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
44 views

What is the definition of the tensor product $U⊗ V$ of two vector spaces $U$ and $V$?

All I want to know is the exact definition of the vector space that is the tensor product of two vector spaces, say $U$ and $V$, i.e. I want to know: - how its vectors are defined, - how does the $+$ ...
1
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1answer
48 views

Gradient of vector field notation

Working in 3D. I know that the gradient is a vector operator defined as $\nabla = [\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}]$. The gradient of a scalar ...
1
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1answer
29 views

cardinality of orbits

Let $G=\mbox{GL}(\mathbb{Z}/p\times\mathbb{Z}/p )\times \mbox{GL}(\mathbb{Z}/p\times\mathbb{Z}/p ) \times \mbox{GL}(\mathbb{Z}/p\times\mathbb{Z}/p )$, $p$ prime, act on $(\mathbb{Z}/p\times ...
0
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1answer
64 views

group ring Z(G) is isomorphic to tensor algebra T(G)

Well, I've been learning Tensor algebra and related topics. So, I wonder tensor algebra is related to somehow familiar with me, so called group ring. $\mathbb{Z}$: a ring of integers. Let G be a ...
4
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1answer
68 views

Divergence operator of higher order and intrinsic point of view

Let $\underline{u}$ be a $1$ - order tensor (say a column vector) I want to prove that : $\underline{\operatorname{div}} \left( (\underline{\underline{\operatorname{grad}}} \, ...
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0answers
37 views

Nice way of proving that the pullback of a tensor field is smooth?

I'm going through some routine exercises in studying smooth manifolds. This one is 12.27 (d) from Lee, Introduction to Smooth Manifolds If $F:M\to N$ is smooth, and $B$ is a covariant $k$-tensor ...
4
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1answer
52 views

Map Laplacian in terms of covariant derivatives

I stumbled upon the following definition: Let $\mathcal{M}$ be a manifold, $g_{ij}$, $h_{ij}$ be two Riemannian metrics on $\mathcal{M}$, $\psi : \mathcal{M} \to \mathcal{M}$ be a ...
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0answers
26 views

Rotation of eigenvectors of a time-dependent tensor

I have a symmetric, real tensor in ${\mathbb R}^3 $ where the components are continuous functions of time: $${\mathbf D} = \left( \begin{matrix} d_{1,1}(t) & d_{1,2}(t) & d_{1,3}(t) \\ ...
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0answers
31 views

Kronecker delta representation of a matrix (Quantum raising / lowering operators)

The Kronecker Delta is commonly used to represent a diagonal matrix: $$ a_i \delta_{ij}=\left( \begin{array}{ccc} a_1 & 0 & 0\\ 0 & a_2 & 0\\ 0 & 0 & a_3 \end{array}\right) $$ ...
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1answer
26 views

divergence theorem applied to a tensor dotted with a vector

Is my expression for the divergence theorem correct? $\int_{V}\underline{v}.div(\underline{\underline{\tau}})dV=\int_{S}\underline{v}.\left(\underline{\underline{\tau}}.\underline{n}\right)dS$ and ...
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2answers
47 views

Does the Divergence Theorem hold for arbitrary tensor fields?

So, a heads up, this is my first post and I'm a fairly new user. Additionally, my math knowledge tops out at vector calculus and ODEs, but don't shy away from answering beyond my understanding should ...
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1answer
18 views

Show $\delta_{KL}$ is a Cartesian tensor

By using the definition of kronecker delta $\delta_{KL}$, show that $\delta_{KL}$ is a Cartesian tensor, that is $$\delta ' _{MN}=L_{MK}L_{NL} \delta_{KL}$$ under the rotation $X_K=L_{MK}X' _M$. I ...
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0answers
38 views

Derivative of dual basis vectors in terms of Christoffel symbols

How can I demonstrate from $$ \frac{\partial \mathbf{e_j}}{\partial x^i} \equiv \Gamma_{ij}^k \mathbf{e_k} $$ what the value of $$ \frac{\partial \mathbf{e^j}}{\partial x^i} $$ (with the index now ...
0
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1answer
41 views

Show that $\epsilon_{ijk}\epsilon_{ljk}=2\delta_{il}$

Question: Show that $\epsilon_{ijk}\epsilon_{ljk}=2\delta_{il}$ where $\epsilon_{ijk}$ is the Levi-civita symbol and $\delta_{ij}$ is the Kronecker delta symbol. My attempt: $\epsilon_{ijk}$ assumes ...
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0answers
35 views

Easy Tensor Notation: What is $\partial_\mu\partial^\nu x_\nu$

Part of a far larger piece of work... I know it should simplify $\partial^\nu x_\nu\partial_\mu$ but I don't know how. Similarly, does it change for the reverse? In other words I want to simplify: ...
2
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5answers
177 views

How to Prove $(A \times B) \otimes C + (B \times C) \otimes A + (C \times A) \otimes B = [(A \times B) \cdot C] \bf{I}$?

