# Tensor algebra Definition and 46 Discussions

In mathematics, the tensor algebra of a vector space V, denoted T(V) or T•(V), is the algebra of tensors on V (of any rank) with multiplication being the tensor product. It is the free algebra on V, in the sense of being left adjoint to the forgetful functor from algebras to vector spaces: it is the "most general" algebra containing V, in the sense of the corresponding universal property (see below).
The tensor algebra is important because many other algebras arise as quotient algebras of T(V). These include the exterior algebra, the symmetric algebra, Clifford algebras, the Weyl algebra and universal enveloping algebras.
The tensor algebra also has two coalgebra structures; one simple one, which does not make it a bialgebra, but does lead to the concept of a cofree coalgebra, and a more complicated one, which yields a bialgebra, and can be extended by giving an antipode to create a Hopf algebra structure.
Note: In this article, all algebras are assumed to be unital and associative. The unit is explicitly required to define the coproduct.

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1. ### B Attempted proof of the Contracted Bianchi Identity

My Attempted Proof ##R^{mn}_{;n} - \frac {1} {2} g^{mn} R_{;n} = 0## ##R^{mn}_{;n} = \frac {1} {2} g^{mn} R_{;n}## So, we want ##2 R^{mn}_{;n} = g^{mn} R_{;n} ## Start w/ 2nd Bianchi Identity ##R_{abmn;l} + R_{ablm;n} + R_{abnl;m} = 0## Sum w/ inverse metric tensor twice ##g^{bn} g^{am}...
2. ### SH2372 General Relativity (8X): Components of tensor operations

3. 4. ### Showing that the gradient of a scalar field is a covariant vector

In a general coordinate system ##\{x^1,..., x^n\}##, the Covariant Gradient of a scalar field ##f:\mathbb{R}^n \rightarrow \mathbb{R}## is given by (using Einstein's notation) ## \nabla f=\frac{\partial f}{\partial x^{i}} g^{i j} \mathbf{e}_{j} ## I'm trying to prove that this covariant...
5. ### Total Momentum Operator for Klein Gordon Field

As $$\hat{P_i} = \int d^3x T^0_i,$$ and $$T_i^0=\frac{\partial\mathcal{L}}{\partial(\partial_0 \phi)}\partial_i\phi-\delta_i^0\mathcal{L}=\frac{\partial\mathcal{L}}{\partial(\partial_0 \phi)}\partial_i\phi=\pi\partial_i\phi.$$ Therefore, $$\hat{P_i} = \int d^3x \pi\partial_i\phi.$$ However...
6. ### I Anyone knows why musical isomorphism is called so?

Anyone knows why musical isomorphism is called so? Why is it musical? https://en.wikipedia.org/wiki/Musical_isomorphism

14. ### Riemann curvature coefficients using Cartan structure equation

To calculate the Riemann coefficient for a metric ##g##, one can employ the second Cartan's structure equation: $$\frac{1}{2} \Omega_{ab} (\theta^a \wedge \theta^b) = -\frac{1}{4} R_{ijkl} (dx^i \wedge dx^j)(dx^k \wedge dx^l)$$ and using the tetrad formalism to compute the coefficients of the...
15. ### I Ricci scalar for FRW metric with lapse function

I need the Ricci scalar for the FRW metric with a general lapse function ##N##: $$ds^2=-N^2(t) dt^2+a^2(t)\Big[\frac{dr^2}{1-kr^2}+r^2(d\theta^2+\sin^2\theta\ d\phi^2)\Big]$$ Could someone put this into Mathematica as I don't have it?
16. ### Find the tensor that carries out a transformation

I got stuck in this calculation, I can't collect everything in terms of ##dx^{\mu}##. ##x'^{\mu}=\frac{x^{\mu}-x^2a^{\mu}}{1-2a_{\nu}x^{\nu}+a^2x^2}## ##x'^{\mu}=\frac{x^{\mu}-g_{\alpha \beta}x^{\alpha}x^{\beta}a^{\mu}}{1-2a_{\nu}x^{\nu}+a^2g_{\alpha \beta}x^{\alpha}x^{\beta}}##...
17. ### I Problem with vanishing tensor equation and raising all indices

I have an equation $$\chi_\nu\nabla_\mu\chi_\sigma+\chi_\sigma\nabla_\nu\chi_\mu+\chi_\mu\nabla_\sigma\chi_\nu=0$$so we also have$$g_{\nu\rho}g_{\mu\tau}g_{\sigma\lambda}\left(\chi^\rho\nabla^\tau\chi^\lambda+\chi^\lambda\nabla^\rho\chi^\tau+\chi^\tau\nabla^\lambda\chi^\rho\right)=0$$Does...
18. ### I Beginner question about tensor index manipulation

