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 ...

learn more… | top users | synonyms

1
vote
0answers
36 views

Example of two modules M, N where the set of the $m\otimes n$ is not a submodule

If I remember my Algebra class correctly, this is in general not true because, for example, the addition isn't closed. So if I take: $B:\mathbb{R}^2 \times \mathbb{R}^2\to \mathbb{R}^{2^2}\cong ...
0
votes
1answer
17 views

Solution to tensor/matrix equation

I need to find a real, symmetric matrix, $A$, that satisfies: $\sum_{i,j} c_i c_j A_{ki}A_{jl} = A_{lk}$ I believe this is an equation of the form: $c^T B c = A$, where $c$ is $\mathbb{R}^{N \times ...
2
votes
0answers
26 views

Does $\mathfrak T^r(\Bbb R^m)$ count as an vector space?

Here $\mathfrak T^r (\Bbb R^m)$ denotes all the $r$-th tensors (multi-linear functions) acting upon the elements $(u_1,\cdots,u_r)$ from the product space $\displaystyle \prod^r \Bbb R^m$. And the ...
2
votes
1answer
29 views

Understanding Symmetric tensor field

I am reading an article in which author calls some basic tensor analysis result. He states in general we define on $\mathbb R^N$ that $$ \mathcal T^k(\mathbb R^N):=\{\xi:\,\mathbb ...
0
votes
0answers
25 views

Jaumann deviatoric stress rate.

Being a bit cheeky as I asked this question over on Physics but didn't get a response. http://physics.stackexchange.com/questions/196393/jaumann-deviatoric-stress-rate Background about terms in this ...
1
vote
1answer
23 views

How to denote a tensor in terms of matrices product?

How to write a tensor in terms of a product of matrices? For example, I have $a \times b$ matrix $F$, and I want to create a 3D $a \times a \times a$ tensor $T$, where $T_{i,j,k} = \sum_{m=1}^{b} ...
4
votes
1answer
60 views

Coordinate-Free Definition of Trace.

$\DeclareMathOperator{\tr}{trace}$ I am reading the wikipedia article on the trace operator. The section titled Coordinate-Free Definition defines the trace as follows. Let $V$ be a finite ...
1
vote
2answers
16 views

The Riemannian Curvature in a solid sphere

Is the Riemannian Curvature at the centre of a solid sphere zero?
2
votes
1answer
28 views

Couple stress tensor reference.

Can someone give me a good mathematical reference for couple stress tensor in its most basic form. Thank you.
2
votes
0answers
24 views

Natural bilinear map $B\colon Alt^p(E^*)\times Alt^p(E)\rightarrow\mathbb R$

$Alt^P(E^*):=\{ u\colon \overbrace{E^*\times\cdots\times E^*}^{p- times}\rightarrow \mathbb R\ \ , u \text{ is alternating multilinear map}\}$ $Alt^P(E):=\{ \alpha\colon ...
2
votes
1answer
34 views

Defining a partial derivative with respect to an antisymmetric tensor/matrix

I'm looking at some nonlinear electrodynamics, and have been following a textbook which contains a primer on some of the stuff I'm interested in following up. However, I seem to have fallen at the ...
0
votes
1answer
43 views

Components of a vector product as an antisymmetrical rank 2 tensor

So I'm reading Landau and Lifshitz' Theory of Elasticity (https://archive.org/details/TheoryOfElasticity) and they have done, among (many) other things, something I simply don't understand. On page ...
1
vote
0answers
35 views

The space of alternating multilinear maps and existence of a bilinear map [duplicate]

Before, I ask similar this. But here I change question settings since it was incomplete. I hope receive good ideas. Let $E$ be a finite dimensional vector space over field $\mathbb R$ with $E^*$ as ...
4
votes
2answers
57 views

Differential at a point and differential (Differential Geometry)

Given $f\in C^\infty(U)$, $U$ open set of $\mathbb{R}^n$, we define the differential of $f$ at $p$ $$ (df)_p:T_p\mathbb{R}^n\to\mathbb{R}\\ (df)_p(v):=v(f) $$ and the differential of $f$ $$ df:U\to ...
0
votes
1answer
29 views

Some calculations with skew forms and wedge product

I have some problems with the language of multilinear forms. I have to prove that if $dim(V)\le 3$, then every $\omega\in\Lambda^q(V^\ast)$ is such that $\omega\wedge\omega=0$. I consider the case ...
1
vote
0answers
27 views

$G$-structure defined by a tensor

Let $M$ be an $n$ dimensional manifold with its bundle of linear frames $\pi:L(M)\to M$. Suppose $T_0$ is a tensor on $\mathbb R^n$ and $u\in L(M)$. We may view $u$ as a linear map $u:\mathbb R^n\to ...
1
vote
1answer
29 views

How can point group symmetry operations be used to reduce the number of independent crystal properties?

