3
votes
1answer
50 views

Double dot product in Cylindrical Polar coordinates - Strain energy

I'm working with a problem in linear elasticity, and I have to calculate the strain energy function as follows: $$ 2W = σ_{ij}ε_{ij} $$ Where σ and ε are symmetric rank 2 tensors. For cartesian ...
0
votes
0answers
33 views

Let $K$ a field, $\operatorname{char}(K)=0$. Let $V$ a vector space over $K$, $\dim(V) \geq 1$, and be $f$ a $n$-tensor. Prove that $f \wedge f =0$

Let $K$ a field, with $\operatorname{char}(K)=0$. Let $V$ a vector space over $K$, $\dim(V) \geq 1$, and be $f$ a $n$-tensor ($f \in {\mathcal T}_n(V):=\Lambda^{n}(V)$), i.e., $f$ is an multilinear ...
2
votes
1answer
72 views

Introductory questions about tensors

I am trying to understand the concept of tensors. I seem to understand that they are generalization of vectors: They are subject to similar basis transformations with vectors but I am somewhat ...
1
vote
0answers
33 views

Time Evolution of Deformation Gradient Tensor in Lagrangian Frame

I found the following proof in a paper: $\frac{D\mathbf{F}}{Dt} = \frac{D\frac{\delta\mathbf{x}}{\delta\mathbf{X}}}{Dt} = \frac{\delta\frac{D\mathbf{x}}{Dt}}{\delta\mathbf{X}}=\frac{\delta ...
1
vote
1answer
47 views

$(V^*)^{\otimes n} \cong (V^{\otimes n})^*$

We assume that $V$ is finite dimensional. Make $\theta: (V^*)^n\to (V^{\otimes n})^*$ by $$ \theta(\alpha_1,\cdots,\alpha_n)(v_1 \otimes \cdots \otimes v_n ) := \prod_{i=1}^n \alpha_i(v_i). $$ Then, ...
1
vote
0answers
20 views

Tensor Algebra and Isomorphism

Consider the standard tensor algebra over a vector space: $$T(V)=\bigoplus_{n\in N}V^{\otimes n}$$, I am trying to define a linear map from $T(V)\times T(V)\rightarrow T(V)$. Up to this time, I have a ...
0
votes
0answers
23 views

tensor derivative of a symmetric second order tensor mapped onto itself, with respect to itself

In a continuum mechanical context, let $\mathbf{A}$ be a symmetric second order tensor. How can I calculate the derivative $$\cfrac{\partial(\mathbf{AA})}{\partial\mathbf{A}}$$ using indicial ...
3
votes
1answer
103 views

Tensors and General Relativity

I've recently begun self studying general relativity, using mostly the material found in Robert Wald's "General Relativity", and almost right out of the gate one encounters the notion of a tensor. In ...
5
votes
1answer
63 views

The derivative of the determinant of a Kronecker product

For an invertible matrix $A$, we have the identity \begin{align} \dfrac{\partial \det A}{\partial A} = \det A (A^{-1})^T \end{align} where the $T$ denotes the transpose operation. How does this ...
0
votes
2answers
45 views

Tensor Product over a ring

Given Two Fields $F,K$, and two vector spaces $V,W$ over $F$, what does tensor product $$V\otimes_{K} W$$ mean? I am not certain whether this is defined in general. I came across it in cases wheh ...
0
votes
1answer
47 views

Another matrix of a given operator $A \otimes A$

Let $V$ be a $4$-dimensional vector space with an ordered basis $e_{1}$, $e_{2}$, $e_{3}$, $e_{4}$, and $A: V → V$ be a linear operator given by its matrix relative to the ordered basis $(e_{1}, ...
1
vote
1answer
34 views

Matrix of a given operator $A \otimes A$

Let $V$ be a 3-dimensional vector space with an ordered basis $e_{1}$, $e_{2}$, $e_{3}$, and $A: V → V$ be a linear operator given by its matrix relative to the ordered basis $e_{1}$, $e_{2}$, $e_{3}$ ...
1
vote
0answers
31 views

How is the multiplication between a multidimensional tensor with a matrix defined?

I am thinking this calculation in the following way but I am wondering if it is correct. Can anybody explain to me please? For example, I have a 3-way tensor $T^{u×i×t}$. How do I multiply this ...
1
vote
1answer
228 views

Trace of tensor product vs Tensor contraction

I have come across various sources that talk about traces of tensors. How does that work? In particular, there seem to be such an equality: $$ \text{Tr}(T_1\otimes ...
0
votes
1answer
24 views

Is tensor product commutative on orthonormal basis?

In general the tensor product $\varphi\otimes\psi$ is not commutative, but I was thinking that if I have tensor product on two orthonormal bases of Hilbert spaces are they commutative i.e is ...
2
votes
2answers
46 views

Cartesian & Tensor Product

What is the difference between a cartesian product and tensor product of two vector spaces $V_1$ and $V_2$ defined over same field $F$ ?
2
votes
1answer
99 views

using the kronecker product and vec operators to write the following least squares problem in standard matrix form

I have a least squares problem with the following form: $$ \min_\mathbf{X} ~ \left\| \sum_{i=1}^n \mathbf{u}_i^\top \mathbf{X} \mathbf{v}_i - b_i \right\|^2 $$ where $\{\mathbf{u}_i\}_{i=1}^n$ and ...
-2
votes
1answer
55 views

Is it true to say that every tensor is an element of a monoid?

