Notation Definition and 1000 Threads

Questions For the two passages in the Background below highlighted in bold, I am having trouble figuring how the maps are defined. I understand that we have commutative ring ##R## admitting a total quotient ring ##F##, a, subset ##M\subset R## which is also a regular multiplicative system. A...

The following is from the paper "A Group Epimorphism is Surjective" by C.E. Linderholm Background Note: where it says the function from ##A/H## to ##A/H##, should it not be ##H/A## to ##H/A?## Questions There are some points about the above proof I am not clear on. I understand that...

The following is taken from Abstract Algebra: A First Course by Stephen Lovett Background Exercise Let ##D=\{2^a 3^b\mid a,b,\mathbb{N}\}## as a subset of ##\mathbb{Z}##. Prove that ##D^{-1}\mathbb{Z}## is isomorphic to ##\mathbb{Z}[\frac{1}{6}]## even though ##D\neq\{1,6,6^2,\ldots\}##...

The following is taken from a chapter from Algebraic geometry and commutative algebra by S Bosch Background and ##\mathrm{Spec}R=\{\mathfrak{p}\subset R:\mathfrak{p}\text{ prime ideal in }R\}##. For reference, I included a screenshot of the typed out quoted passage above below: Questions...

$$u(\nu, T) = \frac{8 \pi \nu^2}{c^3} \cdot \frac{h \nu}{e^{h \nu / k T} - 1}$$ Now using the relation ##c=\nu \lambda##, $$u(\lambda, T)=8 \pi h \frac{1}{\lambda^3} \cdot \frac{1}{e^{hc / (\lambda k T)} - 1}$$ Now this is still per unit frequency. To get it to per unit wavelength, then $$...

I have attached a copy of my work as a pdf file. Please excuse my non-rigorous usage of notation. Thank you so much!

Hi, I'm simply searching for some standard symbol (in place of an equals sign) to indicate that a matrix is a representative of a tensor in some given basis. Is there any standardized symbol like this, or how is this usually written in literature? E.g. say we have a O(1,1) metric tensor gμν...

I'm reading Liang's book on General Relativity and Differential Geometry, and came across this part: I just want to have a crystal clear understanding of why this notation is chosen. Basis transformation would be an automorphism from ##V## to ##V##, and there's a result saying that the set of...

So in my textbook there's a basic problem where you solve for the final velocities of two hockey pucks, which happen to have different colors which are red and blue, using conservation of momentum. The notation that the textbook uses to express the final velocities of the pucks is ##v_{1,f}##...

My question is simply about the notation used here. What does $$F_s=-\frac{\partial V}{\partial s}$$ mean exactly?

Penrose demonstrates in his book "The Road to Reality" a "diagrammatic tensor notation", e.g., As I haven't seen it anywhere else, I wonder if anybody else uses it.

The gravitational potential leads to velocity in downward direction, but spring potential does in upward direction. So should these energies have different signs (plus and minus or vice-versa)?

In the 1975 Wu–Yang paper on electromagnetism=fiber bundle theory Table 1: https://journals.aps.org/prd/pdf/10.1103/PhysRevD.12.3845 Wu & Yang use the notation ##\mathrm{U}_1(1)## for the bundle of electromagnetism and ##\mathrm{SU}_2## for the isospin gauge field. I am unfamiliar with this...

All I know is that e subscript r must be a vector cos the book says so, but what does it mean, is it, a konstant in vector form? I'm confused by it (page one, chapter one spacetime and geometry by SeanCaroll) Help is appreciated Edit. Is vector r describing the curvature that takes place ?

In my introductory modern physics class, I was asked to compute the Taylor Series for exp(-ax^2) about the value x = 0 to second order in x. I am unfamiliar with the what "exp" before the function means, despite having approximated functions with Taylor Series before. I think there was some gap...

Hi PF, $$\int \cos ax\,dx,\quad a\in{\mathbb R-\{0\}}\quad x\in{\mathbb{R}}$$ Let's make $$u=ax,\quad du=adx$$ and apply $$\int \cos u\,du=\sin u+C$$ $$\frac{1}{a}\int \cos ax\,adx=\frac{1}{a}\sin u+C$$ Substituting the definition of u $$=\frac{1}{a}\sin ax+C$$ Doubts: (i) Have I written well...

