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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Pullback maps and an equallity

Let $\Lambda^p (E)$ be the set of $p$-covariant exterior tensors on linear space $E$ over field $K$ (dim$E=n$ and $ 0\leq p\leq n $ , $\Lambda^0E:=K$). We define linear map ...
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1answer
25 views

How do I extrac the anisotropic part of a tensor?

Given the elements $\chi_{ij}$ of a tensor in cartesian coordinates, with \begin{pmatrix} \chi_\bot& 0 &0 \\ 0 & \chi_\| &0\\ 0&0 & \chi_\| \end{pmatrix}, where the ...
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1answer
11 views

Proving a $k$-multilinear symmetric map is invariant iff a condition is satisfied

In Huybrecht's book on complex geometry, he states the following lemma on page 193: Lemma 4.4.2: The $k$-multilinear symmetric map $P$ is invariant if and only if for all $B,B_1,\ldots, B_k \in ...
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34 views

Demonstrate that the matrix $\pmatrix{ x_2^2 && -x_1x_2 \\ -x_1x_2 && x_1^2}$ represents a $2^{\text{nd}}$ order tensor.

Demonstrate that matrix $T$ represents a $2^{\text{nd}}$ order tensor. $T = \pmatrix{ x_2^2 && -x_1x_2 \\ -x_1x_2 && x_1^2}.$ To show that $T_{ij}' = L_{ik}L_{jn}T_{kn}$, I would ...
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1answer
56 views

Kernel of the Symmetrizing Map $Sym:\bigotimes^k V\to \bigotimes^k V$

$\DeclareMathOperator{\sym}{Sym}$ Let $V$ be a finite dimensional vector space over a field of characterisitc $0$ and $\sym:\bigotimes^k V\to \bigotimes^k V$ be the map given by $$ ...
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22 views

Definition of Tensors Over Complex Numbers

My question is two part. First, how does the definition of tensors and tensor spaces change when the vectors that the tensors act upon are elements of a complex vector space as apposed to when they ...
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1answer
22 views

Evaluation of polynomials at tensor products

Let $S,T$ be $R$-Algebras, $f \in S[X]$ a polynomial. in my notes it says you can easily lift $f$ to a ploynomial $f'$ in $(S \otimes T)[X]$. But I have no idea what $f'(s \otimes t)$ is. My guess is ...
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1answer
56 views

Electromagnetic tensor - I need help with a tensor calculation

First of all: this is not about the physics behind it. It's about the tensor calculation I've written down below. I know this kind of calculation is exhausting but I would be thankful if someone could ...
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2answers
29 views

How to visualize a second-order tensor in a rectangular coordinate system?

I can visualize a first-order tensor as a segment (a vector), but I'm not sure how to visualize a second-order tensor. The book that I'm trying to study is "Vector and tensor analysis with ...
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13 views

Curvature of $K$-invariant connection (principal bundles)

Here is a proposition from Kobayashi & Nomizu's Foundations of Differential Geometry. I don't understand how they obtain the final line of the proof. They write: \begin{align} ...
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1answer
46 views

Are affine transformation matrices tensors?

Since affine transforms involve a matrix, if the transform matrix is a tensor, it would be of rank two. But, the real question is whether or not a change of basis, or transformation of the underlying ...
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33 views

Second contracted Bianchi identity

Considering the Riemann tensor and the onece contracted second Bianchi identity $g^{ls}\nabla_sR_{ijkl}=-\nabla_iR_{jk}+\nabla_jR_{ik}$ why should it hold true that $g^{ls}\nabla_sR_{ijkl}=0$? In ...
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1answer
44 views

Integral of the exponential of a homogeneous quartic - reference request

For a calculation I am doing, I have to calculate an integral of the form $$ I = \int_{\mathbf{R}^n} \exp[-Q(\mathbf{x})] d^n\mathbf{x} \text,$$ where $Q(\mathbf{x})$ is a homogenous, degree-4 ...
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2answers
27 views

Tensor manipulation

I am very new at manipulating tensors and I have the following equation: $$A_{\mu \nu\tau} b^\mu c^\nu = g_{\tau \rho} d^\rho$$ where $\tau$ is the independent index and $g_{\tau \rho}$ the metric ...
2
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0answers
21 views

Hodge star of second-rank antisymmetric tensor

Say we have a tensor $F$ which just for familiarity's sake, we take to be a second rank antisymmetric tensor. I understand that given the Hodge star operator defined as ...
3
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1answer
73 views

What is the relation between $C^\infty$-linear and tensorial?

