Tensors Definition and 386 Threads

Hi. I am currently studying about representations of Lie algebras. I have two questions: 1. As I understand, when we say a "representation" in the context of Lie algebras, we don't mean the matrices (with the appropriate Lie algebra) but rather the states on which they act. But then, the...

After my studies of metric tensors and Cristoffel symbols, I decided to move on to the Riemann tensor and the Ricci curvature tensor. Now I noticed that the Einstein Field Equations contain the Ricci curvature tensor (R\mu\nu). Some sources say that you can derive this tensor by simply...

Does anyone have a proof of what a isotropic cartesian tensor should look like in three or four dimensions?

I'm working on the electromagnetic stress-energy tensor and I've found this in a book by Landau-Lifshitz: T^{i}_{k} = -\frac{1}{4\pi} \frac{\partial A_{\ell}}{\partial x^{i}} F^{k\ell}+\frac{1}{16\pi}\delta^{k}_{i} F_{\ell m} F^{\ell m} Becomes: T^{ik} = -\frac{1}{4\pi}...

From my understanding, an arbitrary (0,N) tensor can be expressed in terms of its components and the tensor products of N basis one-forms. Similarly, an arbitrary (M,0) tensor can be expressed in terms of its components and tensor products of its M basis vectors. What about an (M,N) tensor...

Hi, let S be any set and let ##Z\{S\}## be the free group on ##S##, i.e., ##Z\{S\}## is the collection of all functions of finite support on ##S##. I am trying to show that for an Abelian group ##G## , we have that : ## \mathbb Z\{S\}\otimes G \sim |S|G = \bigoplus_{ s \in S} G ##, i.e., the...

Why is it the case that dual field tensors, e.g. \widetilde{F}^{\mu\nu}=\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}F_{\rho \sigma}, aren't being included in the Lagrangian? For example, one doesn't encounter terms like -\frac{1}{4}\widetilde{F}^{\mu\nu}\widetilde{F}_{\mu\nu} in QED or...

Homework Statement American hardwood quebracho has thermal conductivity ##14W/mK## in axial direction, ##10W/mK## in radial direction and ##11W/mK## in ##z## direction. From a large block of wood we cut out a 10 cm long stick with radius ##4mm## in direction that forms the same angles with all...

Homework Statement I'm learning a bit about tensors on my own. I've been given a definition of a tensor as an object which transforms upon a change of coordinates in one of two ways (contravariantly or covariantly) with the usual partial derivatives of the new and old coordinates. (I...

Homework Statement Hi I have a vector v. According to my book, the following is valid: \frac{1}{2}\nabla v^2-v\cdot \nabla v = v\times \nabla \times v I disagree with this, because the first term on the LHS I can write as (partial differentiation) \frac{1}{2}\partial_i v_jv_j =...

I have recently begun the task of trying to understand tensor products, but I must admit, I have found the going difficult. I have been working (mainly) from Dummit and Foote, which I have previously found to be a fairly "friendly" text for the person engaged in self study ... but the treatment...

Homework Statement A general rotation through angle ##a## about the axis ##\underline{n}##, where ##\underline{n}^2 = 1## is given by $$R(a,\underline{n}) = \exp(-ia\underline{n} \cdot \underline{T}),$$ where ##(T_k)_{ij} = -i\epsilon_{ijk}##. By expanding the exponential as a power series in...

Hiya, I am a grad student who has had a couple semesters of GR. I am currently perusing a book about Two Spinors in Spacetime by Penrose and Rindler, as background for an essay on Spinor Methods in GR. My question relates to the concept of taking the Hodge Dual of a antisymmetric tensor. I...

I'm trying to find examples of stress-energy tensors from exact solutions of the EFE corresponding physically to matter-that leaves out all vacuum solutions(including electrovacuum and lambdavacuum) and pure radiation(null dust)-, I'm finding hard to find any other than the usual SET from...

Besides the dimensionality (4 vs. 3), how would you go about explaining the difference between tensors in GR and in continuum mechanics? I was asked by an engineer friend that finds GR too "esoteric" and complex to get into.

Hello all, I will preface this post with an apology for not putting it in the math/science learning materials section. This would have been the best place to post my question, but for some reason I can't post there. My question is the following: what depth of understanding must I have of...

Ok folks, I've taken a stab at the Latex thing (for the first time, so please bear with me). I've mentioned before that I'm teaching myself relativity and tensors, and I've come across a question. I have a few different books that I'm referencing, and I've seen them present the ordinary...