If I am correct, the wave function is presented as a vector in Hilbert Space. Alternatively this vector can be multiplied by the identity operator. Is there a preference for one notation or the other? Are they both possible representations of the same wave function?

So, first of all, why and how are we taking the derivative of the vector r or s as d/dt if t is not a parameter of the equations? Second question is what is the difference between d/dt(r) and d(theta)/dt(r) and also between d/dt(s) and d(theta)/dt(s)? Like, both of these appear at the bottom of...

For this, What was wrong with the notation I used for showing that I has swapped the rows? The marker put a purple ? Any help greatly appreciated! Many thanks!

Hi. I'm self-studying functions which relate to calculus. Let me post what I feel I know and what I'm not grasping yet. Please correct any mistakes I'm making. I'm just talking real numbers: A function is a rule that takes an input number and sends it to another number. We can describe it...

Hello, I realize this might sound dumb, but I'm having such a hard time understanding Einstein notation. For something like ∂uFv - ∂vFu, why is this not necessarily 0 for tensor Fu? Since all these indices are running through the same values 0,1,2,3?

I started my research in statistical digital signal processing two years ago, so I need to familiarize myself with all the notations people use in probability and statistics. I come from a deterministic science background. I name my variables based on what they mean. A velocity is a v , a...

Hi If A is a linear operator but not Hermitian then the expectation value of A2 is written as < ψ | A2| ψ >. Now if i write A2 as AA then i have seen the expectation value written as < ψ | A+A| ψ > but if i only apply the operators to the ket , then could i not write it as < ψ | AA | ψ > ? In...

I suspect it will help if you know about my background: I did some linear algebra in university but never used it and am now in my mid 60s. I am interested in understanding the mathematics of quantum physics. I have read a number of layman's texts on quantum mechanics, but they all gloss over...

In hp 50g emulator, I performed this computation ## \lim\limits_{x\to\infty} [2^{2x} \times (\frac13)^x + (\frac12)^{2x}\times (\frac23)^x ]## What is the meaning of +:0 ?

This is probably a stupid question but, ## \frac{d\partial_p}{d\partial_c}=\delta^p_c ## For the notation of a 4D integral it is ##d^4x=dx^{\nu}##, so if I consider a total derivative: ##\int\limits^{x_f}_{x_i} \partial_{\mu} (\phi) d^4 x = \phi \mid^{x_f}_{x_i} ## why is there no...

Until now in my studies - matrices were indexed like ##M_{ij}##, where ##i## represents row number and ##j## is the column number. But now I'm studying vectors, dual vectors, contra- and co-variance, change of basis matrices, tensors, etc. - and things are a bit trickier. Let's say I choose to...

So, I have recently been trying to learn how to work with tensors. In doing this, I have come across Einstein Notation. Below is my question. $$(a_i x_i)_{e}= (\sum_{i=1}^3 a_i x_i)_r=(a_1 x_1+a_2 x_2+a_3 x_3)_r$$; note that the following expression is in three dimensions, and I use the...

I know that if ##\eta_{\alpha'\beta'}=\Lambda^\mu_{\alpha'} \Lambda^\nu_{\beta'} \eta_{\alpha\beta}## then the matrix equation is $$ (\eta) = (\Lambda)^T\eta\Lambda $$ I have painstakingly verified that this is indeed true, but I am not sure why, and what the rules are (e.g. the ##(\eta)## is in...

In this post [Observables][1] By Urs Schreiber he denotes the space of distributional sections in defenition 7.9 by ##\Gamma_{\Sigma}^{\prime}\left(E^*\right) ## That is if ##u \in \Gamma_{\Sigma}^{\prime}\left(E^*\right)## than ##u## is a linear functional that takes as argument...

When specifying DC vs AC voltage excitation, is it appropriate to type 50 VDC with DC as a subscript? OR should it be "V DC"? The same concept if it were AC.

I'm a beginner in the quantum mechanic and reading E. hill's paper explaining the intensity distribution of the doublet state for diatomic molecule (D->P). To calculate the intensities, D|(n;n')| and/or q(n,a,j;n'a',j') must be calculated, but I have never seen it before in linear...

What's the difference between a bra vector and ket vector in specifying spin states except for notational convenience when calculating probablility amplitudes? Are they equivalent?

Hi Physics Forum, I want to ask if there is an "appropriate" notation for the infinite self-iteraction of an analytic function ##f(x)##, that is ##f(f(f(...)))##. For example I know ##f^{(+\infty)}(x)## can be a way, but there is an operator notation as for the infinite sum...

