Tensor Definition and 1000 Threads

Homework Statement Hi all, I'm having trouble evaluating the product g_{αβ}ϵ^{αβγδ}. Where the first term is the metric tensor and the second is the Levi-Civita pseudotensor. I know that it evaluates to 0, but I'm not sure how to arrive at that. The Attempt at a Solution My first thought...

Hello, I am studying general relativity right now and I am very curious about the Ricci tensor and its meaning. I keep running into definitions that explain how the Ricci tensor describes the deviation in volume as a space is affected by gravity. However, I have yet to find any quantitative...

Hi everyone, I have the following problem in my hands, which I don't know how exactly to address. Let's assume that from any CAD(Solidworks, Catia), I obtain the inertia tensor of my model (impossible to calculate by hand btw). I_full=[Ixx Ixy Ixz Ixy Iyy Iyz Ixz Iyz Izz] I...

Hi PF, I posted this in HW a week ago and got no response. Might be a bit beyond the typical HW forum troller. So, please excuse the double-post. Homework Statement I'm trying to derive the rate-of-strain tensor in cylindrical coords, starting with the Christoffel symbols. Homework...

From what I've understood, 1) the metric is a bilinear form on a space 2) the metric tensor is basically the same thing Is this correct? If so, how is the metric related to/different from the distance function in that space? Some other sources state that the metric is defined as the...

I am not sure whether this needs to be transported in another topic (as academic guidance). I have some preliminary knowledge on tensor analysis, which helps me being more confident with indices notation etc... Also I'm accustomed to the definition of tensors, which tells us that a tensor is an...

In "A Student's Guide to Vectors and Tensors" by Daniel Fleisch, I read that the covariant metric tensor gij=ei°ei (I'm leaving out the → s above the e's) where ei and ei are coordinate basis vectors and ° denotes the inner product, and similarly for the contravariant metric tensor using dual...

Hello, i don't know if my question is well posed, if i have a symmetric tensor Sij = (∂ixj + ∂jxi) / 2 with xi cartesian coordinates, how can i transform it in a spherical coordinates system (ρ,θ,\varphi)? (I need it for the calculus of shear stress tensor in spherical coordinate in fluid...

Homework Statement Calculate the moments of inertia I_1, I_2, and I_3 for a homogeneous cone of mass M whose height is h and whose base has a radius R. Choose the x_3 axis along the axis of symmetry of the cone. Choose the origin at the apex of the cone, and calculate the elements of the...

Homework Statement Hello I'm trying to self study A First Course in General Relativity (2E) by Schutz and I've come across a problem that I need some advice on. Here it is: Use the identity Tμ\nu,\nu=0 to prove the following results for a bounded system (ie. a system for which Tμ\nu=0...

Hello everyone, I have recently read a puzzling statement on my Electromagnetism (Chapter on Special Relativity) material regarding the Field Strength Tensor, F^{\mu\nu}, and its dual, \tilde{F}^{\mu\nu}. Since I've been thinking about this for a while now, and still can't understand it, I...

(this is not a hw) Assume you have a magnet of dimensions x_m, h_m, remanent flux density Br, and coercive field density Hc. The magnet is placed in a magnetic "C" structure (perfect iron) such that it is connected on one side but there is an airgap on the other side. xxxxxxxx xx... xx...

Hello, Hi There I am trying to obtain the relations of the conserved charges of the stress tensor, it has 4, one is the hamiltonian and the other three are the momentum components. \vec{P}=-\int d^3y \sum_i{(-\pi_i(y) \nabla \phi_i(y))} And i have to prove the conmutators...

I am trying to understand the magnetic gradient tensor which has nine components. There are three magnetic field components, but there are also three baselines. These nine gradients are organised into a 3x3 matrix. I have read that only 5 of these terms are independent. What exactly does this...

suppose we consider the measurement operator A=diag(1,-1). Then the tensor product of A by itself is in components : A\otimes A=a_{ij}a_{kl}=c_{ijkl} giving c_{1111}=c_{2222}=1, c_{1122}=c_{2211}=-1 and all other component 0. to diagonalize a tensor of order 4, we write ...

Hello All, Sorry if my question seems to be elementary. I am trying to find out a little bit details of the Riemann metric tensor but not too much in details. I found out the metric (g11, g12, g13, g14...). It provides information on the manifold and those parameters have the information...

