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
78 views

Tensors and rotation matrix

$a_{ij}$ is a rotation matrix that satisfies $\hat{e}'_i=a_{ij}\hat{e}_j$. Show that $\epsilon_{lmn}a_{mi}a_{nj}=\epsilon_{ijk}a_{lk}.$ Using the result from above, how can I show that ...
3
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1answer
54 views

Integrating tensors on manifolds

When/how can you integrate tensors on manifolds and what does it mean? I imagine that line integrals of tensors make sense when you have a connection, since you can uniquely parallel transport all ...
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0answers
17 views

Tensor decomposition: How to determine the number of components (rank) of a CP decomposition?

I need to compute a best $rank-R$ CP decomposition of a tensor built from real world 3-dimensional data, of sizes approximately $100×200×300$. The best rank-R is determined by a given metric, which ...
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0answers
17 views

What is the relation between Kruskal tensor and CP decomposition?

In Matlab Tensor Toolbox there is a tensor type called "Kruskal tensors", I found its form is similar to the CP decomposition. Wikipedia mentioned "As such, many of the methods have been ...
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1answer
78 views

Cauchy Momentum Equation - Stress Tensor

I've been trying to understand the derivation for the Cauchy Momentum Equation for so long now, and there is one part that every derivation glides over very quickly with practically no explanation ...
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2answers
85 views

How does the representation of co-vectors change if we change the basis of a vector space $V$?

I'm trying to understand how vectors, differential forms and multi-linear maps in general transform under change of coordinates. So I start with the simplest case of vectors. Here's my own attempt, ...
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1answer
46 views

Manipulation of Tensors

I have an expression: $\eta^{\mu \nu} F_{\alpha \beta, \nu} F^{\alpha \beta}$ Where $\eta^{\mu \nu}$ is the Minkowski metric, F is an antisymmetric tensor, and the comma on the middle tensor denotes ...
2
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1answer
25 views

Evaluating the Lie derivative of the metric

From the Wikipedia definition of the Lie derivative of a tensor along a vector field, we have, $$\mathcal{L}_X g_{\mu\nu} = X^\lambda \nabla_\lambda g_{\mu\nu} + (\nabla_\mu X^\lambda)g_{\lambda \nu} ...
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1answer
14 views

Gradient of a function with base vectors

\begin{align} \nabla\left(x_1x_3\hat{e}_1+x_2x_3\hat{e}_2\right)&=x_3\left(\hat{e}_1\otimes\hat{e}_1\right)+x_3\left(\hat{e}_2\otimes\hat{e}_2\right)\\&+\mathbf{x_1\left(\hat ...
1
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1answer
32 views

Show $A_{ab}$ are the components of a tensor.

The question asks: "If $v_a$ are the components of a vector, show that in an arbitrary coordinate system that $A_{ab}$ are components of a rank-2 tensor, where:" $$A_{ab}= ...
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1answer
24 views

Progresive diagonalisation of a symmetric matrix

Given a symmetric real matrix $A$ there is an orthogonal transformation that brings $A$ to a block diagonal form, where $a \in \mathbb{R}$: $$ RAR^{T} = \left[ {\begin{array}{cc} a & 0 \\ 0 ...
1
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1answer
32 views

using a symmetric matrix to simplify expression involving antisymmetric matrices

I am trying to simplify the following expression --$$(\vec{A} \times \vec{B})\cdot\underline{\underline{S}}\cdot(\vec{C} \times \vec{D}) = (A_jB_k\epsilon_{jki})S_{im}(\epsilon_{mnp}C_nD_p)$$ $S$ is a ...
1
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2answers
23 views

Is this equation consistent with index notation?

Is the equation $$a_i+b_j=c_k$$ consistent with index notation? I think that the answer is yes, since the free index within each term is arbitrary, so it doesn't matter which one we pick. Is my ...
0
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1answer
22 views

Proving that $(\vec a \times \vec b) \times (\vec a \times \vec c)=\vec a(\vec a \cdot \vec b \times \vec c)$ using index notation.

I'm trying to prove that $$(\vec a \times \vec b) \times (\vec a \times \vec c)=\vec a(\vec a \cdot \vec b \times \vec c)$$ using index notation (i.e. Einstein sumnmation notation). Here's what I've ...
2
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2answers
46 views

How can Ishow that $(\vec a \times \vec b) \cdot (\vec a \times \vec b)=|\vec a|^2|\vec b|^2-(\vec a \cdot \vec b)^2$ using index notation?

I'm trying to use index notation (i.e. Einstein summation notation) in order to show that $(\vec a \times \vec b) \cdot (\vec a \times \vec b)=|\vec a|^2|\vec b|^2-(\vec a \cdot \vec b)^2$. Here's ...
0
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1answer
61 views

How to identifiy $V \wedge V$ with the space of all alternating bilinear forms

Let $\{ e_i \}$ be a basis for $V$, then the space of tensors $V \otimes V$ could be identified with the space of all formal sums $\sum_{ij} \alpha_{ij} (e_i, e_j)$ (I know a base independent approach ...
2
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0answers
49 views

Unnecessary Elements in the Tensor Product?

