Identity Definition and 1000 Threads

since euler's identity states e^pi*i+-1 is it okay to write down e^pi*= e * -1/e = -1 and would for integers the same rule apply where a^pi*i= a * -1/a = -1 or is it only for the constant e? this might be a stupid question, but i cant find an answer and im curious

Hi, I'm missing the jump between these steps. Why should integrands with different variables be able to use the pythagorean identity? The substitution used was u = 4 - x and the bounds remain the same. Any help appreciated thanks.

My approach, ##\cos^2 \frac{1}{2} A = \dfrac{s(s-a)}{bc}## ... then it follows that, ##\dfrac{1}{a} \cos^2 \dfrac{1}{2} A+ \dfrac{1}{b} \cos^2 \dfrac{1}{2} B +\dfrac{1}{c} \cos^2 \dfrac{1}{2} C =\dfrac{s(s-a)+s(s-b)+s(s-c)}{abc}## ##=\dfrac {3s^2-s(a+b+c)}{abc}## ##=\dfrac{3s^2-2s^2}{abc}##...

Let ##P(n)## be the statement that $$ F_n \text{ is even} \iff (3 \mid n) $$ Now, my base cases are ##n=1,2,3##. For ##n=1##, statement I have to prove is $$ F_1 \text{ is even} \iff (3 \mid 1) $$ But since ##F_1 = 1## (Hence ##F_1## not even) and ##3 \nmid 1##, the above statement is...

Hi, I am a physics graduate but now I am teaching middle school science, I have been struggling because it seems that everyone is looking at me as a teacher rather than a physicist and the fact that I lost a lot of my ability to do physics only makes things worse. was anyone in a similar...

Consider this proof: Is it a valid proof? When we divide by ##z##, we assume that ##z \neq 0##. So, we cannot put ##z=0## on the next step. IOW, after dividing by ##z## we only know that $$c_1+c_2z+c_3z^2+...=d_1+d_2z+d_3z^2+...$$ in a neighborhood of ##0## excluding ##0##.

Hi all, I'm trying to verify the following formula (from Fetter and Walecka, just below equation (12.38)) but it doesn't quite make sense to me: where and The authors are using the fact that ##\delta(ax) = |a|^{-1}\delta(x)## but to me, it seems like the...

Hi together! Say we have ## \Lambda_q{(A)} = \{\mathbf{x} \in \mathbb{Z}^m: \mathbf{x} = A^T\mathbf{s} \text{ mod }q \text{ for some } \mathbf{s} \in \mathbb{Z}^n_q\} ##. How can we proof that this is a subgroup of ##\mathbb{Z}^m## ? For a sufficient proof we need to check, closure...

Since ##AB = B##, so matrix ##A## is an identity matrix. Similarly, since ##BA = A## so matrix ##B## is an identity matrix. Also, we can say that ##A^2 = AA=IA= A## and ##B^2 = BB=IB= B##. Therefore, ##A^2 + B^2 = A + B## which means (a) is a correct answer. Also we can say that ##A^2 + B^2 =...

Hey all, I saw a formula in this paper: (https://arxiv.org/pdf/physics/0011069.pdf), specifically equation (22): and wanted to know if anyone knew how to derive it. It doesn't seem like a simple application of BCH to me. Thanks.

My Attempted Proof ##R^{mn}_{;n} - \frac {1} {2} g^{mn} R_{;n} = 0## ##R^{mn}_{;n} = \frac {1} {2} g^{mn} R_{;n}## So, we want ##2 R^{mn}_{;n} = g^{mn} R_{;n} ## Start w/ 2nd Bianchi Identity ##R_{abmn;l} + R_{ablm;n} + R_{abnl;m} = 0## Sum w/ inverse metric tensor twice ##g^{bn} g^{am}...

There is a proof that shows by induction (and by contradiction) that the identity permutation decomposes into an even number of transpositions. The proof is presented in the first comment here...

Hello. Does anybody know a proof of this formula? $$J_{2}(e)\equiv\frac{1}{e}\sum_{i=1}^{\infty}\frac{J_{i}(i\cdot e)}{i}\cdot\frac{J_{i+1}((i+1)\cdot e)}{i+1}$$with$$0<e<1$$ We ran into this formula in a project, and think that it is correct. It can be checked successfully with numeric...

I don't understand why the identity is mentioned in the group's definition and how I am supposed to incorporate it into the table. I honestly have missed some lectures on Linear Algebra, and I can't find any examples or definitions for this in the prof's notes. I'd appreciate some help for sure...

I would appreciate help walking through this. I put solid effort into it, but there's these road blocks and questions that I can't seem to get past. This is homework I've assigned myself because these are nagging questions that are bothering me that I can't figure out. I'm studying purely on my...

I'm trying to use the following trigonometric identity: $$ a \cos ( \omega t ) + b \sin ( \omega t ) = \sqrt{a^2+b^2} \cos ( \omega t - \phi )$$ Where ##\phi = \tan^{-1} \left( \frac{b}{a} \right)## for the following equation: $$ x(t) = -\frac{g}{ \omega^2} \cos ( \omega t) + \frac{v_o}{...