How can point group lattice symmetry operations be applied to reduce the full second-rank elasticity tensor (in Voigt notation) from: to, for example, in the cubic case, this: A reference would ...
3
votes
1answer
58 views

Star operator in the simplest form

Let $E$ together with $g$ be a inner product space(over field $\mathbb R$) , $\text{dim}E=n<\infty$ and $\{e_1,\cdots,e_n\}$ is orthonormal basis of $E$ that $\{e^1,\cdots,e^n\}$ is its dual ...
5
votes
2answers
49 views

Inner Product on $\Lambda^n(E)$

Let $E$ together with $g$ be a inner product space(over field $\mathbb R$) , $\text{dim}E=n<\infty$ and $\{e_1,\cdots,e_n\}$ is orthonormal basis of $E$ that $\{e^1,\cdots,e^n\}$ is its dual basis. ...
1
vote
0answers
23 views

Norm of $n \times n \times n$ Tensor

Given a real positive semi definite matrix, $A\in R^{n\times n}$ and a real matrix $F \in R^{b\times n }$, we have the following inequality: $\|FAF^T\| \le \|FF^T\| \|A\|$. However, I am wondering ...
3
votes
0answers
59 views

Understanding tensors

Locally in a chart, a tensor field looks like $$T= T^{i_1,...i_n}_{j_1,...,j_m} dx^{j_1} \otimes...\otimes dx^{j_m} \otimes \partial_{i_1} \otimes ... \otimes \partial_{i_n},$$ where ...
0
votes
1answer
42 views

Significance of 'faces' in Stress tensor components?

I am trying to understand what the significance is of the face for which a force is acting on when talking about a stress tensor. Say we consider the components $T_{xx}$ and $T_{zx}$ of the stress ...
0
votes
1answer
26 views

Permutation of kronecker products

I would like to be able to compute a re-ordered kronecker product from the result of another kronecker product. For example, consider $$F=A\otimes B\otimes C\otimes D\otimes E$$ from the result F and ...
7
votes
0answers
82 views

Vector valued 2-forms which satisfy Jacobi Identity

Motivated by this MO question we ask the following two questions: 1)What is an example of a compact manifold $M$ which does not admit any smooth (1,2) tensor $\omega$ which restriction to each ...
1
vote
0answers
45 views

Matrix of $f^p:\Lambda^p(E)\rightarrow \Lambda^p(E)$

Let $\Lambda^p (E)$ be the set of $p$-covariant exterior(alternative) tensors on linear space $E$ over field $K$ (dim$E=n$ and $ 0\leq p\leq n $ , $\Lambda^0E:=K$). We define linear map ...
1
vote
1answer
62 views

Is there a physical interpretation of the alternating property?

A map from lists to list-elements is called "alternating" if any list with repeated elements is mapped to zero. This has statistical significance: regressions on collinear data are bad, dependent ...
2
votes
0answers
34 views

Pullback maps and an equallity

Let $\Lambda^p (E)$ be the set of $p$-covariant exterior tensors on linear space $E$ over field $K$ (dim$E=n$ and $ 0\leq p\leq n $ , $\Lambda^0E:=K$). We define linear map ...
2
votes
1answer
39 views

How do I extrac the anisotropic part of a tensor?

Given the elements $\chi_{ij}$ of a tensor in cartesian coordinates, with \begin{pmatrix} \chi_\bot& 0 &0 \\ 0 & \chi_\| &0\\ 0&0 & \chi_\| \end{pmatrix}, where the ...
1
vote
1answer
14 views

Proving a $k$-multilinear symmetric map is invariant iff a condition is satisfied

In Huybrecht's book on complex geometry, he states the following lemma on page 193: Lemma 4.4.2: The $k$-multilinear symmetric map $P$ is invariant if and only if for all $B,B_1,\ldots, B_k \in ...
0
votes
0answers
35 views

Demonstrate that the matrix $\pmatrix{ x_2^2 && -x_1x_2 \\ -x_1x_2 && x_1^2}$ represents a $2^{\text{nd}}$ order tensor.

Demonstrate that matrix $T$ represents a $2^{\text{nd}}$ order tensor. $T = \pmatrix{ x_2^2 && -x_1x_2 \\ -x_1x_2 && x_1^2}.$ To show that $T_{ij}' = L_{ik}L_{jn}T_{kn}$, I would ...
2
votes
1answer
61 views

Kernel of the Symmetrizing Map $Sym:\bigotimes^k V\to \bigotimes^k V$

$\DeclareMathOperator{\sym}{Sym}$ Let $V$ be a finite dimensional vector space over a field of characterisitc $0$ and $\sym:\bigotimes^k V\to \bigotimes^k V$ be the map given by $$ ...
0
votes
0answers
24 views

Definition of Tensors Over Complex Numbers

My question is two part. First, how does the definition of tensors and tensor spaces change when the vectors that the tensors act upon are elements of a complex vector space as apposed to when they ...
1
vote
1answer
29 views

Evaluation of polynomials at tensor products

Let $S,T$ be $R$-algebras, $f \in S[X]$ a polynomial. In my notes it says you can easily lift $f$ to a polynomial $f'$ in $(S \otimes T)[X]$. But I have no idea what $f'(s \otimes t)$ is. My ...
1
vote
1answer
59 views

Electromagnetic tensor - I need help with a tensor calculation

First of all: this is not about the physics behind it. It's about the tensor calculation I've written down below. I know this kind of calculation is exhausting but I would be thankful if someone could ...
1
vote
2answers
31 views

How to visualize a second-order tensor in a rectangular coordinate system?