If we consider that: by definition, a tensor is an element of the tensor product of two algebraic structures the most abstract algrebraic structure on which the tensor product is defined are ...
0
votes
2answers
52 views

Is every tensor an element of a vector space?

As, the tensor product of two vector spaces $V$ and $W$ over a field $K$ is another vector space over $K$, is it true to say that every tensor is an element of a vector space ? (if we do not consider ...
0
votes
0answers
19 views

Alternative operator is a homomorphism?

Let $V$ be a real vector space of dimension $n$, for a (real valued) tensor $f$ of order $r$, define the alternative operator $A$ by $$(Af)(v_1,\cdots, v_r)=\frac{1}{r!}\sum_{\sigma\in ...
2
votes
1answer
39 views

Calculating a certain tensor

Consider the $\mathbb{Z}$ module $\mathbb{Z}/n\mathbb{Z}$. What is $\mathbb{Z}/n\mathbb{Z} \otimes_{\mathbb Z} \mathbb{Z}/n\mathbb{Z}$?
14
votes
5answers
336 views

What, Exactly, Is a Tensor?

I've repeatedly read things that reference tensors, and despite reading the wiki page and other answers here on stackexchange I still don't know what a tensor is. I'm fine with hearing things in the ...
1
vote
0answers
44 views

Question about tensor product of homomorphisms

I've come to think about this problem when reading a proof in Commutative Algebra by N. Bourbaki. Say, let $R$ be a commutative ring, given 3 $R-$modules $A$, $B$, $C$, and the $R$-homomorphism $f:B ...
2
votes
1answer
54 views

Tensor Product definition (help with certain step)

I'm going over some notes I took from the blackboard, and reached a slight hitch. I thought that maybe someone could help. Let $E,F$ be vector spaces over a field $\mathbb K$. A tensor product of $E$ ...
0
votes
1answer
76 views

Tensor product and valence of a tensor?

Given a vector space $V$, its dual vector space $V^{*}$ and a tensor $\mathbf{T}$: $\mathbf{T} \in \underbrace{V \otimes\dots\otimes V}_{n\text{ copies}}\otimes \underbrace{V^{*}\otimes\dots\otimes ...
1
vote
1answer
149 views

Trace of tensor product

I am currently struggeling with the following problem: Imagine that you have 4 vectors $v_1,v_2,v_3,v_4$ in $\mathbb{R}^2$ and then you want to calculate $\hbox{trace}(v_4 \otimes v_3 \cdot ...
1
vote
1answer
41 views

When is a simple tensor equivalent to natural multiplication?

Let $R$ be a commutative integral domain; let $M, N$ be $R$-modules. Let $x_1 \in M$ and let $x_2 \in N$. When does the following equivalence hold? $$x_1 \otimes x_2 = x_1x_2$$ The textbook I'm ...
0
votes
2answers
76 views

Matrix/Tensor Operations

Suppose $A$ is an $m \times n$ matrix, and $B$ is an $n \times k$ matrix. Let $C$ be a tensor, where $$C(i,j,k) = A(i,j) + B(j,k)$$ What is a suitable (tensor) algebraic operation that summarizes ...
1
vote
1answer
63 views

Trouble understanding Tensor product in context of Torsion Tensor

I know that there are quite a few threads dealing with this question already. I have pored through them for quite some time and they have been informative. However there are still some clarifications ...
0
votes
0answers
42 views

Show that the tensor product is followed by a rotation and a dilation?

I am thinking mjqxxxx's sentence (a⊗b) [is] a projection followed by a rotation and dilation. We know that \begin{equation} (\textbf{a} \otimes \textbf{b}) \textbf{c} = (\textbf{b} \cdot ...
0
votes
1answer
58 views

Show that the spherical tensor (1)_ij is the Kronecker's delta?

I am thinking the problem that show that \begin{equation} (\textbf{1})_{ij} = \delta_{ij} \end{equation} My attempt The unit tensor is a spherical tensor \begin{equation} \textbf{1} = ...
2
votes
1answer
117 views

Where is the tensor product of two unit vectors projection onto?

I know that $\bar{e} \otimes \bar{e}$ is a projection onto $\bar{e}$. Then, I start to think where is then $\bar{e}_{i} \otimes \bar{e}_{j}$ projection onto. Where is the expression $\bar{e}_{i} ...
0
votes
1answer
41 views

Show vector mapped onto plane perpendicular to unit vector?

