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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5
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1answer
112 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 ...
0
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0answers
18 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 ...
0
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0answers
59 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 ...
0
votes
1answer
27 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 ...
0
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1answer
36 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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0answers
30 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 ...
1
vote
1answer
30 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 ...
2
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0answers
84 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 ...
2
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1answer
114 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 = ...
1
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0answers
97 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
votes
1answer
67 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} ...
0
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2answers
105 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 $\varepsilon_{kij}b_ic_j$
0
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2answers
416 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 ...
1
vote
0answers
56 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 ...
0
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0answers
40 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
votes
1answer
33 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
votes
1answer
42 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
votes
0answers
57 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 ...
1
vote
3answers
52 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 ...
0
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0answers
101 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
votes
1answer
87 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 ...
1
vote
1answer
136 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
votes
1answer
39 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 ...
5
votes
1answer
115 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
vote
1answer
59 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 ...
1
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2answers
76 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? ...
1
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0answers
47 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 ...
0
votes
2answers
84 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
33 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 ...
1
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0answers
22 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 ${\bf{J}} _{\Omega}$? ...
0
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0answers
12 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 ...
2
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0answers
132 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 proof of the 2-D hairy ball theorem. When trying to generalize this proof to higher dimensions I get the term $$\nabla_{[i}(x^i ...
0
votes
1answer
23 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 ...
4
votes
3answers
1k 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
76 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 ...
0
votes
1answer
58 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
votes
1answer
119 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; ...
1
vote
1answer
70 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 ...
1
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2answers
85 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 $Im(\varphi)=U$. ...
1
vote
1answer
82 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
111 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 ...
0
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0answers
37 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 ...
0
votes
1answer
124 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} = ...
-2
votes
1answer
48 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 ...
3
votes
3answers
297 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*} ...
0
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1answer
41 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 ...
1
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2answers
60 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 ...
4
votes
3answers
473 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 ...
1
vote
2answers
65 views

Rank of a Decomposable Tensor

I'm independently studying Stephen Roman's Advanced Linear Algebra, and I came across a line of reasoning that appears obvious but that I don't understand, and was hoping someone might help me ...
2
votes
1answer
134 views

Relationship between Levi-Civita symbol and Grassmann numbers?

The multiplication rule for Grassmann numbers $\theta_i$ is $$ \theta_i\theta_j = - \theta_j \theta_i $$ so that $\theta_i\theta_i = 0$. Multiplying three Grassmann numbers yields $$ ...