Question Assume that $A$, $B$, $C$, and $D$ are four vectors in $\mathbb{R}^3$. Then I want to show that $$ {\bf{M}} \equiv (A \times B) \otimes C + (B \times C) \otimes A + (C \times A) \otimes ...
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0answers
37 views

Programming nested sums in Matlab for graph-based statistic

I have an undirected graph $G=(E,N)$, where $E$ is the set of edges and $N$ is the set of nodes, of which $|N|=n$. It's convenient to represent the edges via a (symmetric) adjacency matrix $B$. I ...
0
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1answer
45 views

Tensor algebra and symmetric algebra

Reference Book : Dummit - abstract algebra 3rd edition (page 455). But, I think $ $ $=$ $ $ notation is not correct. It may be corrected to $\cong$ (an isomorphic notation). Because $S^2 (V)$, ...
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2answers
57 views

When is a vector a tensor? Is invariance under affine transformation required?

Several resources, such as this (pages 11 & 15) and this, state that position vectors are not rank one tensors while displacement vectors are, since only displacement vectors are invariant under a ...
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0answers
19 views

SU(3) tensor methods in representations [duplicate]

I am trying to understand the Georgi chapter of tensor methods in $SU(3)$ representations, and I don't know how to resolve the tensor product of 2 matrices in a 2 heavy quark + 2 light antiquark ...
2
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0answers
30 views

Tensor formula in SU(3) representations

I am trying to understand the Georgi chapter of tensor methods in $SU(3)$ representations, and I don't know how to resolve the tensor product of 2 matrices in a 2 heavy quark + 2 light antiquark ...
1
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1answer
30 views

Confusion about tensor derivation in wiki article

I can't follow the derivation of the gradient of a vector field in cylindrical coordinates from this wiki page My problem is with the $e_r \otimes e_\theta$ term. Expanding the definition from a ...
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2answers
100 views

Matrix notation in tensor transformations

In special relativity one looks at coordinate transformations that consist of combinations of Lorentz boosts, rotations and reflections - members of the Lorentz group. Under an arbitrary ...
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0answers
45 views

variation metric tensor

$\newcommand{\Tr}{\operatorname{Tr}}$(I asked this on physicsexchange but no reply) I have the metric tensor $g_{\mu\nu}$. I want to make the variation of $\sqrt{-g}$ where $g=\det g_{\mu\nu}$. Can ...
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0answers
17 views

Transforming an infinitesimal line element, dx, to 1/2(curl(u)/\dx)? What does this mean physically?

Consider transforming an infinitesimal line element,say dx, to 1/2(curl(u)/\dx)? Where curl denoted /\ here, and dx is an infinitesimal 3d vector, and u is the displacement vector --What does this ...
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0answers
35 views

Contraction of $(2k,2l)$tensor

In picture below ,how to get the equality in red box?
1
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1answer
47 views

Can someone help me understand the Euclidean metric?

A Euclidean metric is defined as: $g_{euclid} = g_{ij} = \delta_{ij} dx^i \times dx^j = dx^1dx^1 + \ldots + dx^ndx^n$ Can someone explain the following: why do we use $dx^i$ instead of $x^i$ which ...
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0answers
24 views

Higher order derivatives than Riemann Tensor

Does anyone know of any meaningful tensors that are related to the derivative of the riemann tensor? i.e. in the following picture we can consider Given arbitrary Pseudoriemannian manifold and metric ...
0
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0answers
21 views

Infinitesimal displacement- The inf. rotation tensor.

I have a question related to the infinitesimal rotation part- the skew-sym. part- of the infinitesimal displacement gradient matrix. Let the infinitesimal displacement gradient, say D, and E and W ...
1
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1answer
49 views

How to understand acting one tensor on another tensor to obtain a third tensor?

I've already known the definition of the tensor that a tensor T of type $(k,l)$ is a multilinear map from a collection of dual vectors and vectors to $\mathbf{R}$: $$ T: ...
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3answers
57 views

Tensors as geometric objects

Wikipedia's article on tensors starts with: "Tensors are geometric objects..." https://en.wikipedia.org/wiki/Tensor However there is no definition of "geometric object" in Wikipedia. To my amateur ...
1
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1answer
36 views

Dual Spaces vs Dual Bases

I'm trying to wrap my head around differentiable manifolds and tensors. I partially worked through a question which asked me to use the metric tensor and the line element in spherical polar ...
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0answers
18 views

Positive semidefiniteness of higher order tensor

I am trying to understand how positive semidefiniteness of hessians extends to higher dimensions. Specifically I am considering a case where I have a fourth order Hessian tensor such that each ...
3
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1answer
52 views

raising/ lowering indices

Here is my understanding of tensors: There is more than one way to think about tensors. One way is be thinking about tensors as objects with components which obey some transformation laws. For ...
0
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1answer
42 views