Refreshing on trig. today...a good day it is...ok find the text problem here; With maths i realize one has to keep on refreshing at all times... my target is to solve 5 questions from a collection of 10 textbooks i.e 50 questions on a day-day basis...motivation from late Erdos...

I found this identity: ##x\int e^{-x^2} dx - \int \int e^{-x^2} dx dx = e^{-x^2}/2## by solving the integral of ##x*e^{-x^2}## and then finding its integration-by-parts equivalent. Is this identity useful at all?

I have a question about the Thermodynamic Identity. The Thermodynamic Identity is given by dU = TdS - PdV + \mu dN . We assume that the volume V and that the number of particles N is constant. Thus the Thermodynamic Identity becomes dU = TdS . Assume that we add heat to the system (we see that...

My tests are submitted and marked anonymously. I got a 2/5 on the following, but the grader wrote no feedback besides that more detail was required. What details could I have added? How could I perfect my proof? Beneath is my proof graded 2/5.

My tests are submitted and marked anonymously. I got a 2/5 on the following, but the grader wrote no feedback besides that more detail was required. What details could I have added? How could I perfect my proof? Beneath is my proof graded 2/5.

My intuition about the Lie algebra is that it tries to capture how infinitestimal group generators fails to commute. This means ##[a, a] = 0## makes sense naturally. However the Jacobi identity ##[a,[b,c]]+[b,[c,a]]+[c,[a,b]] = 0## makes less sense. After some search, I found this article...

So, this problem I sort of get conceptually but I don't know how I can possibly rewrite (idX)∗ : π1(X) → π1(X). Does this involve group theory? It's supposed to be simple but I honestly I don't see how. Again, any help is greatly appreciated. Thanks.

Dear All, I am trying to prove the following identity: \[\binom{n}{k}=\binom{n-2}{k}+2\binom{n-2}{k-1}+\binom{n-2}{k-2}\] My attempt was based on transforming the binomial coefficients into fractions with factorials, and then elimintating similar expressions. Somehow it didn't work out. I...

I am reading this book: https://web.stanford.edu/class/math285/ts-gmt.pdf on page 2 in remark 1.5(1), it's written that: ##\cap_{j=1}^\infty A_j = X\setminus (X\setminus \cup_{j=1}^\infty A_j)## this seems totally wrong, shouldn't it be ##X\setminus \cup_{j=1}^\infty (X\setminus A_j)## ? I...

First of all, sorry for the title I don't know the name of this formula and that's part of the problem, I can't find anything on google. I have to show the identity above. Here's what I did. I don't know if this is correct so far. ##\vec{u} + \vec{r}(\vec{\nabla} \cdot \vec{u}) + i(\vec{L}...

This alternating series indentity with ascending and descending reciprocal factorials has me stumped. \frac{1}{k! \, n!} + \frac{-1}{(k+1)! \, (n-1)!} + \frac{1}{(k+2)! \, (n-2)!} \cdots \frac{(-1)^n}{(k+n)! \, (0)!} = \frac{1}{(k-1)! \, n! \, (k+n)} Or more compactly, \sum_{r=0}^{n} (...

Cool article by By Molly Harris (BBC), 23rd April 2019 https://www.bbc.com/travel/article/20190422-the-swiss-language-that-few-know I've wondered about places like Alsace and Lorraine that have moved back and forth among two nations/states/regions, or for that matter, the borders of nations...

Came across this trig identity working another problem and I've never seen it before in my life. I don't need to prove it myself, necessarily, but I would really like to see a proof of it (my scouring of the internet has yielded no results). If someone more trigonometrically talented than myself...

What I mean is the way that a product of cosines in which the angles increment the same amount is equal, with some extra terms, of the sum of the cosines. It is discussed here...

I have accidentally derived a very wrong result from the contracted Bianchi identity and I can't see where the error is. I'm sure it's something obvious, but I need someone to point it out to me as I've gone blind. Thanks! Start with the contracted Bianchi identity, $$ \nabla_a \left(...

Is there a simplifcation of ##g^{\alpha \delta}g_{\beta \gamma} ## or what is equal to ?

All we should need for this problem are the basic rules for the Grassmann algebra \begin{equation*} \{ \theta_i, \theta_j\} = 0, \quad \theta^2_i=0 \end{equation*} \begin{equation*} \int d\theta_i = 0, \quad \int d\theta_i \ \theta_i = 1 \end{equation*} Starting from left to right...

I tried to understand proof of this identity from electromagnetics. but I was puzzled at the last expression. why is that line integral of dV = 0 ? In fact, I'm wondering if this expression makes sense.

The book I'm following (Gallian) basically says: r can't be 1 since then it won't map all elements to themselves. If r=2, then it's already even, nothing else to do. If r>2, Then consider the last two factors: ##\beta_{r-1} \beta_r##. Let the last one be (ab). Since the order of elements...

