Tensors Definition and 386 Threads

My class is starting to cover E&M in Lorentz covariant form, and obviously the subject of tensors came up. The problem is that my prof defines tensors in terms of coordinates, which is ugly and against the spirit of relativity. Is there a way of doing tensors coordinate-free in a physics...

Hairy ball theorem - Wikipedia is not as good or as well-referenced as I'd hoped, and it mainly discusses vector fields on the 2-sphere, the ordinary sort of sphere. In particular, it does not mention the minimum number of zero points of a continuous vector field on a sphere. I would guess...

Hi, I was wondering if someone wouldn't mind breaking down the geometrical differences between the Riemann, Ricci, and Weyl tensor. My current understanding is that the Ricci tensor describes the change in volume of a n-dimensional object in curved space from flat Euclidean space and that if we...

I have been struggling with this homework question for a week and it still makes no sense to me. I am asked to "choose a nontrivial second order tensor in R^2 and determine whether or not it can be identified with a first order tensor in R^4 in a natural way, and if it can be, is every...

Using tensors, I'm supposed to find the usual formula for the gradient in the covariant basis and in polar coordinates. The formula is \vec{grad}=[\frac{\partial}{\partial r}]\vec{e_{r}}+\frac{1}{r}[\frac{\partial}{\partial \vartheta}]\vec{e_{\vartheta}} where \vec{e_{r}} and...

Hi... here it is... sorry for my english... http://sphotos-c.ak.fbcdn.net/hphotos-ak-ash4/297165_4722421629188_964469850_n.jpg

What is the rule for orienting tensors? In the above image we have the components of the Cauchy Stress Tensor in 2 dimensions. In the bottom left you see the cartesian co-ordinates are oriented as normal. How do I know \sigma_{xy} is oriented upwards on the right face and downwards on the...

Homework Statement using Einstein notation, show the following identities are true (A · B) : C = A^T · C : B = C · B^T : A Homework Equations The Attempt at a Solution (A · B) : C=(A_{ij} · B_{jk} ) : C = D_{ik} C_{ik} = C_{ik} D_{ik}...

Hello All, I am quiet new to this subject.I am unable to get over tensors and ranks. (1) Does tensor order has to do something with the rank of the matrix? (2) What doe tensor order 2,3...means? If I have a 2 by 2 matrix, does that means it is a 2nd.order tensor and 3 by 3 matrix...

Homework Statement I am stuck at this point where I have to prove that the kronecker delta is isotropic tensor. Homework Equations δij=δji The Attempt at a Solution I know that to prove this I have to show that under coordinate transfor mation it does not change..but it's a bit...

I am trying to understand the notion of a covariant derivative and Christoffel symbols. The proof I am looking at starts out with defining a tensor, Tmn = ∂Vm/∂xn where V is a covariant vector. I am having a mental block with regard to the indeces. How is it that the derivative of a...

Hey all! I am a senior in college pretty much done with my mathematics major, but have had minimal physics. I'm currently self-studying special relativity with guidance from my advisor. Most of the books that I have come across use the algebraic/calculus approach such as Spacetime Physics by...

Hi, I've seen in some texts where a tensor is only supplied with one(or two) of it's arguments when it has more than that, and produce a tensor with a lower order than the original. Is this a formal operation? For example, the moment of inertia tensor has 2 arguments, supplying it with an...

Studying and looking through fluid tensors used in GR and have a question to make sure I understand correctly: If I had an isotropic and homogeneous perfect fluid \Omega g_{\mu\nu} and within this fluid I had a generic stress energy tensor \kappa T_{\mu\nu}^{generic} but defined it so that...

Hi Friends I am reading the following paper http://arxiv.org/abs/hep-th/9705122 In the page 4 he says that \tilde{W}_{\mu\nu}=0\Rightarrow V_{\mu}=\partial_{\mu}\lambda Where \tilde{W}^{\mu\nu}\equiv\frac{1}{2}\epsilon^{\mu \nu\rho\sigma}W_{\rho\sigma} and...

consider t is arbitrary tensor and [x,y] is Lie derivative how can we show that L[x,y]t=Lx Ly t - Ly Lx t

I was wondering how the indices of tensors work. I do not understand how the indices of tensors in can be used. For example, \eta _{\mu \nu }, the metric tensor, is like a matrix, and x^{u} is a contravector. How does this extend to notations such as T{_{a}}^{bc} and T{_{ab}}^{c}?

