Notation Definition and 1000 Threads

I was studying mathematical logic and came across this statement of group theory I'm having a hard time in understanding it. I have concluded that ##G## is any set but not an empty one, ##\circ## is a function having input as two variables (both variables are from set...

I have an identity $$\vec{\nabla} \times (\frac{\vec{m} \times \hat{r}}{r^2})$$ which should give us $$3(\vec{m} \cdot \hat{r}) \hat{r} - \vec{m}$$ But I have to derive it using the Einstein summation notation. How can I approach this problem to simplify things ? Should I do something like...

If we want to expand a function ##f(x)## up to first order around ##x = 0## say, we usually write ##f(x) = f(0) + (df/dx)|_0 x + \mathcal O(x^2)##. But what if I want to expand ##f(x)## in the whole series, and showing only the first order term in x? What notation do you use for that? (Aside...

Here is his paper. I don't see what the big deal about it is. https://arxiv.org/pdf/1911.00892.pdf

I've stumbled over this article and while reading it I saw the following statement (##\xi## a vectorfield and ##d/d\tau## presumably a covariant derivative***): $$\begin{align*}\frac{d \xi}{d \tau}&=\frac{d}{d \tau}\left(\xi^{\alpha} \mathbf{e}_{\alpha}\right)=\frac{d \xi^{\alpha}}{d \tau}...

I was playing around with a graphing program and sketching polar graphs involving tall power towers, when I noticed that ##sin(\theta) \uparrow \uparrow a## has an alternating appearance depending on whether ##a## is odd or even. I also noticed that the area enclosed by these alternating graphs...

If I see ##f(x_ie_i)## I assume it means ##f(\Sigma x_ie_i)## (summing in the domain of f) but what if I instead wanted to write ##\Sigma f(x_ie_i)## (summing in the range)? Is there a way to distinguish between these in Einstein’s summation notation?

A bit vague, I know, but I guess asking is worth a shot.

What does it mean when you see an equation that has some terms then there is a O(λ) or O(π) in it? For example N(λ) = Cm Vol(M, g)λ m + O(λ m−1 ) . I self study and see this often but it is never explained in the books I use and it drives me nuts. Is this some sort of error function?

I'm new to classical mechanics. I've done enough work with vectors to get the basics. But, I'm having trouble understanding the notation on this MIT presentation I found on circular motion: http://web.mit.edu/8.01t/www/materials/Presentations/Presentation_W04D1.pdf On slide 23, for example, I...

I worked out the expectation values of the components of a 1/2 spin particle. However, I'm confused about Griffiths notation for the x and y components. For the x component I got, ## \left< S_x \right> = \frac \hbar 2 (b^*a+a^*b)## which is correct, but Griffiths equates this to ##...

Hello everyone, I'm stuck on the question which I have provided below to do with Dirac notation: In these questions |a>, |b> and |c> can be taken to form an orthonormal basis set Consider the state |ξ> = α(|a> − 2|b> + |c>). What value of α makes |ξ> a normalised state? I'm brand new to Dirac...

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So I know the formal definition of Big-O, which states that ##f(n) = O(g(n))## if and only if there exists ##{C > 0, n_0 > 0}## such that ##|f(n)|\leq{C}{g(n)}~\forall{n>n_0}##. Here's what I think the proof should go (please bear with me, I have no idea what I'm doing): Suppose there exists a...

So if ##P_{0}## is an event, and I have ##\mathcal {g_{\mu\nu}(P_{0})}=0## and ##\mathcal {g_{\mu\nu,\alpha\beta}(P_{0})}\neq0##, does this notation mean ##\partial\alpha\partial\beta## or simply ##\partial(\alpha\beta)##? And what is the significance of it? Why can't it be zero in curved spacetime?

Suppose I have a system of two (possibly interacting) spins of 1/2. Then the state of each separate spin can be written as a ##\mathbb{C}^2## vector, and the spin operators are made from Pauli matrices, for instance the matrices ##\sigma_z \otimes \hat{1}## and ##\hat{1} \otimes \sigma_z##...

What is the notation to show element-wise square root of a vector or matrix?

Can someone clarify the use of semicolon in I know that semicolon can mean covariant derivative, here is it being used in the same way (is expandable?) Or is a compact notation solely for the components of?