I am reading Gurtin book about Continuum Mechanics and Tensors, and I do not see directly that the vector $\mathbf u$ is mapped to the plane perpendicular to $\mathbf e$. Only looking on the formula, ...
1
vote
1answer
120 views

Why is this true:$ \nabla \cdot (\vec V \otimes \vec V)=(\vec V\cdot \nabla ) \vec V +\vec V(\nabla\cdot \vec V) \;\;? $

Can someone help me why the following is true: $$ \nabla \cdot (\vec V \otimes \vec V)=(\vec V\cdot \nabla ) \vec V +\vec V(\nabla\cdot \vec V) \;\;? $$ I've thought of the following relation to be ...
1
vote
1answer
40 views

problem with permutation symbol

Given $\varepsilon_{ijk}T_{ij} = 0$. Prove that $T_{ij} = T_{ji}$ I can prove it by expanding summation. It is very cumbersome. May be there is more compact solution?
0
votes
1answer
98 views

Component-free formula for the determinant of a tensor

Consider a unit vector $\mathbf{a}\in\mathbb{R}^3$ and the associated second-order tensor $\mathbb{A}=\mathbf{a}\otimes\mathbf{a}$. Is there a component-free formula for the determinant of this ...
1
vote
2answers
348 views

Identity tensor as a tensor product of two vectors

Any second order tensor in a given basis can be expressed as a matrix. Also, as any second order tensor can be expressed a tensor product of two first order tensors (or vectors), I would like to find ...
3
votes
1answer
61 views

Help with notation in second order tensor.

I have seen recently this notation: $$F=F_{ij}\,e_i\otimes e_j $$ Where $F_{ij}=\frac{\partial x_i}{\partial X_j}$ is the tensor matrix: $$ F=\left[ {\begin{array}{cc} ...
0
votes
1answer
108 views

Linear System where Coefficient Matrix is a Kronecker Product

I have a system of linear equations where the coefficient matrix and right hand side is given by a Kronecker product: $(A_1 \otimes A_2) u = f_1 \otimes f_2$ My question: Is the solution simply ...
3
votes
1answer
75 views

Non commutative tensor products / simple example?

In O'Neil's "Semi-Riemannian Geometry" it is stated that if $A$ is a $(r,0)$-tensor field and $B$ is $(0,s)$-tensor field then it is ""from the definition'' that $A \otimes B = B \otimes A$. I still ...
4
votes
1answer
92 views

O'Neil's problem 2.10 on Tensor Derivations. Literature on the subject.

I am having real trouble trying to understand a problem in O'Neil's "Semi-Riemannian Geometry" and I can't find much literature on the subject. I will expose the problem and I will be grateful to ...
1
vote
1answer
974 views

Divergence of stress tensor in momentum transfer equation

Let suppose that we work in a 2D cartesian coordinates. what will be x and y components of $\nabla.\left[-p I+\mu \left(\nabla \text{u}+(\nabla \text{u})^T\right)-\frac{2}{3} \mu (\nabla.\text{u}) I ...
1
vote
0answers
364 views

What are these physicists talking about? Dyadic green function?

I am interested in a mathematical explanation(in the sense that you say for example: is it a mapping from A to B) what a dyadic green function and the unit dyad actually is? I am reading this as ...
3
votes
1answer
141 views

Lie Derivative for Wedge Product of Vector Fields

I am having trouble here. The context is: Let $X$, $Y$ and $S$ be vector fields ina a manifold (we can assume it's $\mathbb{C}^2$ though I'm pretty sure this should work in any manifold), and we can ...
0
votes
1answer
242 views

Double dot product of two tensors [duplicate]

I have a problem that makes me very confused... I have two tensors that must be multiply. A is second order tensor and B is fourth order tensor. I know when multiplying two tensor with double dot ...
0
votes
1answer
56 views

Suficient condition for tensor product of vector spaces..

Can anyone help me showing the following: Let $E$, $F$, $G$ and $H$ vector spaces and $\varphi:E\times F\rightarrow G$ a bilinear map. If for every $\psi:E\times F\rightarrow H$ bilinear there is an ...
4
votes
2answers
147 views

For covariant tensors, why is it $\bigwedge^k(V)$, not $\bigwedge^k(V^*)$?

In learning the very basics of differential geometry, I have seen the exterior product defined a couple of ways: First, I have seen it as the image of the covariant tensors (which I believe are ...
3
votes
1answer
149 views

Alternating tensors vs $p$-vectors

Is there a reason to differentiate between alternating tensors and $p$-vectors? More precisely, is the exterior algebra always isomorphic to the subalgebra of alternating tensors? Thanks.
4
votes
1answer
264 views

Tensors as mutlilinear maps

I am aware that many books on differential geometry define tensors as multilinear maps. Namely $$ V\otimes W := L_2(V^*\times W^*,\Bbb F) $$ I am also aware that this space is isomorphic to the tensor ...
0
votes
1answer
44 views

The operator in Tensor algebra.

Let $V$ be a vector space over a field $K$. We define the $k^{th}$ tensor power of $V$: $$T^kV = V \otimes V \otimes ... \otimes V$$ We contruct $T(V)$ as the direct sum of $T^kV$ for $k=0,1,2,...$ ...