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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40 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 ...
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18 views

Trace of a tensor in a differential equation

If $Z$ is a rank-2 tensor, does the following differential equation mean anything to anyone: $\nabla^2Z+\frac{1}{c^2}\frac{\partial^2}{\partial t^2}tr(Z)=0$ The presence of this trace really blurs ...
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2answers
117 views

Annihilator of a Tensor

This is a question I have trouble understanding, hope you can clarify this to me. Problem: Find the annihilator of the tensor $e_1\wedge e_2+e_3\wedge e_4$ in ...
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45 views

Structure Tensor for the Algebra of 2x2 Matrices

Here's a question I need help understanding. Hope you can provide me some insight. Problem: Write the structure tensor for the algebra A of triangular $2\times 2$ matrices with real coefficients. ...
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2answers
71 views

Covariant derivative in abstract index notation

Spose $f,h$ functions, where $\nabla _af = \epsilon _{ab}\nabla ^bh$. Then $\nabla ^af=g^{ac}\epsilon _{cb}\nabla ^bh$. My question is then does $\nabla _a\nabla ^af=\nabla ^c\epsilon _{cb}\nabla ^bh$ ...
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74 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 ...
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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 ...
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1answer
101 views

Is invariance of a multi-linear form required for co/contra variance?

I'm reading the book: The Absolute Differential Calculus by Levi-Civita to get an idea of the history behind the development of tensor calculus. On page 71 he states: An m-fold covariant is an ...
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56 views

Factoring tensors by linear transformation

Let $T_n$ be tensors $\left( V^n \rightarrow \mathbb{R}\right)$ of order $n$. Now let $\sim$ be equivalence on $T_n$ that $A\sim B$ iff there is regular matrix $M$ that $A\circ M = B$. By $A\circ M$ ...
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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 ...
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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} = ...
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1answer
115 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} ...
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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, ...
2
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1answer
63 views

Definition of divergence of a tensor

How do you formally define the divergence of an arbitrary $(p,q)$ tensor? And what does it geometrically signify?
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1answer
133 views

How to translate between differential forms and tensor index notation

The books on Manifold theory & geometry that I studied introduce connection and curvature in the language of differential forms. But Physics books on the other hand like (General Relativity by ...
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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 ...
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1answer
151 views

Tensors = matrices + covariance/contravariance?

I have read several topics on tensors but it is still not clear to me. Tensors are different from matrices because they contain additional information about how do they transform. To fully specify a ...
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1answer
48 views

Non-square tensors?

I learnt tensor algebra for physics and I never saw a non-square (or non-cubic...) tensor. But, from a mathematical point of view, do non-square tensors exist? And if so, are they used in some area in ...
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3answers
111 views

What is the definition of rotation tensor R?

The book A First Course in Continuum Mechanics says the rotation tensor, R, is implicit in F. The matrix presentation of rotation is here. However, I am interested in its tensor representation. How ...
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2answers
103 views

Intuition behind $\vec{e_i} \times \vec{e_j}=\epsilon_{ijk} \vec{e_k}$ (Levi Civita)

Let $\vec{e_i}$ denote a unit vector. Then we can write: $\vec{e_i} \times \vec{e_j}=\epsilon_{ijk} \vec{e_k}$, where $\epsilon_{ijk}$ is the Levi Civita symbol. Can someone intuitively explain me ...
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29 views

When can I recover a (full) tensor-contraction after normalising partial contractions.

I have rank-${1 \brack n}$ tensor $R_{abc....}$ and a rank-${n \brack 1}$ tensor $S^{ABC...}$. Obviously their contraction $$u = R_{abc....}S^{abc...}$$ is a scalar (complex in my case). I can ...
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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?
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120 views

Tensor Projection

I'm currently reading "Vector and Tensor Analysis with Applications" by A.I. Borisenko and I.E. Tarapov, and I'm having trouble following a particular mathematical step in where the author projects ...
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3answers
132 views

What is the mathematical nature of a rotation matrix?

I have a naive question: what is the mathematical nature of a rotation matrix? Is a rotation matrix a tensor ? EDIT: if a rotation matrix is fundamentally a tensor, what is its (n, m) notation?
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22 views

Covariant form of a tensor.

I understand why stress-energy tensor for a comoving observer at rest relative to the fluid is diag$\{\rho, -P,-P,-P\}$ How does this lead to the generalized covariant form, often quoted in ...
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1answer
43 views

Prove: ∇⋅ϕF = ϕ∇⋅F + F⋅∇ϕ

I am asked to prove this identity using tensor notation. However, I am not sure where to even begin the problem.
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1answer
43 views

Example for Higher Order Tensors

Could you please give some examples of higher order tensors? What would be an instance of 2nd, 3rd, 4th or even an higher order tensor in engineering science, especially in mechanics? An example of a ...
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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 ...
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1answer
132 views

A linear connection induces a covariant derivative of tensor fields.