Hi, Homework Statement I have found the tensor of inertia of a rectangle of sides a and b and mass m, around its center, to be I11=ma2/12, I22=mb2/12, I33=(ma2 + mb2)/12. All other elements of that tensor are equal to zero. I would now like to use this result to determine the tensor of inertia...

Hi everyone, :) Xristos Lymperopoulos on Facebook writes (>>link<<);

Dear All, I need some explanations of properties of tensor and the tensor product on different states; σ1ijσij2=_____________ Thank you.

So I have just been introduced to indices, four vectors and tensors in SR and I'm having trouble knowing exactly what I am being asked in some questions. So the first question asks to write explicitly how a covariant two tensor transforms under a lorentz boost. Now I know that it transforms...

The stress energy momentum tensor of the Einstein field equations contains multiple density terms such as the energy density and the momentum density. I know how to calculate relativistic energy and momentum, but none of the websites or videos that I have watched make mention of any division of...

I've been studying the Einstein field equations. I learned that the Ricci curvature tensor was expressed as the following commutator: [∇\nu , ∇\mu] I know that these covariant derivatives are being applied to some vector(s). What I don't know however, is whether or not both covariant...

I think that is a fundamental question of why we need Tensor when dealing with GR? Quoting from the textbook (Relativity, Gravitation and Cosmology: A Basic Introduction) Tensors are mathematical object having definite transformation properties under coordinate transformations. The simplest...

I was thinking... if the modulus of a vector can be calculated by ##\sqrt{\vec{v} \cdot \vec{v}}##, thus the modulus of a tensor (of rank 2) wouldn't be ##\sqrt{\mathbf{T}:\mathbf{T}}## ?

In reexamining chapter 11 of Jackson's Classical Electrodynamics, especially equations 11.148, it seems obvious that in placing the E and B transformation values into the electro-magnetic field-strength tensor one is ignoring the standard rules which do not allow combining polar vectors and...

I am reading Dummit and Foote Section 10.4: Tensor Products of Modules. I would appreciate some help in understanding Example (8) on page 366 concerning viewing the quotient ring $$R/I $$ as an $$ (R/I, R) $$-bimodule. Example (8) D&F page 370 reads as follows: (see attachment)...

I am reading Dummit and Foote Section 10.4: Tensor Products of Modules. I would appreciate some help in understanding Example 2 on page 366 concerning viewing the quotient ring $$R/I $$ as an $$ (R/I, R) $$-bimodule. Example (2) D&F page 366 reads as follows...

Homework Statement I have a question asking me to find the expectation value of S_{12} for a system of two nucleons in a state with total spin S = 1 and M_s = +1 , where S_{12} is the tensor operator inside the one-pion exchange nuclear potential operator, equal to S_{12} =...

I've been working on Ex 5.4 in MTW. The maths is fairly straight forward, but I don't really understand what is going on! In part (b) what are the 'forces' pushing the volume through a distance? If they are forces, they must produce an acceleration but we have a constant velocity. Are these...

Actually this problem really only concerns greatest common denominators. In Section 10.4, Example 3 (see attachment) , Dummit and Foote where we are dealing with the tensor product $$ \mathbb{Z} / m \mathbb{Z} \otimes \mathbb{Z} / n \mathbb{Z}$$ we find the following statement: (NOTE: d is the...

I am reading Dummit and Foote, Section 10.4: Tensor Products of Modules. I am currently studying Example 3 on page 369 (see attachment). Example 3 on page 369 reads as follows: (see attachment) ------------------------------------------------------------------------------- In general, $$...

I am reading Dummit and Foote, Section 10.4: Tensor Products of Modules. I am currently studying Example 3 on page 369 (see attachment). Example 3 on page 369 reads as follows: ------------------------------------------------------------------------------- In general, $$ \mathbb{Z} / m...

I am trying to write the Einstein field equations $$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu} R=\frac{8\pi G}{c^4}T_{\mu\nu}$$ in such a way that the Ricci curvature tensor $$R_{\mu\nu}$$ and scalar curvature $$R$$ are replaced with an explicit expression involving the metric tensor $$g_{\mu\nu}$$...

I am reading Dummit and Foote Section 10.4: Tensor Products of Modules. I am currently studying Example 2, page 363 (see attachment) and I am trying to closely relate the example to Theorem 8 and the D&F text on extension of the scalars preceding Theorem 8 on pages 359-362) In Example 2 (see...

