Tensors Definition and 376 Threads

The fact that physics laws must have the same form in any reference frame (general covariance) is guaranteed by expressing them in tensor notation (, if possible). Considering also non linear coordinate transformations the tensor transformation rules are defined by means of the partial...

When working over a field of characteristic not 2, or otherwise with modules over a ring where 2 is invertible, there is no ambiguity in what one means by symmetric or anti-symmetric rank 2 tensors. All of definitions of the anti-symmetric tensors The module of anti-symmetric tensors is the...

Homework Statement I was wondering how I could prove the following property of 2 antisymmetric tensors F_{1\mu \nu} and F_{2\mu \nu} or at least show that it is correct. Homework Equations \frac{1}{2}\epsilon^{\mu \nu \rho \sigma} F_{1\rho \sigma}F_{2\nu \lambda} + \frac{1}{2}\epsilon^{\mu...

Problem: V a vector space with dimV \le 3, then every homogeneous element in \Lambda(V) is decomposable. So, this exercise doesn't sound very difficult. My problem is, that i don't know the definition of homogeneous and decomposable. Can you please help me? Thank you

Can anyone give me examples of mixed tensors that appear in physics? I'm looking for mixed specifically here: purely covariant or contravariant ones won't do.

I cannot at all understand the theorem on p16 of the notes attached in this thread:Surely seeing as we want an isomoprhism between V and V**, \Phi should act on an element of V i.e. a vector X and take it to an element of V** (i don't know what such an element would look like though!). But...

Hi If i want to express the kinetic energy for some angular momentum, I can write T=\frac{1}{2}\sum_{ij}{I_{ij}\omega_i\omega_j} I cannot quite see why we have wiwj at the end, and not just wi2. It is not that obvious to me. I have read several examples regarding polarization and electric...

I'm taking a continuum mechanics course and we use the 3*3 stress tensor a lot. The problem is that I do not understand what each component mean. What does the tension(Pn normal(Pnn and tangent(Pnt tension mean, or just Pxy?

In special relativity, the metric tensor is invariant under Lorentz transformations: \Lambda^\alpha{}_\mu \Lambda^\beta{}_\nu g^{\mu \nu} = g^{\alpha \beta} Is this the unique rank 2 tensor with this property, up to a scaling factor? How would I go about proving that? I know that two...

I read that we need scalars, spinors, vectors and rank two tensors to describe spin-0, spin-1/2, spin-1 and spin-2 particles, respectively. But then I reacall from quantum mechanics courses that the intrinsic spin of a particle is described by different finite representations of so(3)...

I'm very clearly not understanding something, if someone could help me put my finger on what that something is. So here's what I've got: Problem: Three equall mass points (mass m) are located at (a,a) (a,‐a) and (‐a,‐a) (all have z=0). a) Show I = ma^2 (3ii − ij − ji + 3jj + 6kk)...

Homework Statement Write down the transformation laws under general coordinate transformations for a tensor of type (0,1) and a tensor of type (2,1) respectively The Attempt at a Solution I seem to have two transformation formulas but they could in fact just be the same thing. I'll just do...

Let C_{i_1i_2 \dots i_l} be a symmetric traceless tensor of rank l. Let \hat{x}= \frac{x}{|x|} be a three dimensional unit vector on the unit sphere. Define a tangential derivative such that \nabla_i \hat{x_j} = \delta_{ij} - \hat{x_i} \hat{x_j}. For the spherical harmonic Y_l(\hat{x})=C_{i_1i_2...

I'm trying to teach myself GR from Wald's General Relativity, and it's very tough going. I do have basic knowledge of differential geometry, but I think my geometric intuition is next to nonexistent. I'd very much appreciate some help in understanding several basic questions, or pointers to...

I am having trouble understanding what a bivector is in the context of electromagnetic stress and pressure. What is the purpose of it? Also, what is its connection to tensors?

So the tensor calculus part of my into to GR class is kicking my butt. I'm using the book by D'Inverno, but I feel like he's going too quickly over tensor calculus. Is there a more introductory text on tensors that I missed out on? I've taken courses on linear algebra before, but not...