I am wondering if I am using Einstein notation correctly in the following example. For a matrix ##R## diagonal in ##1##, except for one entry ##-1##, such as ##R = [1,-1,1]##, is it proper to write the following in Einstein notation: ##R_{\alpha} R_{\beta} = \mathbb{1}_{\alpha \beta} ##, such...

Why should it be ##\int_{a}^{x} f(t)dt## ? Couldn't it be like this: Let F(x) = ##\int f(x)dx## so ##\int_{a}^{x} f(x)dx## = F(x) - F(a) Thanks

I am reading Tensor Calculus for Physics by Dwight E. Neuenschwander and am having difficulties in confidently interpreting his use of Dirac Notation in Section 1.9 ... in Section 1.9 we read the following: I need some help to confidently interpret and proceed with Neuenschwander's notation...

I am trying to convert the attached picture into dirac notation. I find the LHS simple, as it is just <ψ,aφ>=<ψIaIφ> The RHS gives me trouble as I am interpreting it as <a†ψ,φ>=<ψIa†Iφ> but if I conjugate that I get <φIaIψ>* which is not equiv to the LHS. *Was going to type in LaTex but I...

hi guys I am trying to learn special relativity and relativistic quantum mechanics on my own and just very confused about the different conventions used for the notation!?, e.g: the four position 4-vector some times denoted as $$ x_{\mu}=(ct,-\vec{r})\;\;or\;as\;x_{\mu}=(ict,\vec{r}) $$ or...

The notation I think best describes it is ## F = \lVert\int^{space}_s|\vec{V}|ds\rVert ## So you have a vector field V in a 3d space. For each point you integrate over all of space (similar to a gravitational or electromagnetic field) *but* vectors in opposite directions do not cancel, they...

In the second paragraph on page 25 of Wald's General Relativity he rewrites T^{acde}_b as g_{bf}g^{dh} g^{ej}T^{afc}_{hj} . Can anyone explain this? I am confused by the explantion given in the book. Especially puzzling is that the inverse of g seems to be applied twice, which I can't make sese...

The wavefunction of ##|\psi\rangle## is given by the bra ket ##\psi (x,y,z)= \langle r| \psi\rangle## I can convert the wavefunction from Cartesian to polar and have the wavefunction as ## \psi (r,\theta,\phi)## What bra should act on the ket ##|\psi\rangle## to give me the wavefunction as ##...

If I have an equation, let's say, $$\mathbf{A} = \mathbf{B} + \mathbf{C}^{Transpose} \cdot \left( \mathbf{D}^{-1} \mathbf{C} \right),$$ 1.) How would I write using index notation? Here [FONT=-apple-system][FONT=-apple-system][FONT=-apple-system]A is a 4th rank tensor...

##\bar{\mathcal{O}}## is moving with a velocity ##v## relative to ##\mathcal{O}## along ##x^{1}## The Lorentz transformations between a Frame ##\mathcal{O}## and ##\bar{\mathcal{O}}## is given by: $$\Delta x^{\bar{0}} = \gamma\left(\Delta x^0 - v\Delta x^1\right)$$ $$\Delta x^{\bar{1}} =...

.

Hello, I would like help to clarify what det( {\delta \over \delta J}) W(J) (equation 15.79) actually means, and why it returns a number (and not a matrix). This comes from the following problem statement (Kaku, Quantum Field Theory, a Modern Introduction) Naively, one would define det...

This is my attempt to re-write the geodesic deviation equation in the special case of 3 dimensions and +++ signature in matrix notation. We start with assuming an orthonormal basis. Matrix notation allows one to express vectors as column vectors, and dual vectors as row vectors, but by...

Goldstein pg 192, 2 edIn a Cartesian three-dimensional space, a tensor ##\mathrm{T}## of the ##N## th rank may be defined for our purposes as a quantity having ##3^{N}## components ##T_{i j k}##.. (with ##N## indices) that transform under an orthogonal transformation of coordinates...

I'am interested in the notation of the approximation in quantum phase estimation algorithm. In the literature there are different definitions, which I divide into two cases here. Both different in their definition of the ##\delta##. In both cases I start with a quote of the source and show an...

I came across this notation when reading the wiki article on Venturi effect, which baffled me. Can someone please tell me its meaning?