For instance, using the vector ##A^\alpha e_\alpha##: ##g_{\mu \nu} e^\mu \otimes e^\nu (A^\alpha e_\alpha) = g_{\mu \nu} (e^\mu, A^\alpha e_\alpha) e^\nu ## ##g_{\mu \nu} e^\mu \otimes e^\nu (A^\alpha e_\alpha) = A^\alpha g_{\mu \nu} \delta_\alpha^\mu e^\nu = A^\mu g_{\mu \nu} e^\nu = A_\nu...
19. ### I Transformation of the contravariant and covariant components of a tensor

I have read many GR books and many posts regarding the title of this post, but despite that, I still feel the need to clarify some things. Based on my understanding, the contravariant component of a vector transforms as, ##A'^\mu = [L]^\mu~ _\nu A^\nu## the covariant component of a vector...

27. ### I Swapping Tensor Indexes

What is the difference between ##{T{_{a}}^{b}}## and ##{T{^{a}}_{b}}## ? Both are (1,1) tensors that eat a vector and a dual to produce a scalar. I understand I could act on one with the metric to raise and lower indecies to arrive at the other but is there a geometric difference between the...
28. ### Divergence of the energy momentum tensor

I need to prove that in a vacuum, the energy-momentum tensor is divergenceless, i.e. $$\partial_{\mu} T^{\mu \nu} = 0$$ where $$T^{\mu \nu} = \frac{1}{\mu_{0}}\Big[F^{\alpha \mu} F^{\nu}_{\alpha} - \frac{1}{4}\eta^{\mu \nu}F^{\alpha \beta}F_{\alpha \beta}\Big]$$ Here ##F_{\alpha...
29. ### Time Derivative of Rank 2 Tensor Determinant

Homework Statement Show that for a second order cartesian tensor A, assumed invertible and dependent on t, the following holds: ## \frac{d}{dt} det(A) = det(a) Tr(A^{-1}\frac{dA}{dt}) ## Homework Equations ## det(a) = \frac{1}{6} \epsilon_{ijk} \epsilon_{lmn} A_{il}A_{jm}A_{kn} ## The...
30. ### I Contracovarient and covarient basis form orthogonal basis

Hi everyone, I am trying to self study some general relativity however I met some problem in the contravarient and covarient basis. In the lecture, or you can also find it on wiki page 'curvilinear coordinates', the lecturer introduced the tangential vector ei =∂r/∂xi and the gradient vector ei...
31. ### I Motivating definitions in calculus on manifolds

Hi I am a person who always have had a hard time picking up new definitions. Once I do, the rest kinda falls into place. In the case of abstract algebra, Stillwell's Elements of Algebra saved me. However, in the case of Spivak's Calculus on Manifolds, I get demotivated when I get to concepts...
32. ### I Index Notation for Lorentz Transformation

The Lorentz transformation matrix may be written in index form as Λμ ν. The transpose may be written (ΛT)μ ν=Λν μ. I want to apply this to convert the defining relation for a Lorentz transformation η=ΛTηΛ into index form. We have ηρσ=(ΛT)ρ μημνΛν σ The next step to obtain the correct...
33. ### A Relationship between metric tensor and position vector

Given the definition of the covariant basis (##Z_{i}##) as follows: $$Z_{i} = \frac{\delta \textbf{R}}{\delta Z^{i}}$$ Then, the derivative of the covariant basis is as follows: $$\frac{\delta Z_{i}}{\delta Z^{j}} = \frac{\delta^2 \textbf{R}}{\delta Z^{i} \delta Z^{j}}$$ Which is also equal...

40. ### Divergence of Gradient inverse

Dear All, I'm doing some tensor calculation on the divergence of gradient (of a vector) inverse. Am I allowed to first use the nabla operator on gradient and then inverse the whole product? In other words, I'm searching for the divergence of a 2nd order tensor which is itself inverse of...
41. ### Transformation rule for product of 3rd, 2nd order tensors

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 =...
42. ### Describe movement of particles with one equation?

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...
43. ### Definition of tensor product

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...
44. ### Tensors in GR

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...
45. ### Tensor coordinate system help

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 =...
46. ### Why can we not do algebraic methods like transposing with tensors

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...