Use this tag for questions about specific tensors (curvature tensor, stress tensor), or questions regarding tensor computations as they appear in multivariable calculus and differential/Riemannian geometry (specifically, when it is amenable to be treated as objects with multiple indices that ...

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37 views

What is meant by “trace on any pair of indices”?

I am reading the book Riemannian Manifolds, written by John M. Lee. On page $53$, the author defines a connection on the tensor bundle $\text{T}_l^k(M)$ and says it satisfies (d) $\nabla$ commutes ...
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1answer
54 views

How to prove that below quantity is a Third Rank Tensor

$F^{ik}$ is an antisymmetric tensor. I want to prove that below quantity is a Third Rank Tensor. $$\dfrac{\partial F_{ik}}{\partial x^{l}} + \dfrac{\partial F_{kl}}{\partial x^{i}} + \dfrac{\partial ...
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1answer
47 views

Notation in symmetric and alternating products of forms

In Lee's Riemannian Manifolds book, we see that a Riemannian metric $g$ can be expressed locally in coordinates $(x^1, \ldots, x^n)$ by $g = g_{ij} dx^i \otimes dx^j$. Introducing the symmetric ...
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1answer
32 views

How to prove the skew symmetry of the Riemannian tensor?

I parallel transported a vector around a parallelogram formed by another two vectors to get the Riemannian tensor: $$R^m{}_{lkj} = \left( \partial_k \Gamma^m{}_{jl} + \Gamma^m{}_{kn} \Gamma^n{}_{jl} ...
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1answer
30 views

Are there any 3rd order tensors satisfying $e_{ijk} e_{lmk} = \delta_{il} \delta_{jm}$ in dimensions higher than three?

My question is simply wrote on the title. (I'm using Einstein's contraction rule.) In the case of three dimensions, I can construct the Levi-Civita-like tensor as follows. \begin{align} e_{ijk} = ...
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1answer
64 views

Minimize Product of Sums of Squared Distances

The Question Given two sets of vectors $S_1$ and $S_2$,we want to find a unit vector $s$ such that $$\{\sum_{u\in S_1}(\|u\|^2-\langle u, s \rangle^2)\} \cdot \{\sum_{v\in S_2}(\|v\|^2 - \langle v, ...
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65 views

Determinant of a tensor

Is there such a thing as the determinant of a tensor of rank $\gt 2$? I'm tried to think how it might be defined -- potentially like, the determinant of the tensor $A=a_{ijk}$ is ...
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0answers
37 views

Difference of two connections is a tensor

I am currently reading through Jost's Riemannian Geometry and Geometric Analysis and am seeking clarification of the statement in the title. Jost abstractly defines a connection on a vector bundle ...
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0answers
32 views

Why the following is not a tensor?

Say we have an arbitrary coordinate system, in which a position vector is represented by: $$\vec V=Z^i\vec Z_i$$ Where $\vec Z_i$ is a covariant basis. Now $\partial {\vec Z_i}/\partial Z^j$ term has ...
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2answers
50 views

Proof of a tensor identity involved in the derivation of the Einstein field equations?

On the wikipedia page for the einstein-hilbert action, the section for the derivation of the einstein field equations cites this identity: $$ \sqrt{g} \nabla_\mu A^\mu = \partial_\mu (\sqrt{g} A^\mu) ...
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0answers
31 views

Showing two tensors are equal

Let $T_{ijkl}$ be a 3-d tensor satisfying $T_{ijkl}=-T_{jikl}=-T_{ijlk}$, and $T_{ijij}=0$. Let $S_{pq}=-T_{rprq}$. I'm trying to show that $$T_{ijkl}=\varepsilon_{ijp}\varepsilon_{klq}S_{pq}$$ Now, ...
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0answers
23 views

Full time derivative of the function (Frank-Oseen energy). Calculus applied to physics problem

This question is about physical theory, but my question is pure nathematical, so I post it here. I don't think you have to know physics in order to answer it. I am studying liquid crystal theory with ...
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0answers
25 views

Tensor notation in vectors

I have the following expression $\partial_{x_a}(\partial_{x_b} \rho \partial_{x_b}\rho) - \partial_{x_b}(\partial_{x_a}\rho\partial_{x_b}\rho)$ How do I write this in vector notation? At least the ...
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1answer
32 views

Symmetries of the space form of riemann curvature tensor

We have $R_{abcd}=(g_{ac}g_{bd}-g_{ad}g_{bc})$ I need to establish the symmetry: $-R_{bacd}=R_{abcd}$ What I thought was just interchange a and b in the expression to get: ...
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0answers
20 views

Why is the border rank and rank different for order 3 tensors and above?