Why stress tensor must be symmetric?

I am learning about tensors. Can somebody give me some help. Thanks.

Hi there. I have this problem, which says: In the cartesian system the tensor T, twice covariant has as components the elements of the matrix: \begin{bmatrix}{1}&{0}&{2}\\{3}&{4}&{1}\\{1}&{3}&{4}\end{bmatrix} If A=e_1+2e_2+3e_3 find the inner product between both tensors. Indicate the type and...

Homework Statement This exercice is in a Chapter named Introduction to Cartesian tensors. The following is the original question of the exercise: Homework Equations Compute the vector: (x1^2 + 2x1*x2^2 + 3x2^2*x3), i The Attempt at a Solution Plz help me, i don't understand what...

One of the definitions of the tensors says that they are multidimensional arrays of numbers which transform in a certain form under coordinate transformations.No restriction is considered on the coordinate systems involved.So I thought they should transform as such not only under rotations but...

Let A, B be matrices with components Aμν , Bμν such that μ, ν = 0, 1, 2, 3. Indices are lowered and raised with the metric gμν and its inverse gμν. Find the trace of ABA-1 in component form? Since A and B are generalized versions of tensors, finding their inverse becomes very tedious if we try...

What is the conceptual difference between and matrix and a tensor? To me they seem like the same thing...

I am a beginner in theory of GR and am trying to understand it better. I have a problem with understanding tensors. I got the algebriac idea, incliding covariance, contravariance and transformations etc of tensors. But not the geometric. Tensors are abstract but can I not have geometric...

Is it correct to view tensors as multi-variable functions? For example, it seems the permutation tensor is a function of three variables and the metric tensor is a function of two variables. Of course, these "functions" turn into constants when i,j, and k (the indices) are known, but it seems...

...is what I am looking for, to understand what is written in GR books. Schutz' First Course In GR is the simplest I could find which has a part dedicated to their explanation but I am looking for something simpler than that. I am looking for something which is not a long mathematics textbook...

Hello I've sometime read physics texts that mention tensor densities (or pseudo-tensors). I find they are quite an ugly notion and I'm not sure to understand their necessity in physics. I have realized that tensor densities with an integer weight can be expressed differently with standard...

Hello. I am trying to get the hang of tensors. I saw this written in http://mathworld.wolfram.com/MetricTensor.html I just wanted to make sure it was correct. g^{\alpha\beta} =\widehat{e}^{i}\ast \widehat{e}^{j} Which says that the dot product of two unit vectors equals the metric...

I'm having a lot of problems with tensors. Here is what the professor in class told us in the lecture notes In three spacetime dimensions (two space plus one time) an antisymmetric Lorentz tensor F^{\mu\nu} = -F^{\nu\mu} is equivalent to an axial Lorentz vector, F^{\mu\nu} =...

Homework Statement Let's say I have (g^{\nu\alpha}g^{\mu\beta} - g^{\nu\beta}g^{\mu\alpha})F_{\nu} The Attempt at a Solution Would this just equal g^{\mu\beta}F_{\alpha} - g^{\mu\alpha}F_{\beta} = \delta^{\mu}_{\alpha}F_{\alpha} - \delta^{\mu}_{\beta}F_{\beta} = 0?

Hi What topics in linear algebra do I need to know to start learning tensors? I know the following topics from linear algebra: 1-equation systems 2-vector spaces(linear independence, span, basis, important subspaces of a vector space) 3-linear transformations(kernel, Image, Isomorphic vector...

My understanding of the Einstein Summation convention is that you sum over the repeated indices. But when I look at the metric tensor for a flat space I know that g^{λ}_{λ} = 1 But the summation convention makes me think that it should equal the trace of the matrix g_{μσ}. So it should...