Let $M$ be a smooth manifold. notation: $\mathcal T(M)^{(k,l)}$ is the $C^{\infty}(M)$-module of all tensor fields of type $(k,l)$ on $M$ ($k$ indicates the covariant part). $\mathcal ...
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1answer
129 views

Prove that decomposition of second order tensors into symmetric and skew components is unique.

Title says it all. $A=A_{sym}+A_{skew}$ $A_{sym}= \dfrac{1}{2}(A+A^T)$ $A_{skew}= \dfrac{1}{2}(A-A^T)$ Anybody can help?
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1answer
55 views

What does “transform among themselves” mean?

I'm reading a script on atomic physics, and there's a chapter on irreducible tensors. I can't understand the meaning of "transform among themselves" in this context: An arbitrary rotation of the ...
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108 views

transformation for Christoffel symbol of first kind

For Christoffel symbol of first kind, I have to show that $$\bar \Gamma _{jkm} = \frac{\partial x^p}{\partial \bar x^j}\cdot \frac{\partial x^q}{\partial \bar x^k} \cdot \frac{\partial x^r}{\partial ...
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143 views

How to prove that symmetric traceless “transverse” tensor rank s in 4 dimensions has 2s + 1 independent components?

How to prove that symmetric traceless "transverse" tensor rank $s$ in 4 dimensions has $ 2s + 1$ independent components? Let's have tensor $$ F^{\mu_{1}\dots \mu_{s}}, \quad {F^{\quad ...
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2answers
346 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 ...
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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} ...
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136 views

What is tensor, really?

How can one understands the definition of tensor from the purely formal point of view? To what abstract structure this concept can be generalized?
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50 views

Synge & Schild Exercise 1.2

$x^1 = a \cos u^1 \\ x^2 = a \sin u^1 \cos u^2 \\ x^3 = a \sin u^1 \sin u^2 \cos u^3 \\ \vdots \\ x^{N-1} = a \sin u^1 \sin u^2 \sin u^3 \cdots \sin u^{N-2} \cos u^{N-1} \\ \displaystyle x^N = a ...
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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 ...
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1answer
116 views

Decomposition of the Curvature operator and Matrix representation

I'm trying do this question from Peter Petersen's Book and I can't do some parts. I know that $$R=\frac{scal}{2n(n-1)}g\circ g+\left(Ric-\frac{scal}{n}g\right)\circ g+W$$ Where, $R$ is the ...
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2answers
106 views

Tensors - need materials to study

I want to study about tensors. Can you indicate me some materials, papers, books which I should begin. I tried last year to study but it seems to hard for me. Thanks for help :)
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1answer
65 views

Summing over tensor indices

How can I prove that the product of two rank-2 tensors, one of which is symmetric and one is antisymmetric, must =0 when their indices are summed over?
3
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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 ...
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1answer
64 views

Self-dual and anti-self-dual decomposition

Please take a look at the following: Let $(M,g)$ be a four-dimensional oriented Riemannian manifold. The Hodge star operator $*$ obeys $**=Id$ acting on 2-forms. This allow us to decompose the ...
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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 ...
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1answer
170 views

Prove that the trace of a dyad uv is the dot product of u and v

$$ I'm\quad trying\quad to\quad demonstrate\quad that\quad the\quad trace\quad of\quad a\quad dyad\quad (tensor\quad product)\quad is\\ equal\quad to\quad the\quad dot\quad product\quad of\quad ...
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101 views

Rank-2n tensor algebra eigenvalue equation

Im interested in resources and work done on the eigenvalue equation for rank-2n tensors: $$ M_{ij}A_{j} = \lambda A_{i} \\ $$ $$ M_{ijkl}A_{kl} = \lambda A_{ij} \\ $$ $$ M_{ijklmn}A_{lmn} = \lambda ...
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1answer
80 views

Problems with tensor notation

I've got a question for the mathematically more educated for I am a humble engineer having a hard time: $\kappa = \left( \delta_{ij}-n_in_j\right)\displaystyle\frac{\partial u_i}{\partial xj} - ...
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2answers
84 views

How to reduce an order 3 tensor to an order 2 tensor?

Are there any techniques to reduce an order 3 tensor to an order 2 tensor? For example, I have an $m \times m \times p$ tensor and I want to reduce it to a $m \times m \times 1$ tensor. Thanks
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1answer
98 views

Basic property of a tensor

In Jost's Riemannian Geometry and Geometric Analysis (6th ed.) on page 142 there is the following remark concerning the torsion tensor. Remark. It is not difficult to verify that [the torsion ...
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114 views

Tensors: intrinsic versus index notation

I consider the following equality: $$ \bar{\bar{T}}=T_{ij}\mathbf{e}_i\otimes\mathbf{e}_j \tag{1}$$ The double bar notation is used to say the quantity is a second rank tensor. Is there more ...