Tensors Definition and 386 Threads

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 =...

For vectors we can define the Joint Guasian as follows: f_X(x_1, \dots, x_N) = \frac {1} {(2\pi)^{N/2}|\Sigma|^{1/2}} \exp \left( -\frac{1}{2} ( x - \mu)^\top \Sigma^{-1} (x - \mu) \right) Now what if (x - \mu) is a matrix A and \Sigma is an order four covariance matrix Q between ellements...

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...

What is the general rule behind why for any given lagrangian (QED/QCD show this) that any vectors or tensors contract indices? I know it must be something simple, but I just can't think of it offhand. QED : F_{\mu\nu}F^{\mu\nu} Proca (massive vector): A_\mu A^\mu QCD : G^{\alpha}_{\mu\nu}...

Not really a homework problem. Need some help understanding tensors. Ok, so the chapter in the book I am using, Vector Calculus by Paul C. Matthews introduces first the coordinate transformation and proceeds to say that a vector is anything which transforms according to the rule...

Hello, can anyone explain simply what is a tensor, using the language of Geometric Algebra? Thanks!

Homework Statement Show that the Lagrangian density: L=- 1/2 [\partial_\alpha \phi_\beta ][\partial^\alpha \phi^\beta ]+1/2 [\partial_\alpha \phi^\alpha ][\partial_\beta \phi^\beta ]+1/2 \mu^2 \phi_\alpha \phi^\alpha for the real vector field \phi^\alpha (x) leads to the field equations...

1. (a) Remembering the distinction between summation indices and free indices, look at the following equations and state whether they conform to tensor notation, and if not why not: (i) Tmn=Am^nB (ii) Uij^i=Ai^kDk (iii) Vjk^ii=Ajk (iv) Ai^j=Xi^iC^j+Yi^j (b) (i) Write out in...

What is the benefit of expressing Maxwell's equation in the language of differential forms? Differential forms seem to be inferior to the language of tensors. Sure you can do fancy things with the exterior derivative and hodge star, but with tensors you can derive those same identities with...

Prove that b_{ijkl}=\int_{r<a} dV x_i x_j \frac{\partial^2}{\partial_k \partial_l} (\frac{1}{r}) where r=|x| is a 4th rank tensor. i've had a couple of bashes and got nowhere other than to establish that its quotient theorem. can i just pick a tensor of rank 3 to multiply it with or...

Hello, The concept of contravariance, covariance and invariance are commonly used in the domain of Tensor Calculus. However I have heard that such concepts are more abstractly defined (perhaps) in cathegory theory. Could someone explain shortly the connection between the abstract definitions...

Lets say that I know the inertia tensors for a few different 3D shapes and I want to connect them together into one big composite shape. From what I understand, I first have to find the new center of mass, then using the parallel axis theorem find the new inertia tensors for each body along an...

Show that a tensor T can be written as T_{ij}=\lambda \delta_{ij} + F_{ij} +\epsilon_{ijk} v_{k} for the tensor \[ \left( \begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array} \right)\] find \lambda, F_{ij}, v_k i can't get anywhere whatsoever with this question?

I'm sort of confused about Christofel symbols being called tensors. I thought that to be considered a tensor, the tensor had to obey the standard component transformation law. For example...

Hi there! I'm tring to check some calculations of the propagator of a gauge field using the R_\xi gauge fixing. Since the propagator has a matricial structure, \Delta_{\mu\nu}=\frac{1}{(p^0)^2+E_p^2}\left[\delta_{\mu\nu}-\frac{1-\xi}{(p^0)^2+E_p^2}p_\mu p_\nu\right], I'd like to check it...

Hello Everyone, I didn't know whether to post this here or in the Physics area. Basically I'm trying to get a good understanding of Tensors so that I can apply them to General Relativity. I'm a freshman in college and kind of been teaching myself this advanced physics since i was 14, and...

Hello, I was trying to follow a proof that uses the dot product of two rank 2 tensors, as in A dot B. How is this dot product calculated? A is 3x3, Aij, and B is 3x3, Bij, each a rank 2 tensor. Any help is greatly appreciated. Thanks! sugarmolecule

Can anybody recommend a good introductory book on tensors? I have a B.S. in math. Thanks in advance.

I've been asked to show that \epsilon _{{{\it ijm}}}\epsilon _{{{\it mkl}}} is an isotropic tensor using \epsilon _{{{\it ijk}}}\det \left( M \right) =\epsilon _{{\alpha \beta \gamma }}m_{{i\alpha }}m_{{j\beta }}m_{{k\gamma }} . Then to take the most general form for a fourth rank tensor...

I am not able to find good books on this topic on net so if any 1 can help me i will be grateful .

Is the parallel axis theorem always valid for inertia tensors? We have only seen examples with flat (2d) objects and was wondering if it would also be valid for 3d objects, like a h emisphere, for example. Thanks.

I am a bit confused with tensors here. now i know that \Lambda, the transformation matrix has a different meaning when I write \Lambda^\mu\ _{\nu} and when I write \Lambda_{\nu}\ ^\mu One is the mu-nu th element of \Lambda and the other is the mu-nu th element of \Lambda^{-1}. Is it...