On page 224 of the 5th edition of Classical Dynamics of Particles and Systems by Stephen T. Thornton and Jerry B. Marion, the authors introduced the ##δ## notation (in section 6.7). This notation is given by Equations (6.88) which are as follows: $$\delta J = \frac{\partial J}{\partial...

I am studying QFT from A First Book of QFT. It is a very well-written book. However, due to some personal reasons, I cannot buy the printed book at this moment. So I borrowed this book from a person (who, in turn, borrowed it from his university library), and scanned it. Everything is fine...

Hi! I am studying Shankar's "Principles of QM" and the first chapter is all about linear algebra with Dirac's notation and I have reached the section "The Characteristic Equation and the Solution to the Eigenvalue Problem" which says that starting from the eigenvalue problem and equation 1.8.3...

I started reading QFT recently. With respect to the Dirac spinors, I was introduced to Feynman's slash notation: OR There are sites which say how this can be inserted in LaTeX documents. I checked out the LaTeX guide of PF, but found that it is not listed there. Writing...

Hi there, I'm just starting Zee's Einstein Gravity in a Nutshell, and I'm stuck on a seemingly very easy assumption that I can't figure out. On the Tensor Field section (p.53) he develops for vectors x' and x, and tensor R (with all indices being upper indices) : x'=Rx => x=RT x' (because R-1=RT...

What does the colon notation, :, mean?

I'm revising for my condensed matter exam, and I've never understood the point group notation, in particular of the 32 crystallographic point groups, so let me try and explain what I understand of it and point out where my confusion lies. Please point out any other misunderstandings I have. We...

Hi all, What is the general set notation for specifying a vertical asymptote and domain for a periodic function? For example, if I have a periodic function which has a period of pi/2, and within that period, a vertical asymptote occurs at pi/4. The domain is R, excluding that vertical...

As the title says, I'm looking for a keyboard shortcut to type "I" with a ˜ The option+n shortcut does not work for I or V on MacOS. Thanks!

Hello everyone, I have been working through some research papers on a topic that really interests me, but I believe I am misunderstanding a few things about Dirac Notation. I have expressions that read: \begin{align*} &< \psi_n \mid g(H - E_{n+1}) \mid \psi_n> \text{,} \\ &< \psi_n \mid (H -...

Say ##z = f(x,y)## then I learned that ##dz = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dy## ##dz = f_x*dx + f_y*dy## The question is whether this expression on ##dz## is really proper. The question comes from this: I know the definition of say ##\frac{dz}{dt}## as...

Hi PF! Can someone help me understand the notation here (I've looked everywhere but can't find it): given a function ##f:G\to \mathbb R## I'd like to know what ##C(G),C(\bar G),L_2(G),W_2^1(G),\dot W_1^2(G)##. I think ##C(G)## implies ##f## is continuous on ##G## and that ##C(\bar G)## implies...

Hello, I am an undergrad currently trying to understand General Relativity. I am reading Sean Carroll's Spacetime and Geometry and I understand the physics (to a certain degree) but I am having trouble understanding the notation used as well as the ideas for tensors, dual vectors and the...

in (3) they say any polynomial of degree less than 2, yet the example is degree 2 ? ok this is due tomro, so hopefully I can get a handle on it today I think this is familiar, just is the notation is stumping me. really appreciate any insight... did a screenshot to avoid typos:cool:

Hello I have been going through the cosmology chapter in Choquet Bruhats GR and Einstein equations and in definition 3.1 of chapter 5 she defines the sectional curvature with a Riemann( X, Y;X, Y) (X and Y two vectors) I don't understand this notation, regarding the use of the semi colon, is it...

Hi PF! I am suppose to determine if the following rule is a distribution $$\langle u,\phi \rangle = \int_0^1 \frac{u(x)}{x} \, dx$$ and then also $$\langle u,\phi \rangle = \int_{-\infty}^\infty \phi + 1 \, dx.$$ The notation is throwing me off. At first I thought I had to show ## \langle Au +...

Hi everyone! Sorry for the bad English! I'm trying to read the "entanglement between photons that never coexisted " from 2012. Avaliable at: https://arxiv.org/abs/1209.4191 And there's this equation: ## |φ± \rangle = \frac 1{√2}(|HaHb ± |VaVb\rangle) ## ## |ψ± \rangle = \frac 1{√2}(|HaVb ±...