In http://relativity.livingreviews.org/Articles/lrr-2004-2/" (section 2.1.5.2) the following is the first sentence in the section reviewing spinors: "Spinors are representations of the Lorentz group only; as such they are related strictly to the tangent space of the space-time manifold."...

I think I get the difference between spinors and tensors in the context of algebraic topology and QM but I want someone to scrutinize my understanding before I move on to another topic. I've never had a class in topology so I might be using some math terms incorrectly. The 3D parameterized...

I'm trying to understand the Einstein field equations conceptually, and one of the things that I'd like to understand is why Einstein decided that the left side of the GR equation should be the Einstein tensor instead of the Ricci tensor, I heard that initially he entertained the idea of...

Homework Statement If \epsilon _{ijjk} is the Levi-Civita symbol: 1)Demonstrate that \sum _{i} \epsilon _{ijk} \epsilon _{ilm}=\delta _{jl} \delta _{km} -\delta _{jm} \delta _{kl}. 2)Calculate \sum _{ij} \epsilon _{ijk} \epsilon _{ijl}. 3)Given the matrix M, calculate \sum _{ijk} \sum...

Hi everyone, I recently started a course on continuum mechanics. It started with the mathematical background of transforming tensors with contravariant and/or covariant indices. There is one thing I don't understand and it should be really straight forward. I hope you can give me a hint...

Hi all I am taking my first grad level class on stress and elasticity and ran into a bit of a wall. We are dealing with a 3x3 stress tensor which describes the state of stress of a given point. Now various textbook problems have questions where given one state describe the state if the...

Hi all, first of all I'm sorry if I've posted my question on the wrong section, didn't know where I should post it, I thought this forum might be the right choice. I've recently graduated from high school and I'm highly interested in stuff about General relativity and Quantum mechanics and...

we have studied in Tensor's analysis that there are two kinds of tensors that usually used in transformation. one is Contravariant & covariant. what is the difference between them and and why they are same for Rectangular coordinates?

Homework Statement I have a few introductory problems dealing with proofs of tensor properties, and one about a transformation from rectangular to spherical coordinates. If someone has the time and inclination to help out this week, I can email you the specific problem set. (I'd prefer not...

Hello all. I am a 17-year-old high school student in the United States. I have taken BC Calculus and will be taking Multi-variable Calculus next year (also known as Calc III in the states), but I already know a lot of it for various reasons. I also have mild experience in topology and abstract...

Im not sure why the learning materials section is blocked for me to post in, however: I am stuck between these books: https://www.amazon.com/dp/0486658406/?tag=pfamazon01-20 https://www.amazon.com/dp/0486640396/?tag=pfamazon01-20 https://www.amazon.com/dp/0486638332/?tag=pfamazon01-20...

I am reading the following document entitled: An Introduction to Tensors for Students of Physics and Engineering. This document can be found at the following link: http://www.grc.nasa.gov/WWW/k-12/Numbers/Math/documents/Tensors_TM2002211716.pdf Specifically, I am having trouble with page 11 on...

Hi everybody, I just finished a intro to modern physics recently where we covered SR, but didn't touch on GR. From what I've read, you have to have an understanding of tensors before you can understand Einstein's equations and most of the math behind GR. I go to Stony Brook, and I haven't...

Is it true that the only combination of second order derivative of metric which transforms tensorially is Riemann tensor (and its traces)?

If I measure an angle in one reference frame to be 90 degrees, would it be 90 degrees with respect to all other reference frames? That is, is angle measurement a rank 0 tensor? I'm assuming all other reference systems are at non-relativistic velocities.

HI, could someone please suggest me a good book to learn both Tensors as well as General relativity? It has to be introductory.

I guess this question isn't actually specific to the Ricci tensor, but to rank 2 symmetric tensors. If one is free to choose any local inertial coordinate system, what is the simpliest form we are garaunteed to be able to write any tensor components of a rank 2 symmetric tensor? If we start...

Homework Statement Given g\equiv g_{ij} = [-1 0; 0 1] Show that A= A^{i}_{j} = [1 2 -2 1] is symmetric wrt innter product g, has complex eigenvalues, but eigenvectros have zero length wrt the complex inner product. The Attempt at a Solution Im sure this is just a simple...