My attempt at this: From the general result $$\int \frac{d^Dl}{(2\pi)^D} \frac{1}{(l^2+m^2)^n} = \frac{im^{D-2n}}{(4\pi)^{D/2}} \frac{\Gamma(n-D/2)}{\Gamma(n)},$$ we get by setting ##D=4##, ##n=1##, ##m^2=-\sigma^2## $$-\frac{\lambda^4}{M^4}U_S \int\frac{d^4k}{(2\pi)^4} \frac{1}{k^2-\sigma^2} =...

How do I work this out? Can’t seem to get my head around it! Thanks

Attempt : I could not progress far, but the following is what I could do. $$\begin{align*} \mathbf{\text{LHS}} & = (\tan A+\tan B+\tan C)(\cot A+\cot B+\cot C) \\ & = 3+\tan A \cot B+\tan B \cot A+\tan A \cot C+\tan C \cot A+\tan B \cot C+\tan C \cot B\\ & = 3+\frac{\tan^2A+\tan^2B}{\tan A \tan...

We have some potential that depends on slowly varying parameters ##\lambda_a##. Using the angle-action variables ##(I, \theta)##, the claim is that we can define a two-form$$W_{ab} = \left\langle \frac{\partial \theta}{\partial \lambda_a} \frac{\partial I}{\partial \lambda_b} - \frac{\partial...

I am probably missing a crucial point here, but what does it means that (12)(34) squares to the identity? How do we prove it? ((12)(34))² = (12)(34)(12)(34) = (12)(12)(34)(34) = (12)(34) ##\neq I ## Is not this the algorithm?

Start with the Bianchi identity,$${\nabla_{[a} R_{bc]d}}^e = 0$$$${\nabla_{a} R_{bcd}}^e - {\nabla_{a} R_{cbd}}^e + {\nabla_{b} R_{cad}}^e - {\nabla_{b} R_{acd}}^e + {\nabla_{c} R_{abd}}^e - {\nabla_{c} R_{bad}}^e = 0$$Use definition of Ricci tensor$$ \left[ {\nabla_{a} R_{bcd}}^e + \nabla_b...

$$\nabla \times B(r)=\frac{\mu _0}{4\pi} \int \nabla \times J(r') \times \frac{ (r-r')}{|r-r|^3}dV'$$ using the vector identity: $$\nabla \times (A \times B) = (B \cdot \nabla)A - B(\nabla \cdot A) - (A \cdot \nabla )B + A(\nabla \cdot B)$$ ##A=J## and ##B=\frac{r-r'}{|r-r'|^3}## since...

Let us suppose we have a function such that $$z = e^{1/ab} - 1$$ Where we have two free parameters, a and b. Q1) Can we say that as ##b \rightarrow \infty##, ##z = 0##? Or, since ##a## is a free parameter, there is always some value for ##a## such that ##z \neq 0## for ##b \rightarrow...

Am I(always) legitimized to write ##-(a-b)^n=(b-a)^n##? I don't know why but it's confusing me... can't really understand when and why I can use that identity

We use ##g_{\alpha \beta} = \vec{e}_{\alpha} \cdot \vec{e}_{\beta}## to show that$$\partial_c g_{ab} = \partial_c (\vec{e}_a \cdot \vec{e}_b) = \vec{e}_a \cdot \partial_c \vec{e}_b + \vec{e}_b \cdot \partial_c \vec{e}_a$$then because ##\partial_{\alpha} \vec{e}_{\beta} := \Gamma_{\alpha...

Hi! The topic is electrodynamic but it's a question about Nabla identity. Given $$ F = (p \cdot \nabla)E $$ How does one compute F? Is this correct? $$ F = \sum_{i} p_i \partial_{i} E_{i} e_{i} $$

Theorem “Identities of + are unique”: O₁ = O₂ Proof: O₁ = Left Identity of + O₁ + x I'm a little confused where to begin this proof, I don't know if that is the first step either I think it is. Proofs are not a strength of mine so I struggle to see how to show that O₁ = O₂. Any guidance would...

##(\nabla\times\vec B) \times \vec B=\nabla \cdot (\vec B\vec B -\frac 1 2B^2\mathcal I)-(\nabla \cdot \vec B)\vec B## ##\mathcal I## is the unit tensor

So the Legendre transforms are straightforward; define ##S_1=S-\beta E## and ##S_2= S-\beta E + \beta \mu n## then we get: ##dS_1 = -Ed\beta - \beta \mu dn + \beta PdV## ##dS_2 = -Ed\beta + nd(\beta \mu) + \beta PdV## And so by applying the equality of mixed partials of ##S_1## and ##S_2## we...

I was just wondering what is wrong with the following logic; From the Gibbs-Duhem relation we get, ##\frac{\partial \mu}{\partial P}\Big\rvert_T = v## Now consider, ##\frac{\partial v}{\partial \mu}\Big\rvert_T = \frac{\partial }{\partial \mu}\Big (\frac{\partial \mu}{\partial P}\Big\rvert_T...