Tensor Definition and 1000 Threads

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.1 Introduction to Tensor Products ... ... I need help with an aspect of Theorem 10.2 regarding the basis of a tensor product ... ... Theorem 10.2 reads as follows:I do not...

Homework Statement This is Exercise 15.2 in MTW - See attachment Homework Equations See attachment The Attempt at a Solution [/B] My attempt at a solution is also in the attachment. Are my initial assumptions OK? If not can someone nudge me in the right direction. If my initial...

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.1 Introduction to Tensor Products ... ... I need help with another aspect of the proof of Theorem 10.1 regarding the existence of a tensor product ... ...The relevant part of...

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.1 Introduction to Tensor Products ... ... I need help with another aspect of the proof of Theorem 10.1 regarding the existence of a tensor product ... ... The relevant part of...

Homework Statement As the title says, I need to show this. A conformal transformation is made by changing the metric: ##g_{\mu\nu}\mapsto\omega(x)^{2}g_{\mu\nu}=\tilde{g}_{\mu\nu}## Homework Equations The Weyl tensor is given in four dimensions as: ##...

I am reading Anthony W. Knapp's book: Basic Algebra in order to understand tensor products ... ... I need some help with a further aspect of the proof of Theorem 6.10 in Section 6 of Chapter VI: Multilinear Algebra ... The text of Theorem 6.10 reads as follows: The above proof mentions Figure...

I am reading Anthony W. Knapp's book: Basic Algebra in order to understand tensor products ... ... I need some help with an aspect of Theorem 6.10 in Section 6 of Chapter VI: Multilinear Algebra ... The text of Theorem 6.10 reads as follows: The above proof mentions Figure 6.1 which is...

I understand(or assume understand) that geodesic deviation describes how much parallel geodesics diverge/converge on manifolds while moving along these geodesic. But is not it a definition for intrinsic curvature? If it is same as Riemann curvature tensor in terms of describing curvature, why...

I am reading Anthony W. Knapp's book: Basic Algebra in order to understand tensor products ... ... I need some help with an aspect of Theorem 6.10 in Section 6 of Chapter VI: Multilinear Algebra ... The text of Theorem 6.10 reads as follows: About midway in the above text, just at the start...

I am reading Anthony W. Knapp's book: Basic Algebra in order to understand tensor products ... ... I need some help with an aspect of Theorem 6.10 in Section 6 of Chapter VI: Multilinear Algebra ... The text of Theorem 6.10 reads as follows:https://www.physicsforums.com/attachments/5391...

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.1 Introduction to Tensor Products ... ... I need help with the proof of Theorem 10.1 on the existence of a tensor product ... ...Theorem 10.1 reads as follows: In the above text...

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.1 Introduction to Tensor Products ... ... I need help with the proof of Theorem 10.1 on the existence of a tensor product ... ... Theorem 10.1 reads as follows:In the above...

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.1 Introduction to Tensor Products ... ... I need help with the proof of Lemma 10.1 on the uniqueness of a tensor product ... ... Before proving the uniqueness (up to an...

In the calculation of the precession of Mercury's orbit, why is the stress-energy tensor equal to 0? There is energy and momentum at the location of the planet.

How does one setup the stress tensor for a non-Newtonian fluid? I know that for any fluid the normals should be the pressure and for a power law fluid the shear stress in the direction of flow is related by K(du/dy)^n. Does this mean that all other components are 0 for a symmetric pipe or...

I am trying to build up a kind of mind map of the following: Module (eg. vector space) Ring (eg Field) Linear algebra (concerning vectors and vector spaces, from what I understood) Multilinear Algebra (analogously concerning tensors and multi-linear maps) Linear maps & Multilinear maps The...

Homework Statement Calculate the magnitude of the resolved shear stress on the ## (112)## plane in the ##[11\bar{1}] ##direction in a bcc crystal. Homework Equations Tried to use: ## t^{\alpha} = \frac{1}{2} \sigma_{ij} (m_{i} n_{j} + m_{i} n_{j}) ## where m is the direction and n is the...