I've been self studying Riemannian Geometry through Spivak's and Lee's books, and fairly often I've seen an argument that goes somewhat like this: We have an operator that acts on vectors in the ...
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2answers
89 views

How do the components of a cross product transform?

Let $x^{j}$ and $y^{k}$ be the components of two vectors $x,y\in \mathbb{R}^{3}$. According to the way the compontents of $x$ and $y$ transform when we change the basis, we know they are ...
2
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1answer
34 views

Riemannian tensor and Levi Civita connection

For a riemannian metric $g$ consider the following tensor $T_{rstu}=k(x)g_{rt}g_{su}-k(x)g_{st}g_{ru}$. Which condition has to satisfy $k$ if we want the tensor $T$ to be the Riemann tensor of a ...
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0answers
49 views

Covariant derivative and box operator commutator

I know that the commutator of two covariant derivatives is giving some Riemann tensors as follow: ...
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0answers
21 views

Do tensor norms exist?

Does there exist norms for tensors, as an extension for the ordinary matrix norm? For example, if there is a derivative of a matrix [A] with respect to a vector {x}, does the norm of this derivative ...
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31 views

Trace of a bilinear Form

What's the definition of trace of a $(m,n)$-Tensor $T:\underbrace{V\times \cdots \times V}_{k- \text{times}}\times\underbrace{V^*\times\cdots V^*}_{l-\text{times}}\rightarrow \mathbb{R}$ that ...
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34 views

Tensor transpose notation

I have a rank 3 tensor $\mathbf{Q}$. What notation should I use to denote the transposition of two of the dimensions? For instance, if I want to transpose the first and second dimensions, one way I ...
2
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1answer
34 views

Indices at the left of a tensor in mathematical physics/differential geometry?

I am a mathematician and I am reading a paper in mathematical physics and I found the following notation: Let $Y$ be a two–form on $M$ such that $$\nabla({}_iY_j)_k = 0.$$ Here, $\nabla$ is ...
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1answer
56 views

Cross product between a vector and a 2nd order tensor

I have been searching for quite a long time, and haven't been able to find any good reference about the cross product between a vector and a tensor: $$ \vec{a} \times \underline{T}= ...
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1answer
19 views

Problem about tensor powers of operator

$F_1,\dots,F_m$ - linearly independent endomorphisms of vector space $V$. I want to prove that there is no $\lambda_1,\dots,\lambda_m$(not all $\lambda_i = 0$) such for any $n > 1$ $$\lambda_1 ...
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1answer
55 views

Taylor Expansion of tensor moved by a flow.

I am reading Peter Petersen's notes on manifold theory and he introduces Lie Derivatives in the following way. "Let $X$ be a vector field and $F^t$ the corresponding locally defined flow on a smooth ...
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2answers
106 views

What is the modern approach to tensors?

I have recently started studying tensors a bit from the index notation point of view -- I understand contractions and the metric tensor and such well enough. However I've been told that this approach ...
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1answer
65 views

What are higher derivatives?

From Wikipedia: Higher derivatives can also be defined for functions of several variables, studied in multivariable calculus. In this case, instead of repeatedly applying the derivative, one ...
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1answer
33 views

Multivariable Calculus with Tensors

I'm looking for a book at the undergraduate level on multivariable calculus (for a 2nd course of multivariable calculus) that introduces and makes use of tensors to describe higher order derivatives ...
5
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1answer
111 views

Divergence theorem for a second order tensor

I want to integrate by part the following integral in cylindrical coordinates $$\int \vec{r} \times (\nabla \cdot \overline{T}) ~d^3\vec{r} $$ where $\overline{T}$ is a second order symmetric tensor ...
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1answer
34 views

Tensor contraction

Given that: $T_{i,j}=\lambda\theta\delta_{i,j} + 2\mu E_{i,j}$ Show that: $T_{i,i} = 3\theta \lambda + 2\mu E_{i,i}$ I didn't get the intuition behind tensor contraction, thus i can not solve this ...
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55 views

Matrix Calculus

My school offers Matrix Calculus class next semester. I have never heard about this subject before and got intrigued. After a short chat with professor I found myself unable to get rid of suspicion ...
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1answer
22 views

Lorentz transformation of electromagnetic field tensor

I need to calculate: $f^{\mu'\nu'}=L^{\mu}_{\kappa}L^\nu_\lambda f^{\kappa\lambda}$ Where $L^\nu_\lambda$ is the usual Lorentz transformation matrix I thought that I just needed to do some normal ...
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1answer
20 views

Relationship between euclidean metric in sphere of radius $r$ and the unit sphere.