For vector spaces $U, V$ there exits a unique (up to isomorphism) vector space, denoted by $U \otimes V$, and a bilinear map $\eta : U \times V \to U \otimes V$ such that for every bilinear map $\xi : ...
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0answers
16 views

Questions about a special tensor transformation

Suppose tensor $U_{i\alpha\beta}$ with dimension $M*N*N$ satisfy following condition: $$U_{i\beta\alpha}=W^1_{\alpha\alpha'}W^2_{\beta\beta'}U_{i\alpha'\beta'}$$ where $W^1$ and $W^2$ are $N*N$ ...
1
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1answer
81 views

Partial derivative with respect to metric tensor $\frac{\partial}{\partial{g_{kn}}}(g_{pj}g_{ql})$

$$-\frac{1}{4\mu_0}F^{pq}F^{jl} \frac{\partial}{\partial{g_{kn}}}(g_{pj}g_{ql})=+\frac{1}{4\mu_0} F^{pq} F^{lj} 2 \delta^k_p \delta^n_j g_{ql}$$ I need to know how to derive ...
3
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1answer
88 views

Rank of tensors in terms of ranks of associated linear maps

Let $V$ be a vector space over a field $k$, let $w \in \otimes^l V$ be a tensors. We call $w$ a simple tensor if it can be written as $$ w=w_1 \otimes w_2 \otimes \ldots \otimes w_l, $$ where $w_i \in ...
1
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1answer
21 views

Nested square brackets in tensor indices

I know that using square brackets on tensor indicies denote the anti-symmetric part $$ T_{[ab]} = \frac{1}{2} \left( T_{ab} - T_{ba} \right)$$ I now have to prove that $$ T_{a [[bc]d]} = T_{a ...
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0answers
27 views

How does one express $\theta_{i_1\dots i_k}$ as a linear sum of $\theta_{i_{1'}\dots i_{k'}}$?

Let $\{dx^{i_1}\wedge\cdots\wedge dx^{i_k}\mid 1\leq i_1<\cdots<i_k\leq n\}$ and $\{dx^{i_1'}\wedge\cdots\wedge dx^{i_k'}\mid 1\leq i_{1'k}<\cdots<i_{k'}\leq n\}$ be two basis for the ...
0
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0answers
27 views

How to show a coordinate system is orthogonal by using metric tensor

For example, the paraboloidal coordinate \begin{align} x &= uv\cos\theta\\ y &= uv\sin\theta\\ z &= \frac{1}{2}(u^2-v^2) \end{align} the covariant metric tensor I calculated is $$ ...
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0answers
16 views

k-space tensor integral in statistical mechanics [duplicate]

k is the modulus of the vector k. Please help me to integrate the above tensor expression in the infinite domain of the vector k. I have tried to let u in the direction of kz and then transform the ...
1
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1answer
18 views

If $\mu(e_1,…e_n)=1$, then how to show that $\mu=f^1\wedge f^2…\wedge f^n$?

Let V be a n dimensional vector space, $\mu$ be an antisymmetric n tensor.(i.e, a real valued multilinear functional with n inputs) If there exists a basis for $V$, say, {$e_1,e_2,...,e_n$}, such that ...
1
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1answer
36 views

$k$-space tensor integral in statistical physics

$$Q=\int_{\text{all space}} \frac{\hbar \nu_g \mathbf{k}\mathbf{k}}{\exp[(\hbar \nu_g |\mathbf{k}|-\mathbf{k}\cdot\mathbf{u})/k_B T]-1}d\mathbf{k} $$ Please help me to integrate the above tensor ...
3
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0answers
60 views

Computing volume element in spherical coordinates

Suppose $y = (r, \theta^1, \theta^2)$ are spherical coordinates in $(\mathbb{R}^3,g)$. What is the $d\text{vol}$ in these coordinates? I solved it but I don't know if it's right. My solution: We ...
0
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1answer
55 views

How can we show that $f_1f_2…f_k=0$ iff $\exists j$ st $f_j=0$?

Assume $V$ is an n dimensional vector space. $f_1,...f_k\in V^*,v_1,...,v_k\in V$ Define the symmetric k tensor $f_1f_2...f_k(v_1,..,v_k)=\Sigma_{\delta\in S_k}f_{\delta 1}(v_1)...f_{\delta_k}(v_k)$ ...
0
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1answer
26 views

Then, how can we show that $\forall i,j\in \mathbb Z, s.t.:1\leq i<j\leq n$, $e_i\wedge e_j $ is a basis vector for $\wedge^2(V) $?

Let $V$ be a n dimensional vector space. Suppose $x,y\in V, f,g\in V^*$. Define $f\wedge g(x,y) = det \left( {\begin{array}{cc} fx & fy \\ gx & gy \\ \end{array} } \right)$ Then, how can we ...
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2answers
20 views

Then, if $k>n$, can it be shown that $\psi$ is the trivial one which sends every element to $0 $?