Recall the definition of border-rank of a tensor T: border-rank(T) = the minimum r such that $\forall \epsilon > 0$ there exists an approximate tensor $T' = \sum^r_{i=1} u_i \otimes v_i \otimes ...
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2answers
34 views

Action of universal R-matrix of U_q(sl_2)

My question is really simple but requires a few definitions. No special knowledge of quantum groups should be needed, it is more about tensor algebra. Let $q \in \mathbb{C}$ with $q \neq 0, \pm 1$. ...
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1answer
23 views

Multiplying metric tensors.

Suppose I have the metric $g_{ab}$ in a k-dimensional manifold. Firstly, do metric tensors like this always commute? Is it always necessarily true that $g^{ab}g_{bc}=\delta^a_c$? What happens when I ...
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33 views

Why cant we define the Einstein Tensor in an easier way?

The Einstein tensor is defined as: $G_{ab}=R_{ab}-\frac{1}{2}g_{ab}R$ where $R=g^{ab}R_{ab}$. So why cant we just simplify this like: ...
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3answers
325 views

Why is the Riemann curvature tensor the technical expression of curvature?

According to my textbook on general relativity (Sean Carrol's book) and differential geometry, the Reimann curvature tensor is the technical expression of curvature. What makes the tensor so special? ...
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1answer
20 views

Tensor equations. Can I change an equation from covariant to contravariant?

Say I have a tensor equation like $G_{ab}=R_{ab}-\frac{1}{2}g_{ab}R$. Does this also imply that $G^{ab}=R^{ab}-\frac{1}{2}g^{ab}R$?
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1answer
21 views

Need definition of symmetric and antisymmetric tensor representations of a Lie algebra

I couldn't find a definitive answer online. Suppose we have a representation of a Lie algebra $(\pi,V)$. Consider the symmetric and antisymmetric vector subspaces of the $k$-th tensor product of ...
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2answers
58 views

Levi-Civita tensor

Show that $\epsilon_{ijk} A_{il} A_{jm} A_{kn} = \det(A) \epsilon_{lmn}$ where $\epsilon$ epsilon is the standard Levi-Civita symbol and A is a three dimensional matrix. I found the above ...
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0answers
39 views

What is this notation $\odot$ for?

(Note that symmetric algebra and symmetric tensor do not coincide when the characteristic is not $0$.) I'm reading this aricle:http://en.m.wikipedia.org/wiki/Symmetric_tensor And here it defines ...
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1answer
45 views

Are all 2D tensors in a specified flat metric equal to that same metric conformally scaled?

I have a tensor $T_{mn}$ where its indices coorespond to a flat metric $g_{mn}$. I want $T_{mn}$ to be a new metric $\tilde{g}_{mn}$, such that $T_{mn}(g_{rs}) = \tilde{g}_{mn}$. A theorem says that ...
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1answer
11 views

Is it okay to define k-th symmetric power of $M$ in this way?

I want to define the tensor algebra and related algebras in a very formal way. I will illustrate how I tried to define algebras below. Let $R$ be a commutative ring and $M$ be an $R$-module. ...
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0answers
19 views

What's actually $S^k(M)$?

Let $R$ be a commutative ring and $M$ be an $R$-module. Let $T(M)$ be the tensor algebra of $M$. Then, what is $S(M)$ (symmetric algebra and $S^k(M)$? Some articles define $S(M)$ as a quotient of ...
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17 views

Under what condition, does this universal property of tensor algebra hold?

Let $R$ be a commutative ring and $M$ be an $R$-module. Note that the tensor algebra $T(M)$ is a unital associative $R$-algebra. Below is the universal property of the tensor algebra. Theorem: ...
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1answer
58 views

What does shear mean?

As I understand it, the gradient of a vector field can be decomposed into parts that relate to the divergence, curl, and shear of the function. I understand what divergence and curl are (both ...
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0answers
17 views

What is shear in the context of vector fields? [duplicate]

As I understand it, the gradient of a vector field can be decomposed into parts that relate to the divergence, curl, and shear of the function. I understand what divergence and curl are (both ...
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0answers
36 views

Matrix into a 4-tensor

I've been trying to figure out a way to add a dimension to a matrix with a certain rule: Let our matrix X have dimensions NxM, L be some positive number, and the desired resulting 4-tensor (3d ...
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1answer
19 views

Does there exists a proof that any divergentless tensor can be decomposed into the sum of divergentless symmetric and antisymmetric tensors?

A friend and I attempted to work out the proof on the board that any divergentless asymmetric tensor can be written as the sum of divergentless symmetric and antisymmetric tensors. We wrote down the ...
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1answer
27 views

Checking alternating tensors

How do I check that $$f(x,y)=x_1y_2-x_2y_1+x_1y_1$$ is an alternating tensor? I did check that f is a tensor, but how do I know if it is alternating by direct computation? Thanks in advance!
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Are the Factor Matrices that are obtained after a CP Decomposition of a Tensor in the same Row Space?