Hi PF, As you may know, is the the elasticity/stiffness tensor for isotropic and homogeneous materials characterized by two independant material parameters (λ and μ) and is given by the bellow representation. C_{ijkl} = \lambda\delta_{ij}\delta_{kl} + \mu(\delta_{ik}\delta_{jl} +...

I am trying to show that if (C^ab)(A_a)(B_b) is a scalar for arbitrary vectors A_a and B_b then C^ab is a tensor. I want to take the product of the two vectors then use the quotient rule to show that C^ab must then be a tensor. This lead to the question of whether or a not the product of two...

Can anyone show me an example of applying the Ricci curvature tensor to something other than GR? I also ask the same for the curvature scalar. Lately I've been trying to truly increase my understanding of curvature, so that I can see exactly how solutions of the EFE's predict the existence and...

I am doing my M.sc in physics.. In my course I have classical field theory and electrodynamics... I need to learn tensor notations to understand the above subjects... Please tell me about some good introductory books to learn tensor notation to handle things in electrodynamics and classical...

Using Ray D'Inverno's Introducing Einstein's Relativity. Ex 6.31 Pg 90. I am trying to calculate the purely covariant Riemann Tensor, Rabcd, for the metric gab=diag(ev,-eλ,-r2,-r2sin2θ) where v=v(t,r) and λ=λ(t,r). I have calculated the Christoffel Symbols and I am now attempting the...

If I ever say anything incorrect, please promptly correct me! The state of a system in classical mechanics is specified by point in phase space, the point giving us the position and velocity at a given instance. Could we rephrase it by saying a vector in phase space specifies the system? If...

1. Does anyone know why for an isotropic distribution function, pressure tensor reduces to a scalar pressure? For instance, for a Maxwellian distribution P=A ∫ vx vy exp-(vx2 + vy2 + vz2) dvx dvy dvz is not zero. I think everybody should realize how bogus some of the authors are. Google...

What is the trace of a second rank tensor covariant in both indices? For a tensor covariant in one index and contravariant in another ##T^i_j##, the trace is ##T^k_k## but what is the trace for ##T_{ij}## because ##T_{kk}## is not even a tensor?

Is the metric tensor a tensor of rank two simply because the line element (or equivalent Pythagorean relation between differential distances) is "quadratic" in nature? This would be in opposition to say, the stress tensor being a tensor of rank two because it has to deal with "shear" forces. I...

Can these tensor be seen as operators on two elements. So given two elements of something they produce something, for instance a scalar ?

Homework Statement hello, i want to calculate the inertia tensor of the combination of a point mass and a sphere in the object's frame, the center of mass is at the origin. The point mass remains at the surface of the phere The sphere is uniform, radius r and mass M, and the point mass has mass...

Homework Statement In En the quantities Bij are the components of an affine tensor of rank 2. Construct two affine tensors each of rank 4, with components Cijkl and Dijkl for which ∑k ∑l Cijkl Bkl = Bij + Bji ∑k ∑l Dijkl Bkl = Bij - Bji are identities. Homework Equations The Attempt at a...

I know by definition that if T is a 2nd order tensor and v is a vector, curl(Tv)=curl(T)v but what if instead of constant vector v, I have w=grad(u), not constant but obviously an irrotational vector field. Is this still true: curl(Tw)=curl(T)w ? My guess is yes since curl(w)=0 but have no...

Homework Statement The Lorenz gauge ∂Φ/∂t + ∇. A = 0 enables the Maxwell equations (in terms of potentials) to be written as two uncoupled equations; ∂2Φ/∂t2 - ∇2Φ = ρ 1 and ∂2A/∂t2 - ∇2A = j 2 The tensor version using the Lorenz gauge is, i am told, ∂μ∂μ Aα = jα 3 expanded this is...

Homework Statement The gauge ∂tχ - A =0 enables Maxwell's equations to be written in terms of A and φ as two uncoupled second order differential equations. However, when the lorentz condition div A = 0 is applied, we are told the equation can be encapsulated as: one tensor equation ∂μFμA = jμ...

