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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1answer
18 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
26 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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12 views

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
26 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
40 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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19 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 ...
2
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1answer
53 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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0answers
35 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
46 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
49 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
48 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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0answers
47 views

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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0answers
31 views

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 ...
0
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1answer
31 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: ...
0
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1answer
35 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} ...
2
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0answers
39 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
35 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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0answers
58 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 ...
2
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1answer
73 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
82 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 )] = ...
2
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1answer
24 views

contravariant and covariant basis

My text book defines covariant (1) and contravariant (2) basis as follows. $$ \epsilon_i=\frac {\partial x}{\partial q_i} \hat e_x + \frac {\partial y}{\partial q_i} \hat e_y + \frac {\partial ...
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1answer
88 views

What is the intuition behind tensors?

Can someone please explain the intuition behind tensors? Like an example or something of the similar kind that I should keep in mind reading the theorems about it? I can't visualize it Thanks in ...
1
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1answer
42 views

(1,s) tensor fields

I'm having a hard time understanding what's going on with tensor fields. I understand that $A$ is a smooth covariant tensor field of order $s$ (or a $(0,s)$ tensor field) on a smooth manifold $M$ if ...
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2answers
60 views

Meaning of $(\nabla_x + \nabla_\xi)^n$

what does $(\nabla_x + \nabla_\xi)^n$ mean? (In the context, $x,\xi\in \mathbb{R}^d$). In my notes, the author states this should be a tensor, but what exactly does he mean. Is this clear notation? ...
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0answers
46 views

Introduction to tensor (for graph analysis)

I am starting a PhD program in social network analysis and I would like to have some suggestions about introductory books and online material to Tensor calculus. I am a complete newbie in algebra. ...
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0answers
27 views

How to rigorously show tensor identities using symmetry arguments?

I am wondering how to rigorously show tensor identities such as the following. Let $n$ denote the radial unit vector in $D$ dimensions. Then $\langle n_i n_j \rangle = \frac 1 D \delta_{ij}$ and ...
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2answers
58 views

Self-studying differential forms and tensors

I am interested in understanding the generalized Stokes' Theorem. From my understanding, this theorem involves differential forms and exterior algebra, and tensors (to some extent). I'm not ...
0
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1answer
31 views

Covariant Stared Vector Spaces?

I reading john Lee's book entitled "Introduction to Smooth Manifolds" and on page 311 $$T^{k}\!\!\left(V^{*}\right)=V^{*}\!\!\times\!\!V^{*}...V^{*}\;\;k\text{-times}$$ is defined as the vector ...
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0answers
18 views

How to compute the inertial tensor ${\bf{J}} _{\Omega}$?

Suppose that $\Omega$ is a body of revolution of the function $y=f(x), a\le x \le b$ around the $x$-axis, where $f(x)>0$ is continuous. How to compute the inertial tensor ${\bf{J}} _{\Omega}$? ...
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0answers
10 views

Why the sense of orientation in the graphic representation of a 2-form does not cancel each other?

In this article, it is said that the graphic representation of the 2-form ${\bf F}=B_z{\bf d}y\wedge{\bf d}z$ are tube with a sense of a circular orientation. How does this circular orientation show ...
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2answers
66 views

Prove $\omega\wedge\eta=0$ iff there exist a $\theta\in A^3(\mathbb{R}^5)$ such that $\eta=\omega\wedge\theta?$ [closed]

Suppose $\omega\in A^1(\mathbb{R}^5)$ and $\eta\in A^4(\mathbb{R}^5)$. prove $\omega\wedge\eta=0$ iff there exist a $\theta\in A^3(\mathbb{R}^5)$ such that $\eta=\omega\wedge\theta?$
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0answers
114 views

Geometric meaning of $\nabla_{[i}(x^i \nabla_{j]}x^j)$ and $(\nabla_{[i}x^i )\nabla_{j]}x^j$

While teaching myself tensor calculus I have come up with [this] http://mathematica.stackexchange.com/a/71613/12306 {The proof of the 2-D hairy ball theorem). When trying to generalize this proof ...
0
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1answer
17 views

Scalar Curvature on a given surface

I have a very basic understanding of how to compute the ricci curvature tensor...I know a considerable amount about it, but don't know how to compute it. Could someone give me an example of how to ...
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3answers
477 views

Are there any differences between tensors and multidimensional arrays?