I finally got my library fine paid off last week and I Picked up Schaum's Outlines Tensor Calculus by David C. Kay. I figured I really need to learn about tensors because every time I read a book or paper about certain subjects such as relativity, nonlinear optics, aerodynamics, etc., I see...

Ok I have T_{ij}=μS_{ij} + λ δ_{ij}δS_{kk}. I am working in R^3. (I am after S in terms of T) . I multiply by δ_{ij} to attain: δ_{ij}T_{ij}=δ_{ij}μS_{ij} + δ_{ij} λ δ_{ij}δT_{kk} => T_{jj}=δ_{jj}λS_{kk}+μS_{jj} * My question is , for the LH term of * I choose T_{jj} rathen than T_{ii}. I...

Has anyone actually gone through this book? I was looking for something that explained tensors a bit clearer and came to this book. It has pretty good reviews, but I was wondering if anyone here has anything to add or suggestions. https://www.amazon.com/dp/0521171903/?tag=pfamazon01-20

Given a function f(x(t, s) y(t, s)), if is possible to compact \frac{∂f}{∂t}=\frac{∂f}{∂x} \frac{∂x}{∂t}+\frac{∂f}{∂y} \frac{∂y}{∂t} by \frac{df}{dt}=\bigtriangledown f\cdot \frac{d\vec{r}}{dt} So, analogously, isn't possible to compact the sencond derivate \frac{\partial^2 f}{\partial s...

Hi everyone, :) I don't quite get what this question means. How do we convert the expression into the bilinear form first? Hope you people can give me some insight. :) Question: Find the rank of the bilinear function \((e^1+e^3)\otimes (e^2+e^4) - (e^2-e^4)\otimes (e_1-e_3)\).

Hi everyone, :) Here's a question I am trying to solve at the moment. I want to know what is meant by decomposable in this context. Really appreciate any input. :) Problem: Let \(V\) be a vector space over a field \(F\), \(\{e_1,\,e_2,\,\cdots,\,e_n\}\) a basis in \(V\) and...

Anyone have a good introduction to Tensor PDF I can read. I don't want to read a 100 pages on Tensors. I'm looking for a short but in depth introduction to tensors. Anyone have any suggestions.

Hey, I was wondering if anyone had any recommendations for books that provide an introduction or a detailed explanation on the whole 4 vectors, tensors and using indicial notation (in the context of General Relativity or Quantum Field) - basically anything that explains to me how to...

I apologize for the sheer volume of questions I am asking. I have never faced this with an assignment. I get 90% of the way then spent 8 hours on the last 10%. This is inefficient. Problem Statement If T is has a non-zero determinant and is second order, show that ##\textbf{T}^\top...

Homework Statement The problem can be found in Jackson's book. An infinitesimal Lorentz transform and its inverse can be written under the form ##x^{'\alpha}=(\eta ^{\alpha \beta}+\epsilon ^{\alpha \beta})x_{\beta}## and ##x^\alpha = (\eta ^{\alpha \beta}+\epsilon ^{'\alpha \beta})...

We know that the Newton binomial formula is valid for numbers in elementary algebra. Is there an equivalent formula for commuting spherical tensors? If so, how is it? To be specific let us suppose that A and B are two spherical tensors of rank 1 and I want to calculate (A + B)4 and I want...

I'm trying to understand the transformation relations for 2d stress and the book doesn't show the derivation of the 2d stress transformation relations from the directional cosines. The 2d stress transformation relations are found by using the transformation equation and the 2d directional...

Homework Statement When we raise and lower indices of vectors and tensors (in representations of any groups) we always use tensors which are invariant under the corresponding transformations, e.g. we use the Minkoski metric in representations of the Lorentz group...

I'm not really sure where I should post this forum in particular so I guess I'll just put it here haha. My questions: What are tensors in general? What are they used for? What Mathematics do I need to understand well, before I begin to learn tensor mathematics? Also does anyone know a good...

This is a question on the nitty-gritty bits of general relativity. Would anybody mind teaching me how to work these indices? **Definitions**: Throughout the following, repeated indices are to be summed over. Hodge dual of a p-form X: (*X)_{a_1...a_{n-p}}\equiv...

Hi all! I've got a short question concerning a minor notational issue about tensor contraction I've run across recently. Let A be an antisymmetric (0,2)-tensor and S a symmetric (2,0)-tensor. Then their total contraction is zero: C_1^1C_2^2\,A \otimes S=0. As a proof one simply computes...

