Index Definition and 831 Threads

This is from Giancarlo Bernacchi's book "Tensors made easy": And this is from Daniel Fleisch's book "A Student's Guide to Vectors and Tensors": The meaning of the two lower indices is exchanged in the two explanations. Is one of them correct and the other one wrong, or is there no consensus...

Proof: Consider the index of a limit cycle. By definition, a limit cycle is an isolated periodic solution of an autonomous system represented in the phase plane by an isolated closed path. The theorem states: If ## \Gamma ## surrounds ## n ## equilibrium points ## P_{1}, P_{2}, ..., P_{n} ##...

If I'm not mistaken, the curl can be expressed like this in index notation: ##(\nabla \times \vec v)^i = \epsilon^{i j k} \nabla_j v_k = \epsilon^{i j k} (\partial_j v_k - \Gamma^m_{j k} v_m) = \epsilon^{i j k} \partial_j v_k ## (where the last equality is because ##\epsilon^{i j...

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...

Hello, When dealing with a dataframe with two columns, X and T where T is time, the time column is often converted to be the index of the dataframe itself (which by default is 0,1,2,3,...). What is the advantage of doing that? I know how to implement that but I am not sure what the main benefit...

Hi, Trying to Latex in an estimator y^ together with an index, y_i . The latter is obvious as ##y_i##, but I can't get the estimator ^ without mangling it all. Any ideas?

I am interested in experimenting with AR lenses and one of the main technologies is a waveguide. They rely on total internal reflection, which needs light to go from a higher to lower refractive index medium. I am reading that current AR lenses use an outer layer with a lower refractive index to...

Hi, I've attached the photo of the diagram, a photo of my drawings on the diagram. for a): Since the prism is an equalateral triangle, all angles inside the prism is 60°. This means the angle adjacent to α is 180° - 60° = 120°, which means the last angle is 180° - 24.5° - 120° = 35.5°. The...

I got this question from my son last night. If you Google "Why is Cherenkov radiation blue", you get this: Somewhat more substantial is the Wiki article on the Frank-Tamm formula. That formula ties the Cherenkov radiation wavelengths to the transmission characteristics at any specified...

For Ω=1, κ=0. Does the value of κ simply follow from the value of Ω, or can its value have an independent existence? So if Ω>1, does κ have to be 1?

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...

Problem statement : If ##\boldsymbol{\displaystyle x^{\displaystyle 2x^{\displaystyle 6}}=3}##, evaluate ##\boldsymbol{\left(x^{\displaystyle x^{\displaystyle x^{\displaystyle 6}}}\right)^{\displaystyle\sqrt 3} = ?}##. Attempt : I copy and paste my attempt using Autodesk...

The answer is this-: Because we can't declare a field of different data types within a same index in different document types. Say there's an index called "college". Then there are document types called "student" "teacher" "administration" "staff". What problem would occur if we allow this...

These are the steps of indexing in Lucene given in our syllabus-: The first step says that it is creating an index whereas the last step says that it's adding document to index. What's the difference between these two? Can I get an example. Here's what I think it should happen-: 1) Collect...

It's been stated that the index of refraction of materials varies with frequency throughout the EM spectrum. What are the index of refraction for various materials in the radio frequency?

Answer: I already found the answer for Sin theta 2 like the pic below But, I am still not sure about how to derive delta here. Can anyone show me they way? thank you

program main implicit none real,dimension(3),target::x=[0,4,5] real,pointer::y call point_to_biggest(x,y) !print *,y ! For testing purposes; to be unmuted only when y points to something contains subroutine point_to_biggest(array,ptr) implicit none...

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...

Hi all, I'm pretty rusty on my index gymnastics, I'm wondering if someone can explain to me the correct way to take the tensor product of a general 4-vector ## g^{\mu} ## and a tensor ## P^{/mu\ /nu} ## in Minkowski space. The part that is troubling me is the fact that both have a ## \mu ## as...

Source of my knowledge-: Encoding-: Decoding-: Other references-: Question-: There is no entry for index 7 in the dictionary while decoding, how do I fix this issue?

https://lh6.googleusercontent.com/W6yfx-b9ja9FUmZdtDcCh5mZCF5WgVu5pPtNdbTSKvel3CAjBY2ju83qcrIvTLM1D3CfyboNVl66aeOOaHgfxUMSlVXdQfoEojqK2wHcAh-yp0zwNXv3BXb0bzrxk4Ys0JdZB1k2...

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 A is a 4th rank tensor B is a 4th rank tensor C is a 3rd rank tensor D is a 2nd rank tensor I wrote it as...

See statement.

I was studying radio waves phenomenons and found this on Wikipedia: "Tropospheric ducting is a type of radio propagation that tends to happen during periods of stable, anticyclonic weather. In this propagation method, when the signal encounters a rise in temperature in the atmosphere instead of...

On the surface, the equation is simply the peak rarefractional pressure divided by the root of the applied frequency: ##MI = \frac{P_{ra}}{\sqrt{f}}## But the pressure is reduced/derated by an attenuation factor/coefficient that is dependent on depth and frequency e.g. ##0.3 \ dB / (cm \cdot...

