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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62 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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108 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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160 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
210 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
56 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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562 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
67 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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0answers
159 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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57 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
114 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
145 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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130 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
69 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
88 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
94 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 ...
4
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1answer
105 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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2answers
213 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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0answers
105 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
106 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
87 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
104 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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1answer
125 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 ...
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1answer
2k 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 ...
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134 views

Multilinear or Tensor Regression?

Given input data $x_t\in \mathbb{R}^n$ and output data $y_t\in\mathbb{R}^m$, the closed form solution to $\min_A \sum_t \|y_t - Ax_t\|^2_2$ is given by $A = (XX^T)^{-1}XY^T$ where $x_t$ form the ...
4
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1answer
316 views

The divergence of the Weyl tensor

First, the Weyl tensor is given by $$W_{ijkl}=R_{ijkl}-\frac{1}{n-2}(g_{ik}A_{jl}-g_{il}A_{jk}-g_{jk}A_{il}+g_{jl}A_{ik})$$ where, $A_{ij}$ is the Schouten tensor, given by ...
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1answer
103 views

Matrix tensor factorisation

Say we have a matrix $A$ expressed as the tensor $$A=\sum_{m=1}^Mx^{(m)}A^{(m)}$$ where $A$ and $A^{(i)}$ are $N\times P$ matrices and $x$ is a $M\times 1$ vector. I would like to decompose $A$ (or ...
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0answers
471 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 ...
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1answer
602 views

What is the practical difference between abstract index notation and “ordinary” index notation

I understand that in "normal" index notation the indexes can be thought of as coordinates of scalar values inside a tabular data structure, while in the abstract index notation they can not. However, ...
2
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0answers
80 views

Tensors: summing over indices

Would anybody mind teaching me how to work these indices? Definitions: Throughout the following, repeated indices are to be summed over. Hodge dual of a p-form $X$: $$(*X)_{a_1...a_{n-p}}\equiv ...
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2answers
205 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 ...
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1answer
35 views

Help with substituting definitions into tensor

I have 4 definitions for the following (Einstein summation) tensor $A^{ijk}A^{*}_{ijk}=A^{111}A^{*}_{111}+3(A^{112}A^{*}_{112})+3(A^{122}A^{*}_{122})+A^{222}A^{*}_{222}$ If I have these 4 ...
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1answer
44 views

How does $A^{123}A^{*}_{123}$ look when expanded?

tensors are a new subject for me. I am trying to expand $A^{123}A^{*}_{123}$ Does it look something like the following? ...
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1answer
134 views

Eigenvalues of a second derivative

I have a function f(r) that describes a Gaussian random field. A second derivative can be formed $\nabla_i \nabla_j f(r)$. I am looking at a paper that claims that in finding the extremum, the ...
3
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1answer
44 views

$\nabla \varphi \overset{?}{=} \nabla \cdot \varphi \bar{\bar{I}}$ where $\varphi$ is scalar, $\bar{\bar{I}}$ is identity tensor

I am trying to determine if these two are equivalent. I have a function written with both terms, and this is the only discrepancy. The gradient increases the rank of the scalar to a vector, while the ...
4
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1answer
283 views

Derivation or Intuition of Formula for Levi-Civita Symbol

http://www.ees.nmt.edu/outside/courses/GEOP523/Docs/index-notation.pdf spouted off and threw out with no motivation $$\epsilon_{ijk} = \frac{1}{2}(i - j)(j - k)(k - i) \, \forall \, \, k \in \{1, 2, ...
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80 views

Independency of the frame of reference of the strain rate tensor

I've got a problem regarding tensors. Premise: we are considering a fluid particle with a velocity $\mathbf{u}$ and a position vector $\mathbf{x}$; $S_{ij}$ is the strain rate tensor, defined in this ...
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631 views

Book on tensors

Can anyone recommend me a book on tensors with an intuitive approach? I have some course notes on that subject, but it's really abstract and theoretical. I want to understand why tensors were ...
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1answer
330 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 ...
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1answer
691 views

A user's guide to Penrose graphical notation?

Penrose graphical notation seems to be a convenient way to do calculations involving tensors/ multilinear functions. However the wiki page does not actually tell us how to use the notation. The ...
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1answer
64 views

The gradient of a function is an alternating one-tensor

I'm currently reading Spivak's Calculus on Manifolds and I seem to have hit a snag in Chapter Four: Integration on Chains. Spivak develops tensors, vector fields, alternating tensors and differential ...
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1answer
59 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 ...
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76 views

(Complex) Projective Space

I followed a course in projective geometry and I'm not sure about 2 things: If I have 6 lines in projective space (IP³) with commun secant, why are the 6 corresponding tensors linearly dependent? ...
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2answers
951 views

Do I understand metric tensor correctly?

So I've been studying vectors and tensors, and I'm trying to understand metric tensors. As I understand them, besides a vast array of explanations, they provide an invariant distance between vectors ...
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2answers
159 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 ...
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1answer
81 views

Simplifing formulas using tensor notation

Im trying to symplify formulas like: $$\operatorname{div}(\operatorname{rot}\vec{F}),\qquad \operatorname{rot}(\operatorname{rot}\vec{F}) $$ or something more strange like: ...
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1answer
156 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.
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1k views

Vorticity equation in index notation (curl of Navier-Stokes equation)

I am trying to derive the vorticity equation and I got stuck when trying to prove the following relation using index notation: $$ {\rm curl}((\textbf{u}\cdot\nabla)\mathbf{u}) = ...
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1answer
39 views

Coordinate-free definition of pseudotensors

How to define pseudotensors (particularly, pseudovectors) in a coordinate-free form? Can it be defined on a manifold (like a tensor field)? Or may be the objects that physicists model via ...
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34 views

Tensor transformation for tensor of a special form

The components of the tensor $A^{ij}$ are $A^{12} = A^{21} = A$, whereas all the other components are zero. I am asked to write $\bar{A}^{ij}$, following a transformation to a new coordinate system, ...
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55 views

Relation of Hodge dual to antisymmetric part of the

I have a question in reaction to an article by M. Born and L. Infeld (cf. [1]) concerning the relation between the hodge dual of the electromagnetic tensor and the antisymmetrization of its ...