I am reading Dummit and Foote, Section 10.4: Tensor Products of Modules. I am studying Corollary 9 and attempting to fully understand the Corollary and it proof. (For details see the attachement page 362 in which Theorem 8 is stated and proved. This is followed by the statement and proof of...

First, thanks to both Deveno and ThePerfectHacker for helping me to gain a basic understanding of tensor products of modules. In a chat room discussion ThePerfectHacker suggested I show that $$ {\mathbb{Z}}_a \otimes_\mathbb{Z} {\mathbb{Z}}_b $$ where a and b are relatively prime integers -...

I am trying (struggling! :() to understand tensor products as developed by Dummit and Foote in Section 10.4 - specifically the early section devoted to the "extension of scalars". I have been reflecting on my attempts to understand the material of Dummit and Foote, pages 359 -362 (see...

I am reading Dummit and Foote, Section 10 on tensor products of modules. I am at present trying to understand the use of Theorem 6 (D&F, page 354 - see attachment) in Theorem 8 (D&F page 362, see attachment). The proof of Theorem 8 in D&F Chapter 10 (see attachment) reads as follows...

I am attempting to understand Dummit and Foote exposition on 'extending the scalars' in Section 10.4 Tensor Products of scalars - see attachment - particularly page 360) [I apologise in advance to MHB members if my analysis and questions are not clear - I am struggling with tensor products! -...

I'm working through A. Zee's new EGR book, and I came to a step on tidal forces I couldn't follow. He presents the gravitational potential V(\vec{x})=-GM/r and asks us to verify that the tensor R^{ij}(\vec{x})\equiv\partial^{i}\partial^{j}V(\vec{x}) is, in this case...

Homework Statement ##D_{ijk}## is an array with ##3^3## elements, which is not known to represent a tensor. If for every symmetric tensor represented by ##a_{jk}## $$b_i = D_{ijk}a_{jk},$$ represents a vector, what can be said about the transformation properties under rotations of the...

I've been watching the Stanford lectures on GR by Leonard Susskind and according to him the metric tensor is not constant in polar coordinates. To me this seems wrong as I thought the metric tensor would be given by: g^{\mu \nu} = \begin{pmatrix} 1 & 0\\ 0 & 0\\ \end{pmatrix} Since...

In Dummit and Foote, Section 10.4: Tensor Products of Modules, on pages 359 - 364 (see attachment) the authors deal with a process of 'extension of scalars' of a module, whereby we construct a left $$S$$-module $$ S \oplus_R N $$ from an $$R$$-module $$N$$. In this construction the unital ring...

I am reading and trying to fully understand Keith Conrad's paper: Tensor Products I. These notes are available at Expository papers by K. Conrad or the specific paper at http://www.math.uconn.edu/~kconrad/blurbs/linmultialg/tensorprod.pdf. Conrad's Theorem 3.3 (see attachment - page 10) is...

If we use the "flux of 4-momentum" definition of the stress-energy tensor, it's not clear to me why it should be symmetric. Ie, why should ##T^{01}## (the flux of energy in the x-direction) be equal to ##T^{10}## (the flux of the x-component of momentum in the time direction)?

I am reading and trying to follow the notes of Keith Conrad on Tensor products, specifically his notes: Tensor Products I (see attachment ... for the full set of notes see Expository papers by K. Conrad ). I would appreciate some help with Theorem 3.2 which reads as follows: (see attachment...

x^2 - y^2 in the \mid n\ell m \rangle basis - tensor op. Homework Statement I must determine the matrix elements of x^2 - y^2 in the \mid n\ell m \rangle basis. "...use the fact that x^2 - y^2 is a sum of spherical components of a rank two tensor, together with the explicit form of the...

I am (trying to :-) ) reading Dummit and Foote Section 10.4 on Tensor Products of Modules and am finding D&F's introduction to the topic of tensor products quite bewildering! ... Can anyone give me a simple definition of a tensor product of modules together with an example to give me a basic...

Hello all, After a brief break from attempting to learn tensor calculus, I'm once again back at it. Today, I started reading this: http://web.mit.edu/edbert/GR/gr1.pdf. I got to about page 4 before things stopped making sense, right under equation 3. Question 1: apparently a "one-form" is a...

hey pf! in reading a book on viscous stresses i found the following: \tau_{ij}=2\mu\Big(s_{ij}-\frac{1}{3}s_{kk}\delta_{ij}\Big) where einstein summation is used. now we have s_{ij}=\frac{1}{2}\Big(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\Big) and then the claim is...