In section 4.4 of gravitational radiation chapter in Wald's general relativity, eq.4.4.49 shows the far-field generated by a variable mass quadrupole: \gamma_{\mu \nu}(t,r)=\frac{2}{3R} \frac{d^2 q_{\mu \nu}}{dt^2} \bigg|_{t'=t-R/c} I have the following field from a rotating binary...

Hello, first off, I'm not sure if I put this question in the right place so sorry about that. Given Bi = 1/2 εijk Fjk how would you find F in terms of B? I think you multiply through by another Levi-Civita, but then I don't know what to do after that. Any help would much appreciated.

Homework Statement We have the following orthogonal tensor in R3: t_{ij} (x^2) = a (x^2) x_i x_j + b(x^2) \delta _{ij} x^2 + c(x^2) \epsilon_ {ijk} x_k Calculate the following quantities and simplify your expression as much as possible: \nabla _j t_{ij}(x) and \epsilon _{ijk} \nabla _i...

Homework Statement A tensor t has the following components in a given orthonormal basis of R3 tij(x) = a(x2)xixj + b(x2) \deltaij x2 + c(x2) \epsilonijk xk (1) where the indices i,j,k = 1, 2, 3. We use the Einstein summation convention. We will only consider orthogonal transformations...

At the risk of sounding ignorant I'd like to propose a question to someone well versed in Homological Algebra and General Relativity. I'm starting to study the tensor product functor in the context of category theory because I'm interested in possibly doing a paper on TQFT for a directed...

How, if at all, would differential geometry differ between the opposite "sides" of the surface in question. Simplest example: suppose you look at vectors etc on the outside of a sphere as opposed to the inside. Or a flat plane? Wouldn't one of the coordinates be essentially a mirror while...

Each set of constant numbers such as ##(v_1, v_2, v_3)## are the components of a constant Cartesian vector because by rotation of coordinates they satisfy the transformation rule. Can we consider each set of constant arrays ## a_{ij};i,j=1,2,3 ## as components of a Cartesian tensor? In other...

Hi there, Over the last couple of weeks, I have been learning about the relativistic description of electromagnetism through Leonard Susskind's Theoretical Minimum lectures, and although I have managed to follow it, there are some parts which I am becoming increasingly confused by, not helped...

I think I've read the the tensor in three dimensions has 10 elements in its matrix(?). Is this related to the 10 dimensions in some forms of string theory?

I want to know the relationship between the optical axis direction of a crystal and the dielectric constants in different directions in an anisotropic material.

we use perfect fluid which is characterized by a energy density and isotropic pressure for general forms of matter. When guessing the values of energy momentum tensor indices we can use the physical insight that they are the flux of four momentum in a constant surface of spacetime. The...

On pages 67 & 68 of Hassani's mathematical physics book, he gives the following definition: "Let ## \mathcal{A} ## and ## \mathcal{B} ## be algebras. The the vector space tensor product ## \mathcal{A} \otimes \mathcal{B} ## becomes an algebra tensor product if we define the product ##...

I have 2 vectors ##\vec {V}=(v_1,v_2,v_3,v_4...v_D)## and ##\vec {X}=(x_1,x_2,x_3,x_4...x_D)## in D-dimensional euclidean space. I want a tensor ,which is crosswise with both of them. I think that the tensor is parallel with (D-2)-dimensional area, am I right? I do not know a lot about tensors...

Hey I'm trying to follow the derivation given here: http://lampx.tugraz.at/~hadley/ss1/studentpresentations/Bloch08.pdf Homework Statement As it says in the pdf: "Based on Noether's theorem construct the energy-momentum tensor for classical electromagnetism from the above Lagrangian. L=-1/4...

In chapter 8 of Padmanabhan's "Gravitation: Foundations and Frontiers" titiled Black Holes, where he wants to explain that the horizon singularity of the Schwarzschild metric is only a coordinate singularity, he does this by trying to find a scalar built from Riemann tensor and show that its...

Hello, I have a question to the singularities of spacetime (where the metric tensor is infinite, but not the coordinate singularities which can be removed be a change of coordinate) It's easy to show that a singularity of the riemanian tensor scalar RαβμνRαβμν leads to a singularity of the...