Hi all, I am taking this math methods course in grad school, and in the lectures we stormed through differential geometry. My geometry is already horrible, I find it hard to understand all these forms, fields, tensors, wedge products etc... I would be glad if you could suggest some books...

Hi, I've decided to learn GR myself recently since it's like the "sexy" side of physics. But I'm getting stuck with the tensors notations already. Maybe my math background is just not sufficient enough to do GR. In general, how do I know that an object is tensorial; for example, objects like...

From Carroll's textbook: 1. The problem statement Imagine we have a tensor X^{\mu \nu} with components X^{\mu \nu} = \begin{pmatrix} 2 & 0 & 1 & -1\\ -1 & 0 & 3 & 2\\ -1 & 1 & 0 & 0\\ -2 & 1 & 1 & -2 \end{pmatrix} Find the components of: (a) {X^\mu}_\nu; (b) {X_\mu}^\nu.2. The attempt at a...

Hello there, I'm currently trying to get my head around General Relativity for a term paper; the twist is that I'm dealing with an arbitrary amount of dimensions, that is 4+d, where d is unspecified. Now the maple tensor package does calculation with some fixed amount of dimensions just...

How would you represent tensors as matrices? I've searched all over, and my book on GR (Wald) only has one example where he makes a matrice from a tensor, and I still don't understand the process.

I'm not sure this is the correct forum section for this question, if not, please move me. Essentially, I'm looking for help understanding what basis vectors, one forms, and basis one forms are. I'm fairly sure I get basis vectors, I would describe them as a description of a co-ordinate system...

Dear friends, How is the divergence in curvilinear coordinates of a second order mixed tensor defined? I mean, shall I contract the covariant or the contravariant index?? And for both cases which is the physical meaning? \nabla_i N^i_j or \nabla_j N^i_j ? Thanks a lot, Enzo

Is there a way bras and kets can be understood in terms of vectors and tensors and coordinate bases? I'm fairly sure that if a ket is thought of as a vector with an upper index, then it's bra is a vector with a lower index, but getting the rest of it all to look like tensors is rather...

Tensors for relativity: http://www.teorfys.uu.se/people/minahan/Courses/SR/tensors.pdf

A vector is drawn as an arrow, a covector (one-form) as a series of parallel lines. Is there a way to pictorially represent a tensor of rank greater than one? I want to have an intuitive/geometric sense of what it means to parallel transport such an object, and without a picture I don’t have one.

Is there any free online tutorial which completely explains General Relativity without concept of Tensors?

I quite often hear that GR is formulated in terms of tensors because laws of physics expressed in terms of tensor equations are indepedent of choice coordinates because they `transform nicely'. I thought the motivation for tensors was that since spacetime is curved, we locally linearize it by...

I'm reading through Pauli's "Theory of Relativity", which has a discussion of tensors in the mathematical tools section of the book. When introducing surface tensors, he states "Such tensors can be obtained by considering two vectors x, y which together span a two-dimensional parallelepiped...

somebody know some link or ebook about tensor with examples solved and execises

What is a good book that introduces tensor notion clearly?

in general, what exactly does it mean for a tensor to be non-degenerate? does it mean that the vector space underlying it all has a zero kernel? I'm still a bit hazy on the degeneracy of bilinear forms in general. They're not exactly like tensors, either, but I am guessing there's some...

I understand that all rank 2 tensors can be decomposed into a symmetric and a skew symmetric part, but I don't really understood how this is done. It has something to do with permutations of the indices, I guess, but I never learned anything about what a permutation is. Can anyone explain how...

Hello folks, During my education I was not exposed to tensor notation much at all. Therefore I never developed an understanding for it. I spend some time on my own now, but often find it quite obtuse and lacking some of the detail I feel I need to reach that point of comfort. Does anyone...

What is the physical significance of tensors? Occasionally, motivating statements are made roughly along the lines of "if an equation can be expressed purely in terms of tensors, then it is true for all observers". But that doesn't seem quite complete because different tensor-users would have...

sow I'm working on learning some of the maple commands for tensors, and I see a lot of the basic structure, however I don't know how to store a tensor in maple for multiple usages, can anybody help?

I was just reading chapter on Cartesian tensors and came across equation for transformation matrix as function of basic vectors. I just do not get it and cannot find a derivation. I am too old to learn Latex, I uploaded a word document with the equation. Thanks, Howard

Homework Statement a) What does the tensor nature of susceptibility mean? Ie. the xxxx in: \overline{P}=\epsilon_0\chi_{xxxx}\overline{E} b) What does the 3 vertical dots mean in: \chi^{(2)}\vdots\overline{E} ?The Attempt at a Solution a) I understand that...

Homework Statement Let us define 4 vector by 4 co-ordinates:(x1,x2,x3,x4) where (x1,x2,x3) are space components (like x,y,z) and x4 is related to time as x4=ict.Express the following equations in tensor notation. (i)The continuity equation: div J+(del*rho/del t)=0 (ii)The wave...

Is the matrix of a second order symmetric tensor always symmetric (ie. expressed in any coordinate system, and in any basis of the coordinate system)? Please help! :blushing: ~Bee