Hi, Homework Statement Given that Aij is a contravariant tensor of rank 2, is the following a contravariant tensor of rank 3: Aijxi/xk?The Attempt at a Solution Using the chain rule, I have found xi/xk to be a contravariant tensor of rank 1: \bar{x}i/\bar{x}k = \bar{x}i/xl * xl/\bar{x}k Is that...

Homework Statement We know that c[ij] = a[i]b[j] is a way to make a rank-2 tensor from two rank-1 tensors. We also know that C[abcxyz]=A[abc]B[xyz] is a way to make a rank-6 tensor from two rank-3 tensors. However, is there a matrix representation of this? I know the idea of a 6-dimensional...

Hi, Homework Statement I recently started delving into tensor calculus and am quite the stuck with the following: Given the tensor Ai = (x+y, y-x, z)i in cartesian coordinates, what would be the second covariant coordinate in cylindrical coordinates? AND Given the tensor Aij = (-1 0, -1 1)ij...

Hi All, I have a question about transformation matrices (sorry about the typo in the title). The background is that I've spent some time learning differential geometry in the context of continuum mechanics and general relativity, but I'm unable to connect some of the concepts. So I have this...

I have been puzzling over a best point of view to comprehend the true algebraic nature of tensors for years now. With vector spaces, I similarly puzzled and concluded that vector spaces are basically sets of abstract members that satisfy a closure on linearity relationship (i.e., any linear...

Homework Statement A horizontal support bar has a downwards force F = 450 N applied near one end, as shown. The radius of the bar is c = 4 cm, and the length L = 1.2 m. The stress tensor σ at any point describes the components of stress in a particular coordinate system. For the coordinate...

By the question in the title, I mean, do the so-called 4-vectors and tensors of SR transform as tangent vectors and tensors (in the sense of differential geometry) with respect to any transformation (local diffeomorphism) of the space-time coordinates or only with respect to Lorentz...

When contracting R^{\sigma}_{ \mbox{ }\mu\nu\rho} to R_{\mu\nu} Should one multiply by g^{\alpha\rho}g_{\alpha\sigma}? I often get confused with ordering of indices and such like

Homework Statement I have a tensor X^{μ\nu} and I want to make this into X_{μ\nu}. Can I do this by simply saying X_{μ\nu}=\eta_{μ\nu}\eta_{μ\nu} X^{μ\nu} ??

Can anyone explain how to contract R_{\alpha \beta} to R^{\rho}_{\alpha\beta\sigma} without multiplying it by 16 i.e g^{\rho\xi}g_{\xi\sigma} It is in a sum with other tensor products and so I obviusly can't just multiply one term by anything ither than 1. Should \eta 's be used...

Why would g^{\alpha \beta} \partial_{\beta} T_{\beta \rho} become \partial^{\alpha} T_{\beta \rho} and not \partial^{\alpha} T_{\rho}^{\alpha} or could it be either?

Hello! In the study of electric and magnetic fields, two equations are called the constitutive relations of the medium (the vacuum, for example): \mathbf{D} = \mathbf{\epsilon} \cdot \mathbf{E}\\ \mathbf{B} = \mathbf{\mu} \cdot \mathbf{H} But in a generic medium (non linear, non...

Am i right in thinking: g^{\mu\nu}g_{\mu\nu}=4 \mbox{ and } g^{\mu\nu}T_{\mu\nu}=T ?

The summation convention for Tensor Notation says, that we can omit the summation signs and simply understand a summation over any index that appears twice. So consider a 3X3 matrix A whose elements are denoted by aij, where i and j are indices running from 1 to 3. Now consider the...

It is often stated that the Kronecker delta and the Levi-Civita epsilon are the only (irreducible) invariant tensors under the Lorentz transformation. While it is fairly easy to prove that the two tensors are indeed invariant wrt Lorentz transformation, I have not seen a proof that there aren't...

I enjoyed this video about tensors very much. I would recommend it to anyone seeking to understand the concept in general and general relativity specifically. https://vimeo.com/32413024 You can fast forward through the repetitive parts and try to place yourself in the role of beginner as you...

Given that an unaccelerated detector detects no Unruh radiation, and an accelerated detector does, it seems to me that whatever the detector is measuring, it can't (by definition) be a tensor, as it doesn't transform properly. I was wondering if there were anything in the literature that went...