Tensor Definition and 1000 Threads

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 gcd...

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 - that is...

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 R is a subring of...

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...

Is there a coordinate independent/geometric definition of a tensor density?

In my note, we have written the field strength tensor as: F^{\mu\nu} =\partial ^\mu A^\nu -\partial ^\nu A^\mu = \begin{pmatrix} 0&E_x &E_y&E_z \\ -E_x&0 &B_z &-B_y \\ -E_y&-B_z &0 &B_x \\ -E_z&B_y &-B_x&0 \end{pmatrix} But if I look into...

Hi, I'm trying to write down the stress-energy tensor for a single photon in GR, but I'm running into trouble with its transformation properties. I'll demonstrate what I do quickly and then illustrate the problem. Given a photon with wavevector p, we write {\bf T} = \int \frac{\mathrm{d}^3...

Is there a simple intuitive description of what the Ricci tensor and scalar represent? I have what seems to me a straightforward understanding of what the Riemann tensor Rabcd represents, as follows. If you parallel transport a vector b around a tiny rectangle, the sides of which are determined...

I've been trying to solve the following problem involving Angular Acceleration and the inertial tensor for about 2 weeks now. I know it's bad ask for a question to be solved, but I'm really at a loss here folks. I'm a high school student who has taken a physics class. What I'm Trying To Do...

How do you find the inverse of metric tensor when there are off-diagonals? More specifivally, given the (Kerr) metric, $$ d \tau^2 = g_{tt} dt^2 + 2g_{t \phi} dt d\phi +g_{rr} dr^2 + g_{\theta \theta} d \theta^2 + g_{\phi \phi} d \phi^2 + $$ we have the metric tensor; $$ g_{\mu \nu} =...

Tensors can be of type (n, m), denoting n covariant and m contravariant indicies. Rank is a concept that comes from matrix rank and is basically the number of "simple" terms it takes to write out a tensor. Sometimes, however, I recall seeing or hearing things like "the metric tensor is a rank 2...

I noticed that sometimes exist a parallel between scalar and vector calculus, for example: ##v=at+v_0## ##s=\int v dt = \frac{1}{2}at^2 + v_0 t + s_0## in terms of vector calculus ##\vec{v}=\vec{a}t+\vec{v}_0## ##\vec{s}=\int \vec{v} dt = \frac{1}{2}\vec{a}t^2 + \vec{v}_0 t + \vec{s}_0##...

If I start with the stress-energy tensor T^{\mu\nu} of the electromagnetic field and then apply energy-momentum conservation \partial_\mu T^{\mu\nu}=0, I get a whole bunch of messy stuff, but, e.g., with \nu=x part of it looks like -E_x \nabla\cdot E, which would vanish according to Maxwell's...

According to Wikipedia, This definition doesn't sit well with me. Flux is defined as the rate that something passes through an infinitesimal surface, divided by the infinitesimal area of that surface. For example, the current flux (or current density), when dotted with a unit vector, gives...

Suppose we have some two-dimensional Riemannian manifold ##M^2## with a metric tensor ##g##. Initially it is always locally possible to transform away the off-diagonal elements of ##g##. Suppose now by choosing the appropriate equivalence relation and with a corresponding surjection we construct...

Homework Statement Let ξm and ηn be vector bundles over a paracompact base space. Show that the stifle-Whitney classes of the tensor product ξm ⊗ ηn (or of the isomorphic bundle Hom (ξm, ηn) can be computed as follows. If the fiber dimensions m and n are both 1 then: w1 (ξ1 ⊗ η1) = w1(ξ1) +...

Homework Statement Problem as stated: Consider a vector A^a. Is the four-component object \left( \frac{1}{A^0},\frac{1}{A^1},\frac{1}{A^2},\frac{1}{A^3}\right) a tensor? Homework Equations Roman indices run from 0 to 3. Einstein summation convention is used. Tensors of rank 1 (vectors)...

Homework Statement Hi, I am not sure if this is the right place for my question but here goes! The stress tensor in the Si coordinate system is given below: σ’ij = {{-500, 0, 30}, {0, -400, 0}, {30, 0, 200}} MPa Calculate the stress tensor in the L coordinate system if: cos-1a33=45°, and...

In the case of swarms of particles, the stress energy tensor can be derived by considering the flow of energy and momentum "carried" by the particles along their worldlines. Is there a way to interpret the field definition of the stress energy tensor from Wald, p455 E.1.26 T_{ab} \propto...