I see lots of references saying things like A tensor is a multidimensional or N-way array But others that say things like it should be remarked that other mathematical entities occur in ...
2
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1answer
56 views

Evaluating contractions of a tensor product

Consider $T = \delta \otimes \gamma$ where $\delta$ is the $(1,1)$ Kronecker delta tensor and $\gamma \in T_p^*(M)$, the co-tangent space over some manifold $M$. Evaluate all possible contractions of ...
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1answer
38 views

Index notation with non-commuting matrix entries

Just a contradiction I came across working with matrix multiplication in index notation: I'm probably using some rule wrong, but I can't figure out which one. Consider the expression $A_{ij} B_{ik}$, ...
3
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1answer
66 views

differential geometry : basic query about tensor notation and tensor products

I have a few very basic queries. I've been studying differential geometry as part of a course on General Relativity, so I don't have a very well grounded understanding of the mathematical formalism; ...
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1answer
35 views

Show that the trace of the operator $S \wedge T$ is zero

I have some difficulties with the following problem: Let $V$ be a finite dimensional vector space over $\mathbb{K}$. Let $S,T \in L(V,V)$. Show that the trace of the operator $S \wedge ...
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2answers
62 views

Question about tensor products, decomposable tensors, …

I need some help with the following problem: Let $V_1,\ldots,V_m$ be finite dimensional vector spaces over $\mathbb{K}$. Let $\varphi \in L(V_1,\ldots,V_m;U)$ such that $Im(\varphi)=U$. ...
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1answer
26 views

Field extension of a vector space

If $V$ is a vector space over the field $k$, and $K$ is a field extension of $k$, then $(V)_K$ over $K$ is a vector space. How this new vector space is constructed? and how are the linear ...
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0answers
20 views

How does the $10$ dimensional irrep (tensor) of $SU(3)$ look like?

We know that for $SU(3)$ the following tensors furnish the $\mathbf{d}$ dimensional irreducible representation: $$\phi^i\hspace{1cm} (\mathbf{3})\\ \phi^{ij}\hspace{1cm} (\text{asymmetric in ...
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0answers
34 views

Cauchy Equations and Navier Stokes

I'm attempting to take the Navier Stokes Equation and coming up with an expression that will allow me to numerically determine the velocity of non-Newtonian fluid flow. The text I'm using is Cengel ...
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0answers
33 views

Does there exists known special cases of a zero Riemann tensor for 3D metrics?

In two dimensions, if one has a flat metric $g_{ab}$, then one can transform $g_{ab}$ to another flat metric $h_{ab}=e^{2\varphi}g_{ab}$, when $\nabla^2 \varphi =0$ and the Riemann tensor remains ...
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1answer
40 views

Do we write a metric tensor as a matrix?

The metric tensor is an (0,2) tensor that is denoted by $g_{\mu\nu}$ in general relativity. I often see people write the metric field in matrix form like \begin{equation} g_{\mu\nu} = ...
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1answer
28 views

Is the finding trace of the Riemann tensor the same thing as contracting two indices?

To form the Ricci curvature tensor, we have to take the trace of the Riemann tensor. But I also know \begin{equation} R_{ij} := R_{kij}^{\phantom{kij}k} \end{equation} Can someone show me why ...
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3answers
192 views

Proof in Hamilton: Divergence theorem for differential forms?

For a vector field $X\in\Gamma(TM)$ on a closed Riemannian manifold $(M,g)$ we have \begin{align*} \int_M\text{div}X\;\mu=0, \end{align*} where \begin{align*} ...
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0answers
30 views

Stiffnes tensor, Hooke's law

Let's have a deformed body of an isotropic homogenous material. How is it possible that we can write the free energy in the form $$F=F_0+\frac12\lambda\left(\sum_i ...
0
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1answer
39 views

coordinate transformation and tensor

A 2 dimensional Euclidean space is represented by two different coordinate systems: the Cartesian system $(x_1,x_2)$ and an alternative system $(\xi^1,\xi^2)$ where ...
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2answers
51 views

Question concerning tensors

As some of you may have seen from my previous question(s), I am working through Spivak's Calc on Manifolds and this happens to be the first time I've been introduced to tensors formally, though I have ...
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3answers
130 views

Why is a linear transformation a (1,1) tensor?

Wikipedia says that a linear transformation is a $(1,1)$ tensor. Is this restricting it to transformations from $V$ to $V$ or is a transformation from $V$ to $W$ also a $(1,1)$ tensor? (where $V$ and ...