Tensor algebra Definition and 69 Threads

1. Problem statement: Assume that u is a vector and A is a 2nd-order tensor. Derive a transformation rule for a 3rd order tensor Zijk such that the relation ui = ZijkAjk remains valid after a coordinate rotation.Homework Equations : [/B] Transformation rule for 3rd order tensors: Z'ijk =...

Hello everybody, I have a new thread to post,it is very important to find a solution for this : -Imagine a box full of air particles.The particles are forced to move to a point A on the edge of the box.My question is now,how can I mathematicly describe the movement of these particles toward...

I'm relatively new to differential geometry and would like to check that this is the correct definition for the tensor product of (for simplicity) two one-forms \alpha,\;\beta\;\;\in V^{\ast} : (\alpha\otimes\beta)(\mathbf{v},\mathbf{w})=\alpha (\mathbf{v})\beta (\mathbf{w}) where...

Homework Statement (a) Show acceleration is perpendicular to velocity (b)Show the following relations (c) Show the continuity equation (d) Show if P = 0 geodesics obey: Homework EquationsThe Attempt at a SolutionPart (a) U_{\mu}A^{\mu} = U_{\mu}U^v \left[ \partial_v U^{\mu} +...

Hi all, I'm fairly new to GR, and I'm also somewhat new to tensors as well. I'm looking for some detailed explanation of a tensor, as I want to begin studying GR mathematically. I watched a video that was posted on PF not too long ago that was pretty good. I'm having trouble remembering who it...

Homework Statement A tensor and vector have components Tαβγ, and vα respectively in a coordinate system xμ. There is another coordinate system x'μ. Show that Tαβγvβ = Tαβγvβ Homework Equations umm not sure... ∇αvβ = ∂vβ/∂xα - Γγαβvγ The Attempt at a Solution Tαβγvβ =...

Hello everyone! Even though I have done substantial tensor calculus, I still don't get one thing. Probably I am being naive or even stupid here, but consider $$R_{\mu\nu} = 0$$. If I expand the Ricci tensor, I get $$g^{\sigma\rho} R_{\sigma\mu\rho\nu} = 0$$. Which, in normal algebra, should...

Hi there. I wanted to demonstrate this identity which I found in a book of continuum mechanics: ##curl \left ( \vec u \times \vec v \right )=div \left ( \vec u \otimes \vec v - \vec v \otimes \vec u \right ) ## I've tried by writting both sides on components, but I don't get the same, I'm...

Homework Statement Hi, I have a problem calculating the variation of the action using tensor algebra because two derivative indices are causing some problem. Homework Equations Generally you have the action S = \int L(A^{\mu}, A^{\mu}_{\;,\nu}, x^{\mu})d^4x where: A ^{\mu}=...

Does anyone know any good reading material on Tensor Algebra? Cannot seem to find good book about it. Thanks Also, I apologise if I post this in the wrong section.

Homework Statement (firstly, Apologies for having to use a picture..) If u^{i} is the 4-velocity of a point on a manifold, then we use affine parameterisation g_{ij}u^{i}u^{j}=1. The attached picture shows our rest frame, ie x^{0}=const and a point ("us") on this surface. If our velocity is...

I am trying to proove that the following relation: A_{\nu} \partial_{\mu} \partial^{\nu} A^{\mu} = A_{\nu} \partial^{\mu} \partial^{\nu} A_{\mu} The only way I found is by setting: A_{\nu} \partial_{\mu} \partial^{\nu} A^{\mu} = A_{\nu} \partial_{\mu} \partial^{\nu} g^{\mu \sigma}...

Homework Statement {u^i} = {g^{kj}} A _{kj}^i just trying to modify it, not sure of my tensor algebra. Is this right? {u^i} = {g^{kj}} A _{kj}^i {u^i} = g_a^j{g^{ka}} A _{kj}^i g_j^a{u^i} = {g^{ka}} A _{kj}^i Just not sure if there should have been a metric contraction, with the resulting D...

Does anyone have a good resource of worked examples on simple (ish) tensor algebra? By way of background I'm a mechanical engineer tackling computatonal solid mechanics and my class notes aren't exactly helpful. I have Holzapfel's 'Nonlinear Solid Mechanics' and while very useful I like to learn...

Homework Statement The angular momentum density in the electromagnetic field is defined in terms of the momentum density (3.6, BELOW) by \textbf{L}_{EM} = \textbf{x}\times\textbf{P}_{EM} = \textbf{x}\times(\textbf{E}\times\textbf{B})/\mu_{0}c^2 Show that if the continuity equation for...

Homework Statement Problem 2, chapter 3 of Wald's General Relativity. The details don't matter much, but it is given a totally anti-symetric tensor field Eab such that EabEab=2(-1)^(s), s being the signature of the metric. I have checked a solution to the exercise, and somewhere during the...

Unfortunately there seems to be a misprint in the paper I'm reading which is an introduction to clifford algebra, it says:(I highlighted in red possible misprint, either one of them has to be true misprint if you know what I mean) The Clifford algebra C(V) is isomorphic to the tensor algebra...

Hi Please tell me if tensor algebra is neccesary for understanding GR. I don't know anything about tensor algebra. Thanks in anticipation Abhishek Jain

Hi, I need some help understanding basic tensor algebra, especially differentiation. The subject I'm studying is quantum field theory, so I'll use examples from there. First let's start with a real scalar field. This has a Lagrangian density given by \mathcal{L} =...