: A blade for a lawnmower consists of two parts made of the same material and joined together as shown: The length OP is one unit in length and MPQN is square in shape. Develop an equation for the cross-sectional area of the blade and...

Homework Statement Prove that: tan^2∅/tan∅ - 1 + cot^2∅/cot∅ - 1 = 1 + sec∅cosec∅ Homework Equations The Attempt at a Solution I have solved the question taking tan∅ = sin∅/cos∅. But I want to solve it some other way.

So it is true that ei∏+1=0. But what does this mean? Why are all these numbers linked?

Homework Statement This was a problem on our final. I played with traits of a bijection to no avail and got a 0%. It's got me completely stumped. I really cannot even figure out a way to start. Let X be a finite set. Let f : X → X be a bijection. For n ε Z>0, set fn = {f°f...°f} n times Prove...

When I was checking my work, Wolframalpha took my trig work a step further with an identity that no one in my Calculus II class has ever seen, including my teacher. csc(2x) - cot(2x) = tan(x) I tried to prove the identity myself and I looked online, but no luck. Please, could someone...

I have been working on showing the equality between N=0 to ∞ Ʃ cos(2nθ)(-1)^n/(2n)! = cos(cos(θ))cosh(sin(θ)) I started by using the standard series for cosine and putting cos(2nθ) in for the x term. I did the same for cosh(sin(θ)). I manipulated the forms every way I could think but...

it's bothering my brain..i thought about it many times...i can't make intuition of it can anyone prove it? oh by the way... C = Sqrt[A^2 + B^2] and theta is equal to arctan(B/A)

Let f be an analytic function defined in an open set containing the closed unit disk and let z in ℂ be fixed. I've simplified a more complicated expression down to this identity, and as implausible as it looks, after some numerical checking it does in fact appear to be true, but I can't find a...

Homework Statement Write the following in terms of the cofunction identity: cos(4pi/3) Homework Equations I know the θ is in Quadrant 3, but my question is, which cofunction identity am I suppose to use? Cofunction Identities: cosx = sin(pi/2 - x) #1 cosx = sin(pi/2 + x) #2...

Hello! I'm want to prove a vector identity for (\nabla \times \vec{A}) \times \vec A using the familiar method of levi-civita symbols and the identity \epsilon_{kij}\epsilon{kmn} = \delta_{im}\delta_{jn} - \delta_{in}\delta{jm}, but I don't seem to come up with any usefull answer. I...

"Proving" the Jacobi identity from invariance Hi all, In an informal and heuristic manner, I have heard that the "change" in something is the commutator with it, i.e. \delta A =[J,A] for an operator A where the change is due to the Lorentz transformation U = \exp{\epsilon J} = 1 + \epsilon J...

If we assume the inner product is linear in the second argument, the polarization identity reads (x,y) = \frac 14 \| x + y \|^2 - \frac 14 \| x - y \|^2 - \frac i4 \|x + iy\|^2 + \frac i4 \| x - iy \|^2. But there is another identity that I've seen referred to in some texts as the...

Homework Statement Prove that the identity map \mathrm{id}_{S^{2k+1}} and the antipodal map -\mathrm{id}_{S^{2k+1}} are smoothly homotopic. Homework Equations N/A The Attempt at a Solution My attempt: Fix k \in \mathbb{Z}_{\geq 0} and let \{e_i\}_{i=1}^{2k+2} be the standard basis for...

Homework Statement cos(x-y)cosy-sin(x-y)siny=cosx a.try to prove that the equation is an identity b. determine a counterexample to show that it is not an identity Homework Equations cos(x-y) = cosxcosy+sinxsiny sin(x-y) = sinxcosy-cosxsiny The Attempt at a Solution a.Left side of...

Homework Statement \frac{cos^{2}t+tan^{2}t -1}{sin^{2}t} = tan^{2}t Homework Equations Here are all the trig identities we know up to this point (the one's that we have learned so far, obviously we derive many others from these when verifying identities). Pythagorean Identities...

prove 1-(cos(x)+sin(x))(cos(x)-sin(x))=2sin^2(x) foil out the center I get 1-cos^2(x)-cos(x)sin(x)+cos(x)sin(x)+sin^2(x) the -cos(x)sin(x)+cos(x)sin(x) cancels to 0 leaving 1-cos^2(x)-sin^2(x) then I'm lost... I know I can switch 1-cos^2(x) to sin^2(x) but that doesn't help...

Could anyone please tell me how are those two expressions equivalent?! what are the identities used?