Electric charge continuity is expressed as ∂tρ + ∂iJi =0. (1) The manifold, M in question is 3 dimensional and t is a parameter, time. ∂iJi is the inner product of the ∂ operator and J. With M a subspace of a 4 dimensional manifold with metric signature -+++, eq. (1) can be written in...

Kronicker Delta, Levi-Civita, Christoffel ... and "tensors" For quick reference in grabbing latex equations: http://en.wikipedia.org/wiki/Levi-Civita_symbol http://en.wikipedia.org/wiki/Kronecker_delta http://en.wikipedia.org/wiki/Christoffel_symbols Wiki warns that the Christoffel...

Homework Statement I just started trying to get into relativity using Schutz's book but I've been getting a little confused with some of the tensor stuff. For starters, what does it mean by a one form maps covectors to real numbers? is it an operation or a transformation? and when it maps to a...

I've been trying to get a grasp on Lie Derivatives. I understand that we can represent a lie derivative acting on a vector as a commutator. What do I do when I act a lie derivative on a tensor? Can I still just write out the commutator?

Under what conditions can the metric tensor be integrated to a global metric function? i.e., a function g(x,y) that gives the distance along a geodesic between x and y? For example, we can do this on the sphere using spherical trigonometry (cf...

I am aware that a vector is a first-order tensor, and that a second-order tensor has nine components in three-space, but can someone tell me more about the directional quantities that are associated with these nine components? Are they still unit vectors? Can a second order tensor be written as...

Alright, a little motivation for my question before I ask it... We have been assigned to read a section of our book (Anadijiban Das' "Tensors", section 1.1) and find the definitions of all words we don't know. The section I have been assigned is all about Fields. It gives the definition of a...

if i have a tensorT^{uv}...i need to calculate the covariant derivate T^u_{v;a} The logical thing is to do T^u_v and next to calculate T^u_{v;a} is also correct to first calculateT^{uv}_{;a} and next T^u_{v;a}=T^{ui}_{;a}g_{iv}?

Hello all, this is my first post on this forum, though I have been perusing it for a while. I am currently re-reading through Carroll's text on SR and there is a curious comment on p24 that intrigues me. Carroll says that the *only* tensors in SR which are invariant are the Kronecker delta...

Hi all, Quick question I haven't been able to find the answer to anywhere: Can I use exterior calculus for symmetric tensors? I'm familiar with the exterior calculus approach to things like Stokes's theorem and Gauss's law, but that's vector stuff. It seems to me the only tensors in...

Suppose a, b are vectors in R^n, R^m resp. What can I say about the length of a\otimes b wrt a and b? You can think of \otimes as the Kronecker product.

Not sure if this is the right place to ask, but this doubt originated when reading on string theory and so here it goes... The general canonical energy-momentum tensor (as derived from translation invariance), T^{\mu\nu}_{C} is not symmetric. Also, the general angular momentum conserved...

Homework Statement Prove the following relationship: \epsilonpqi\epsilonpqj = 2\deltaij Homework Equations The Attempt at a Solution All I have so far is the decomposition using the epsilon-delta \epsilonpqi\epsilonpqj = \epsilonqip\epsilonpqj \epsilonqip\epsilonpqj =...

I'm getting myself up to speed on GR to try to understand a book by John Moffat called "Reinventing Gravity...". So far I've been using Sean Carroll's sort of classic course notes and a fair bit of Wikipediea (sp?). It may be a naive question, but the point I wouldn't mind comments on is the...

I have a general question about index notation. For an arbitrary quantity, a, "a" denotes a scalar quantity. "a_i" denotes a vector. "a_ij" denotes a 2nd-order tensor. So, if I have something like "a_i*e_ij*b_j" Would this be like multiplying an nx1 vector, an mxm matrix, and an Lx1 vector...

Hello. I wasn't sure whether to post this here on in some of the physics sections. I have a rank 2 tensor in one coordinate reference system [x1, x2, x3], the one where only the principal elements are non zero: R=[ a11 0 0; 0 a22 0; 0 0 a33 ]. I want the tensor R in some other...