Lets say we have to decompose a 3-way Tensor X over Real Numbers that has Dimensions I x J x K. Into R Rank-1 Tensors via CP Decomposition model. As a result we will obtain 3 factor matrices, 1 for ...
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1answer
27 views

System of equation involving tensor

I have to solve a system of equation involving tensor: \begin{align} \underline{\underline{a_1}}\cdot\underline{x} + \underline{\underline{\underline{\underline{b_1}}}} \therefore ...
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0answers
42 views

Connections and Ricci identity

Given $\nabla$ a torsionless connection, the Ricci identity for co-vectors is $$\nabla_a \nabla_b \lambda_c - \nabla_b \nabla_a \lambda_c = -R^d_{\,\,cab}\lambda_d.$$ Prove $R^a_{[bcd]} = 0$ by ...
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22 views

what book do you recommend that covers basic tensor stuff explaining like you did

I came across your explanation of covariant and contravariant tensors. Found it very clear. Im tryiing to learn this stuff from some books but so far none explains it that clearly and definitely not ...
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1answer
60 views

Deriving Ricci identity for co-vector fields

Let $\nabla$ be the covariant derivative associated with a torsionless connection. Prove the Ricci identity for covectors: $$\nabla_a \nabla_b \lambda_c - \nabla_b \nabla_a \lambda_c = ...
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43 views

Proof that border rank and the rank of a matrix (order 2 tensors) are equivalent

Recall the definition of border rank for a matrix (order 2 tensor, which can be easily be extended to any order tensor): border-rank(T) is the minimum r such that $\forall \epsilon > 0$ there ...
2
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1answer
48 views

Determining the rank of a sparse tensor in $\mathbb{R}^{2 \times 2 \times 2}$

In this question I will only consider order 3 tensors. Consider the following tensor in $\mathbb{R}^{2 \times 2 \times 2}$ (which I want to prove its rank 3): $$ T' = \begin{bmatrix} \begin{bmatrix} ...
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2answers
53 views

Introducing new indices with tensor/index notation

I understand where the $k$ comes from in $\varepsilon_{klm}$, however why do we need to introduce $l,m$ rather than continuing with $i,j$, i.e $\varepsilon_{kij}b_ic_j$
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2answers
64 views

Notion of contraction in tensor algebra

Assuming a vector space V and it's basis set $\{\vec{e}_\nu\}$. A vector $\vec{v}$ can be written as: $\vec{v}=x^\nu\vec{e}_\nu$ where $x^\nu$ is the corresponding contravariant coordinate. We can ...
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Topology and Smooth Structure on the Bundle of Covariant $k$-Tensors

(All vector spaces are finite dimensional and real). Given a vector space $V$, let $T^k(V^*)$ denote the vector space of all the covariant $k$ tensors on $V$. Following Lee's Introduction to Smooth ...
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Understanding Euler density

I know the definition of Euler density in terms of antisymetrized contractions of products of the Riemann curvature tensor, ie Euler density is the $\mathcal{R}^n$ in these formulae: And I know ...
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1answer
32 views

Step in a proof about alternating operators

The theorem is that if $f$ is a $k$-linear function on a vector space $V$, then the $k$-linear function $Af$ is alternating. $\def\sgn{\operatorname{sgn}}Af=\sum (\sgn \sigma)\sigma f$ Proof: ...
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1answer
36 views

Ortogonal matrix simple identity?

In order to prove the invariance of the trace of a tensor under the transformation $\tilde{T}^{i,j}=\Sigma_{k,l} O^i_kO^j_lT^{k,l}$ where $O\in SO(3)$ I have to prove that $\Sigma_{k,l} ...
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42 views

Natural Isomorphism between $V^*\otimes W^*$ and $\mathcal L^2(V,W; F)$.

I am trying to prove the following. Let $V_1, \ldots, V_k$ be finite dimensional vector spaces over a field $F$. There is a natural isomorphism between $V_1^*\otimes\cdots\otimes V_k^*$ and ...
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3answers
37 views

Basic Question Regarding the Universal Property of the Tensor Product.

(All vector spaces are over a fixed field $F$). Universal Property of Tensor Product. Given two finite dimensional vector spaces $V$ and $W$, the tensor product of $V$ and $W$ is a vector space ...
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67 views

Curvature and Pfaffian forms in terms of the Riemann tensor

I am teaching my self differential geometry, but I am mainly familiar with classic tensor notation. In modern Cartan exterior form notation the curvature forn $\Omega$ and the Pfaffian seem to do the ...
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1answer
76 views

Elementary tensors [duplicate]

I need to determine whether the following function is tensor on $\Bbb R^4$ and express it in terms of elementary tensors. Can someone please help me with it? I do not know what elementary tensor means ...
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1answer
88 views

Connections and tensor fields

Let $T$ be a $(1, 1)$ tensor field, $\lambda$ a covector field and $X, Y$ vector fields. We may define $\nabla_X T$ by requiring the ‘inner’ Leibniz rule, $$\nabla_X[T(\lambda, Y )] = ...