Let $\psi:V\times \cdots \times V\to \mathbb R$ be an antisymetric $k$ tensor on $V$, which is $n$ dimensional. Then, if $k>n$, can it be shown that $\psi$ is the trivial one which sends every ...
0
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1answer
27 views

Confused by indicial notation term $u_{j,ij}$

I am confused by the indicial term $u_{j,ij}$ and cannot find it treated in discussions of tensor/indicial/Einstein notation even though it is an important term in linear elasticity. Working off ...
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0answers
11 views

Finding Polar Components by Raising/Lowering Indices

This is (I think) a simple question—I'm just making sure everything's correct: I'm given a vector field, $v^a$, which has constant Cartesian components $v^x = 0$ and $v^y = 1$. I'd like to find its ...
4
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1answer
74 views

Is there an easy way to reason about expressions involving lots of indices?

I have been reading some Riemannian geometry recently. So far, I think I am understanding the concepts well enough. However, I am finding it difficult to translate some of the notation into meaning. ...
0
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1answer
44 views

Confusion when applying Tensor transformation law to $\partial_{[a,v_b]}$

What I'm trying to show is that, if $v_a$ is a covector field, $\partial_{[a, v_b]} = \frac{1}{2}(\partial_a v_b - \partial_b v_a)$ transforms like a type $(0,2)$ tensor. First of all, a type ...
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2answers
77 views

Question about cross product and tensor notation

I am a bit rusty on tensor algebra and calculus and may use some wrong terminology, but I know that the cross-product can be expressed in tensor notation with the aid of the ...
0
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1answer
42 views

Do Tensors have a determinant property?

We know that only square $n \times n$ matrices have a determinant property! And it can be defined just like this: $$A=\begin{array} & & & \\ ...
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0answers
35 views

Curl of Deviatoric Stress Tensor In Index Notation

I'm taking the curl of the deviatoric stress tensor in index notation, and I've ran across something that I can't seem to be able to simplify. The issue is shown in the following portion of the curl ...
2
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0answers
48 views

How to prove a symmetric tensor keeps symmetric under rotation? [closed]

For example, $T_{ij}=T_{ji}$, prove $R_{il}R_{jm}T_{lm}$ is also symmetric. I know I need to prove $R_{il}R_{jm}T_{lm}=R_{jl}R_{im}T_{lm}$, and the fact that $R$ is antisymmetric might be helpful, ...
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1answer
11 views

When to use transformation of variable and when transformation of differentials

I was reading the book: Mathematical Methods in the Physical Sciences by M. Boas and I came across this statement; I wasn't quite sure why this was the case. Is it because in the curvilinear ...
3
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2answers
176 views

What is the scalar product of tensors?

Given there a vector space $V$ with a scalar product $g(v_1,v_2)$ on it, what is the scalar product on, say, $V \otimes V^*$ ? According to Jeffrey Lee's "Manifolds and Differential Geometry" (see ...
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0answers
86 views

Is the identification between symmetric tensors and homogeneous polynomials useful?

The general question: Given an $n$-dimensional vector space $V$ over a field $k$, there exists an identification $$\mathrm{Sym}^d(V) \sim k[x_1, \dots, x_n]_d$$ between the space of symmetric order ...
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24 views

General Tensor Assistance

Sorry if this is a stupid question, but it might help me grok things if I can connect from something that's intuitive to me. Consider a transformation from Cartesian coordinates to spherical ones: ...
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1answer
22 views

Tensors furnish representations of the group

I'm bad at english, so what exactly does it mean in simple english that Tensors furnish representations of the group?
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27 views

The effect of the Levi-Civita symbol on matrix elements

Suppose the matrix $O$ is orthogonal i.e. satisfies $$\tag{1} O^TO = 1 $$ and is also special $$\tag{2} \det O =1. $$ One can write equation $(2)$ as $$\tag{2'}\varepsilon^{i_1i_2\dots ...
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1answer
90 views

What is contracting a tensor actually doing?

I'm learning about tensors, and have a vague idea regarding what contracting a tensor means—but I'm still not sure of exactly what it's doing. Maybe someone here can put it in more intuitive terms. ...
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41 views

Superscripts and subscripts in tensors - interpretation as a matrix operation?

Consider the equation $$u_k = t_i U_i^k$$ I am told that subscripts represent covectors (row vectors) and superscripts represent column vectors. My intuitive interpretation of the above equation as a ...
5
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1answer
94 views

How can I determine the number of wedge products of $1$-forms needed to express a $k$-form as a sum of such?

This question was motivated by this related one: How "far" a differential form is from an exterior product . Let $\mathbb{V}$ be a vector space of dimension $n$ with underlying field ...
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2answers
81 views

Help with a paper about tensors

I came across something in a paper I am not able to understand jet. Unfortunately the author is kind of short with explanations. Maybe someone here can help me to understand this. $M^d \in ...
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0answers
35 views

Tutorial about tensor

Description: I have not learned the course about tensor. However, when I learning Mathematica software to deal with my problem that came from my specity, there are ...
3
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1answer
56 views

Derivation of the Geodesic Equation

From page 40 of A. Schild and J. L. Synge's "Tensor Calculus", I'm having issues understanding the following mathematical steps ( I feel like it's simple algebra that I'm messing up. We have, $a_{rm}$ ...