Homework Statement I'm given a gas equation, ##PV = -RT e^{x/VRT}##, where ##x## and ##R## are constants. I'm told to find ##\Big(\frac{\partial P}{\partial V}\Big)_T##. I'm not sure what that subscript ##T## means? Homework Equations ##PV = -RT e^{x/VRT}## Thanks a lot in advance.

Homework Statement The question involves using sigma notation of Riemann sums to find the area under the graph of ##x^2+\sqrt {1+2x}##. I managed to calculate most of the values and I have ##16+\frac 8 3 + \Sigma {\frac 2 n \sqrt {9 + \frac {4i} n}}## Homework Equations [/B] ##\Sigma i= \frac...

Hey everyone, I wasn't sure if this belonged in the general math forum or not, so I posted it here instead (mods - feel free to move if it belongs elsewhere). What I want to know is how to properly write out a computer algorithm in proper math notation. Take this code for example: Height...

In this video (the link should take you 23 minutes in, where) Professor Susskind writes a theorem. (He says E and S are functions of T and V.) I am just wondering if what he writes is an abuse of notation? And if so, how would you write it? (Susskind gives his own proof from 26:30 to 36:00...

In Einstein summation convention, the summation occurs for upper indices and its repeated but lower indices. However I have some confusion 1) $${\displaystyle v=v^{i}e_{i}={\begin{bmatrix}e_{1}&e_{2}&\cdots &e_{n}\end{bmatrix}}{\begin{bmatrix}v^{1}\\v^{2}\\\vdots \\v^{n}\end{bmatrix}},\ \qquad...

I think there are different conventions. Once I worked with XPS, and there we used the spin quantum number, the letter for the orbital, and a number for the energy leverl I think, to characterize photoelectrons. Another notation uses, to characterize an atomic state, the convention...

I believe it is also called "einstein notation"? The "notation-thingy" using kronecker delta, levi-civita and etc to simplify expressions with div, grad, curl (i took the course in my native language so i am not entirely sure what the notation or technique is called in english). Looking to get...

Using that ##\hat{a} =a = \sqrt{\frac{mw}{2 \hbar}} \hat{x} +\frac{i}{\sqrt{2mw \hbar}} \hat{p}## and ## a \dagger = \sqrt{\frac{mw}{2 \hbar}} \hat{x} -\frac{i}{\sqrt{2mw \hbar}} \hat{p}## We can solve for x in term of the lowering and raising operator. Now, recently I read a derivation of...

I understand that when a sequence is described recursively, for example: ##a_1=2, a_{n+1} = \sqrt{3a_n}## then we mean that the first term is 2, the second term is ##\sqrt{3*2} = \sqrt{6}##, the third term is ##\sqrt{3*\sqrt{6}}##, and so on. What I do not understand is how to interpret the...

I am learning binomial theorem now on my long journey to calculus. I noticed that in older textbooks, the binomial coefficient looks like C(n on top,k on bottom) I don’t think that I can display it here and in newer ones,they look like ##\binom{n}{k}## is the old notation outdated?or this is...

<Moderator's note: Moved from a technical forum and thus no template.> Hi all, I have attempted this question but have a few queries on how transformers work, and what the dot notation represents. (a) The flux would be clockwise around the iron core. (b) This is the question where it gets a...

In Chapter 7: Hamilton's Principle, in the Classical Dynamics of Particles and Systems book by Thornton and Marion, Fifth Edition, page 258-259, we have the following equations: 1. Upon squaring Equation (7.117), why did the authors in the first term of Equation (7.118) are summing over two...

Hello Everyone, I am trying to write the intersection of a physical problem in the most compact way. I am not really familiar with Set Theory notation, but I think it has the answer. It is about the intersection of two circular areas: - Area 1: A - Area 2: B If I want to write this in Set...

Limh→0+ (f(rh,h))/h Is the f(rh,h) part the same as f(r+h)-f(h)? I have never seen this before and googling for a long time didn't help, there are no videos with this notation and it's not in my book so, am I just to assume it is? because it doesn't look like it should be the same. Anyone know...

First, I don't know if this is the right place so if not, please direct me. Thank you. As for the question, I am in a discrete mathematics class online. The instructor is practically non-existent when asking for help simply saying to "refer to the book for clarification". I have scoured google...