I can visualize a first-order tensor as a segment (a vector), but I'm not sure how to visualize a second-order tensor. The book that I'm trying to study is "Vector and tensor analysis with ...
2
votes
0answers
14 views

Curvature of $K$-invariant connection (principal bundles)

Here is a proposition from Kobayashi & Nomizu's Foundations of Differential Geometry. I don't understand how they obtain the final line of the proof. They write: \begin{align} ...
3
votes
1answer
49 views

Are affine transformation matrices tensors?

Since affine transforms involve a matrix, if the transform matrix is a tensor, it would be of rank two. But, the real question is whether or not a change of basis, or transformation of the underlying ...
1
vote
0answers
33 views

Second contracted Bianchi identity

Considering the Riemann tensor and the onece contracted second Bianchi identity $g^{ls}\nabla_sR_{ijkl}=-\nabla_iR_{jk}+\nabla_jR_{ik}$ why should it hold true that $g^{ls}\nabla_sR_{ijkl}=0$? In ...
2
votes
1answer
51 views

Integral of the exponential of a homogeneous quartic - reference request

For a calculation I am doing, I have to calculate an integral of the form $$ I = \int_{\mathbf{R}^n} \exp[-Q(\mathbf{x})] d^n\mathbf{x} \text,$$ where $Q(\mathbf{x})$ is a homogenous, degree-4 ...
0
votes
2answers
28 views

Tensor manipulation

I am very new at manipulating tensors and I have the following equation: $$A_{\mu \nu\tau} b^\mu c^\nu = g_{\tau \rho} d^\rho$$ where $\tau$ is the independent index and $g_{\tau \rho}$ the metric ...
2
votes
0answers
22 views

Hodge star of second-rank antisymmetric tensor

Say we have a tensor $F$ which just for familiarity's sake, we take to be a second rank antisymmetric tensor. I understand that given the Hodge star operator defined as ...
3
votes
1answer
77 views

What is the relation between $C^\infty$-linear and tensorial?

I've been self studying Riemannian Geometry through Spivak's and Lee's books, and fairly often I've seen an argument that goes somewhat like this: We have an operator that acts on vectors in the ...
2
votes
2answers
94 views

How do the components of a cross product transform?

Let $x^{j}$ and $y^{k}$ be the components of two vectors $x,y\in \mathbb{R}^{3}$. According to the way the compontents of $x$ and $y$ transform when we change the basis, we know they are ...
2
votes
1answer
37 views

Riemannian tensor and Levi Civita connection

For a riemannian metric $g$ consider the following tensor $T_{rstu}=k(x)g_{rt}g_{su}-k(x)g_{st}g_{ru}$. Which condition has to satisfy $k$ if we want the tensor $T$ to be the Riemann tensor of a ...
1
vote
0answers
53 views

Covariant derivative and box operator commutator

I know that the commutator of two covariant derivatives is giving some Riemann tensors as follow: ...
1
vote
0answers
25 views

Do tensor norms exist?

Does there exist norms for tensors, as an extension for the ordinary matrix norm? For example, if there is a derivative of a matrix [A] with respect to a vector {x}, does the norm of this derivative ...
0
votes
0answers
32 views

Trace of a bilinear Form

What's the definition of trace of a $(m,n)$-Tensor $T:\underbrace{V\times \cdots \times V}_{k- \text{times}}\times\underbrace{V^*\times\cdots V^*}_{l-\text{times}}\rightarrow \mathbb{R}$ that ...
2
votes
0answers
38 views

Tensor transpose notation

I have a rank 3 tensor $\mathbf{Q}$. What notation should I use to denote the transposition of two of the dimensions? For instance, if I want to transpose the first and second dimensions, one way I ...
2
votes
1answer
34 views

Indices at the left of a tensor in mathematical physics/differential geometry?

I am a mathematician and I am reading a paper in mathematical physics and I found the following notation: Let $Y$ be a two–form on $M$ such that $$\nabla({}_iY_j)_k = 0.$$ Here, $\nabla$ is ...
1
vote
1answer
59 views

Cross product between a vector and a 2nd order tensor

I have been searching for quite a long time, and haven't been able to find any good reference about the cross product between a vector and a tensor: $$ \vec{a} \times \underline{T}= ...