The formula for the differential of a vector-valued function

If we have a vector, $\,U=U\left(x_1,x_2,x_3\right)$, in the coordinate axis $\left(x_1,x_2,x_3\right)$, then why does the following differential relation hold? $$ dU= \left(\frac{\partial ...
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2answers
84 views

Einstein Tensor Notation: Addition inside a function

Main Question Can I represent addition of multi-dimensional variables in this linear function in Einstein Summation Convention? $$ f(\mathbf{x} + \mathbf{v}) $$ This didn't seem right ...
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1answer
18 views

Understanding contra/covariance

From Hehl and Obukhov's Foundations of Classical Electrodynamics: A transformation from a basis $e_\alpha$ of $V$ to another one $e_{\alpha '} = (e_1', \dots, e_n')$ is described by a matrix $L:= ...
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0answers
151 views

Examples of geometric structures on real manifold that lead to almost complex structure

I have a $M^{2n}$ real manifold. I am interested, if there are any well known geometric structures on manifold that lead in some natural manner to almost complex structure on $M.$ I read on wiki ...
2
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1answer
20 views

Definition of brackets in index notation

I've come across the notation $B_{[ij]}$ and $\partial_{[k}B_{ij]}$. What do these notations mean? The text says that $[ij] := (ij-ij)/2$, but doesn't the $ij$ just specify the which component of ...
1
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1answer
45 views

If $\mathfrak{g}\subseteq\mathfrak{so}(V)$, then $H^{0,2}(\mathfrak{g})=(\mathfrak{so}(V)/\mathfrak{g})\otimes V^*$

Problem I'm learning about $G$-structures and was assigned this exercise (Cartan for Beginners Exercise 8.3.6.1): Let $$ H^{0,2}(\mathfrak{g})=(V\otimes\Lambda^2V^*)/\delta (\mathfrak{g}\otimes ...
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1answer
38 views

How to show that the isotropic tensor of order n is a multiple of the kronecker delta

I have already found this question here but with the property of invariant under rotation. However I don't have this property and I want to prove that $T_{ij} = \alpha \delta _{ij}$ where $T_{ij}$ ...
2
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1answer
76 views

Is this 3 Qubit state Entangled?

Is this 3-Qubit state entangled? http://prntscr.com/8vsg0b $|X\rangle=\frac{1}{\sqrt 2}~~|000\rangle+ \frac{i}{\sqrt 2}|111\rangle$ I've worked with 2-Qubit states and you can turn them into ...
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0answers
31 views

Covariant derivative of a covariant vector

$$\mathbf{A} = A^{1}\mathbf{e_{1}}+A^{2}\mathbf{e_{2}}$$ $$A= \sum_{i=1}^{n}A^{i}\mathbf{e_{i}}$$ Taking the derivative wrt the tangent basis vector and dropping the summation by Einstein convention: ...
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0answers
17 views

How would you express $(\underline{a} \times \underline{b}) \times (\underline{a} \times \underline{c})$ in index notation?

At a guess, I would have said that the answer is $\varepsilon_{ijk}\varepsilon_{jlm}a_{l}b_{m}\varepsilon_{kpq}a_{p}c_{q}$, but I'm guessing that this is incorrect. What is the corrdct expression? ...
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2answers
62 views

What does it mean for a vector to be a derivative?

I'm reading on Killing vectors and Killing vector fields, and one notion that keeps coming up is a derivative being a vector. For example, it's put here(Eq. 5.46) that in 2-dimensional Euclidean ...
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1answer
36 views

Does the set of null vectors of an indefinite scalar product determine the product up to scale?

If $V$ is a vector space with index 1, let $g$ and $\hat{g}$ be scalar products, and denote \begin{align}\Lambda &= \{v \in V \mid g(v,v)=0\} \\ \hat{\Lambda} &=\{v \in V \mid ...
1
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1answer
39 views

Rank one decomposition or elementary tensor decomposition of matrices over commutative rings

I'm facing the following problem: Let $A$ be an $m\times n$ matrix over a commutative ring $R$ (to begin with, a finite field would be sufficient too) and want to compute a decomposition in terms ...
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0answers
32 views

The explicit expression of symmetric tensors, and the Symmetrize gradient of symmetric tensors

This question has been asked here long time ago but I am still confused... Let me repeat some definitions. In general we define on $\mathbb R^2$ that $$ \mathcal T^k(\mathbb R^2):=\{\xi:\,\mathbb ...
0
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1answer
30 views

Corresponding matrix field basis

Hi people, I'm reviewing my notes for an exams and this is a question which I was unable to wrap my head around for many months. It should be fairly simple but I might be lacking a crucial piece of ...
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0answers
16 views

Is there any clue for the supremum of projective norm in an unit ball of injective tensor product space

Is there any information about the supremum, that is in a d-order tensor space $\mathcal{T}=\mathcal{R}^{n_1}\times\mathcal{R}^{n_2}\times\cdots\times\mathcal{R}^{n_d}$, $x\in \mathcal{T}$ what is ...