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Hey all. Was wondering if anyone knew how I would go about determining the amount of reflectance that occurs when there is a gradual change in the refractive index. For example, if I have a material in air whose refractive index begins at e_r=1 (i.e. it matches the refractive index of the air)...

a) I managed to obtain some results that are roughly around what is given in the answers. Because \varepsilon_{st} and \varepsilon_{\infty} are values of \varepsilon_{1}, I used this approximation: n\approx \frac{1}{\sqrt{2}} (\varepsilon_{1}+\sqrt{\varepsilon_{1}^2})^{1/2} -> \varepsilon_{1} =...

> A diffraction grating, ruled with 300 lines per mm, is illuminated with a white light source at normal incidence. (ii) Water (of refractive index 1.33) now fills the whole space between the grating and the screen? What is the angular separation, in the first-order spectrum, between the 400 nm...

Is the curvature index κ necessarily zero in a flat universe with cosmological constant?

Considering the below equality (or equivalency), could someone please explain how the bounds and indices are shifted? $$\sum_{i=2}^{k}(h_i/f_{i-1})=\int_{1}^{k}(h(i)/f(i))di$$

I hate to create a thread for a step in a calculation, by I don't know what else to do. I'm having a lot of trouble reproducing E. Weinberg's index calculation (found here https://inspirehep.net/literature/7539) that gives the dimension of the moduli space generated by BPS solutions in the...

Hello fellow users, I've been given the Lorentz model to calculate the refraction index of a dielectric, the formula in its simplest way states that: ##n^2(\omega) = 1 + \frac {\omega^2_p} {\omega^2_0 - \omega^2}## Where ##\omega_p## is the plasma frequency and ##\omega_0## is the resonance...

So I am not really familiar with lens questions when there's 2 different refraction indexes. I tried using n1/p+n2/q=-(n2-n1)/R but it doesn't seem to work. p would be the actual location of the fly and q would be the virtual location, what the fish sees if I am understanding correctly. n1...

Hello:) i need some help trying to find some formulas that i can retrive the material index from at the end of simplification to use with an ashby chart to select materials. here is an example of deflection equation from a sphere used for a spherical cap and what material index we used to find...

I once naively think that the speed of light is also a constant in a medium in all inertial frames which is not the case. I tried to derive the result yet there is a discrepancy from the results I read in some articles. For example, from [Link to unpublished paper redacted by the Mentors], the...

Hi all, I am having some problems expanding an equation with index notation. The equation is the following: $$\frac {\partial{u_i}} {dx_j}\frac {\partial{u_i}} {dx_j} $$ I considering if summation index is done over i=1,2,3 and then over j=1,2,3 or ifit does not apply. Any hint on this would...

In my studies trying to get the Ricci tensor of 2° order i stuck in this expression: ##h_{\mu}^{\ \sigma},_{\lambda}h_{\sigma}^{\ \lambda},_{\nu}=h_{\mu \lambda},^{\sigma}h_{\sigma}^{\ \lambda},_{\nu}## So, to complete my calculations those quantities should be the same, but i don't understand...

Solution: (delta (ϑ)) = -(minus delta n) /(n*sqrt((n^2-1)))

Hi everyone! So, the problem I'm having has more to do with "how to pose the problem to solve it in some software as Matlab or similar". I have experimentally measured values ##\varepsilon_{exp}^i## with ##i=1,\cdots,6##, that is, I have 6 detectors. Then, I know (from a Monte Carlo...

I want my graph to display the tick values along the x-axis, instead of the index number. Because I use linspace, my data goes from 0 to 1, but my index ranges from 1 to 1000. My plot labels the x-axis from 1 to 1000, by default. I want to change the default (or just program it) to display 0 to...

ok, by direct substitution i know that either ##x=2## or ##x=4## but i would like to prove this analytically, would it be correct saying, ##xln 2= 2ln x## ##xln_{2}2=2 ln_{2}x## ##x=2 ln_{2}x## ##\frac {1}{2}=\frac { ln_{2}x}{x}## ##ln_{2}x^{1/x}##=##\frac {1}{2}## →##2^{1/2}##=##x^{1/x}##...

Do suspended sub-micron sized particles influence the index of refraction of a liquid?

Ok, so i finished my lab for physics 2 today. It was on reflection, refraction, Snells law etc. During my study, i looked up the refraction index of glass via this link: https://en.wikipedia.org/wiki/List_of_refractive_indices i noticed it had refraction indexes for a number of things that do...

I found some parts of Vol II, Chapter 25 basically unreadable, because I can't figure out his notation. AFAICT he's using a (+,-,-,-) metric, but these equations don't really make any sense: The first one is fine, and so is the second so long as we switch out ##a_{\mu} b_{\mu}## for ##a_{\mu}...

Eq35.17 and 35.18 is obtained,but the result of Eq35.21 I obtained has an extra minus sign,I don't know what's wrong of my calculation,so I hope some people much more smarter can figure it out.Thanks.

Here N, a, and b are integer constants. M is also an integer but changes for every value of x, which makes the index of the second summation dependent on the first. The problem is the relationship M(x) is analytically difficult to define. Is there a way to solve/simplify this expression?

For instance, using the vector ##A^\alpha e_\alpha##: ##g_{\mu \nu} e^\mu \otimes e^\nu (A^\alpha e_\alpha) = g_{\mu \nu} (e^\mu, A^\alpha e_\alpha) e^\nu ## ##g_{\mu \nu} e^\mu \otimes e^\nu (A^\alpha e_\alpha) = A^\alpha g_{\mu \nu} \delta_\alpha^\mu e^\nu = A^\mu g_{\mu \nu} e^\nu = A_\nu...