I haven't learned much of advanced mathematics. It seems that we can use metric tensors to lower or raise index of christoffel symbols. But isn't christoffel symbols made of metric tensors and derivatives of metric tensors? How can we contract indices of a derivative directly with metric tensors...

Can someone explain why the covariant derivative of g_{\alpha\beta} with respect to x^{\lambda} is always zero? I am asking for a physical reason why it must be so.

Let r_{\mu} be a tensor in coordinates x^{c} and R_{b} be a tensor in coordinates X^{c}. Then let r_{\mu} = 0. Then {\partialX^{\nu}/\partialx^{\mu}}R_{\nu} = 0. I read in a book that one can divide both sides of the last equation by the partial derivative to get R_{\nu} = 0. I do not...

Homework Statement Hello guys, hope you'll help me out with this! I'm asked to calculate g^{\alpha\beta}g^{\sigma\rho}(g_{\alpha\sigma}g_{\beta\rho}-g_{\alpha\rho}g_{\beta\sigma}) where g is the metric tensor on a n-dimensional manifold but I can't get to the right result, i keep on getting...

I am learning about tensors. Is gαβAβ the same as Aβgαβ ? Thanks for any help.

I'm looking to learn general relativity, but I'm having a hard time. Frankly, I can't find any textbooks that I can understand. There seems to be a gap between the maths I did at uni, and the maths of general relativity. I've done vector calculus, differential equations, linear algebra and...

some, like momentum appear to be, but are all of them?

I'm trying to figure this out but its confusing. I'm going by some notes someone put up online: samizdat.mines.edu/tensors/ShR6b.pdf Look at Exercise 8.3 on page 19. I got no idea how to do this, and actually I'm not even sure what it's asking. Can anyone give me some pointers? If I just...

I'm wondering why 1/k! is needed in Alt(T), which is defined as: \frac{1}{k!}\sum_{\sigma \in S_k} \mbox{sgn}\sigma T(v_{\sigma(1)},\cdots,v_{\sigma(k)}) After removing 1/k!, the new \mbox{Alt}, \overline{\mbox{Alt}}, still satisfies...

If \omega is an alternating tensor, then Alt(\omega)=\omega, where Alt is the mapping that maps any tensor to an alternating tensor. I guess the converse is also true, i.e., if Alt(\omega)=\omega, then \omega must be an alternating tensor. Am I right?

Hey everyone, I am reading a Schaum's Outline on Tensor Calculus and came to something I can't seem to understand. I'm admittedly young to be reading this but so far I've understood everything except this. My question is: what is the difference between a contravariant tensor and a covariant...

So I tried asking this over in the math resource section but it won't let me post a thread there, so I figured here is the next best place to ask. I am rather worried about Tensor mathematics. I have read a bit into the courses I'm going to be taking next year and I'm rather scared... First...

What's a good self-teaching book about tensor calculus? Is Schaum's Outline version good? I heard it suffers from lack of proofs/derivations. I want something that's easy to follow and doesn't just throw things out without at least some sort of derivation/ sketch-of-proof / explanation. Also...

Hello, I'm working on some problems and I want to pose the following, though I am not completely sure it is correct. Can somebody point me to some sources on this? I have tried googling myself, but I only found differentiation identities with either just vectors and scalars on the on hand, or...

Homework Statement This is not really a homework problem. I'm just trying to learn about tensors by myself. I'm new here (This is only my second post). From what I could gather from the forum rules this seems to be the place for my question. I went through literally hundreds of websites...

I have a question about Garrity's formula at the top of p. 125, here, for a function from the set of 2-form fields to the set of tangent vector fields, together with the formula on p. 123 for the exterior derivative of a 1-form field and Theorem 6.3.1 on p. 125 (Garrity: All the Mathematics you...

I've been looking through my notes for the last few weeks and i still do not see the reason for this use of notation that my lecturer uses, for example We denote by M^{*} \otimes M \otimes M^{*} the vector space of all tensors of type M \times M^{*} \times M \rightarrow \mathbb{R}, where M is...