I want to show $g_r=r^2g_1$ where $g_1$ is the (Riemannian) metric in the unit sphere induced by its inclusion in $\mathbb{R}^n$ and $g_r$ is the metric in the sphere of radius $r$ also induced by ...
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26 views

Understanding Extension of Scalars in a Vector Space

$\newcommand{\R}{\mathbf R}\newcommand{\C}{\mathbf C}$ Low-Tech Complexification: Let $V$ be a finite dimensional vector space over $\R$. We can forcefully make $W:=V\times V$ into a complex vector ...
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30 views

Levi-Civita tensor in curved space

In the book "Gravitation and cosmology" by Weinberg at the page 99-100. He defines the Levi-Civita tensor as $\epsilon^{0123}=+1$ from which he writes ...
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1answer
84 views

Multiplying two tensors of the Levi-Civita type

How to multiply two epsilons with one another? We know ...
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1answer
36 views

Covariant metric tensor of a subspace

Suppose $f_1,f_2$ and $f_3$ are vectors in a vector space $V$ with a dot product. Me assume that the vectors are linearly independent. What does it mean to find the covariant metric tensor of ...
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1answer
23 views

Deriving the Geodesic Equation

I found a derivation of the geodesic equation that includes this step as I write it: $$ \frac{d (g_{ab}\dot{x}^b)}{dt}=\frac{1}{2}\partial_ag_{bc}\dot{x}^b\dot{x}^c \Rightarrow \\ \\ ...
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1answer
81 views

Comparing metric tensors of the Poincare and the Klein disk models of hyperbolic geometry

I was trying to compare the metric tensor at the wikipedia pages of the Beltrami Klein model https://en.wikipedia.org/wiki/Klein_disk_model and the metric tensor of the Poincare disk model at ...
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34 views

How does a 4-tensor act as a linear trasformation of 2-tensors?

I'm trying to understand tensors by looking at this table and thinking about the various types of transformations the tensors represent. From the linked table, I tried looking up some of the less ...
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1answer
21 views

Locality of tensors part of definition?

I am wondering whether linearity with respect to scalar functions $f \in C^{\infty}(M, \mathbb{R})$ is part of the definition of a tensor? Let me explain it by referring to the Riemann curvature ...
3
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1answer
27 views

Why should the metrical groundform on a variety be a quadratic form?

I'm learning General Relativity and I can't understand why the distance function on space time is a quadratic form $$\textrm{d}s^2=g_{\mu\nu}\textrm{d}x^{\mu}\textrm{d}x^{\nu}$$ I explain it through ...
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1answer
85 views

Show that $\mathbb{C} \otimes_\mathbb{Z} \mathbb{C} \cong \mathbb{C} \otimes_\mathbb{Q} \mathbb{C}$

Show that $\mathbb{C} \otimes_\mathbb{Z} \mathbb{C} \cong \mathbb{C} \otimes_\mathbb{Q} \mathbb{C}$ This is not homework, it is part of an answer of Show that $\mathbb{A}_\mathbb{C}^2 \ncong ...
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15 views

The space $V^{0}_{p}$ of p times covariant tensors and canonical isomorphisms

I have been studying tensor calculus by myself, but I have found the following claim in my book: The space $V^{0}_{p}=V^{*} \otimes \cdots \otimes V^{*}$ of $p$ times covariant tensors is ...
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1answer
18 views

Showing that two tensors are coaxial

Given two second-order tensors $\mathbf{C} = C_{ij}\mathbf{e}_i\otimes\mathbf{e}_j$ and $\mathbf{U} = U_{ij}\mathbf{e}_i\otimes\mathbf{e}_j$ with the following relation between $\mathbf{C}$ and ...
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2answers
56 views

A question about irreducible representation of symmetric group (permutation group) in tensor space and tensor contraction

In chapter 13 of the textbook of Group Theory in Physics by Wu-Ki Tung, Lemma 2 discusses the equivalence of two irreducible representations of GL(m) on ${T^i}_j$. In its proof, it simply mentioned ...
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2answers
42 views

Is there a way to factor out the middle tensor product?

Given a state $(|1\rangle \otimes |0\rangle \otimes |1\rangle) + (|0\rangle \otimes |0\rangle \otimes |1\rangle)$, is it possible to factorise out the $|0\rangle$ in the middle of both of them? ...
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20 views

Is this proof of the product of determinants in tensor notation correct?

I'll start with the matrix C which is the product of the matrices A and B. $$c^i_k = a^i_jb^j_k$$ The determinant of C is $$\frac{1}{3!}\delta_{ijk}^{rst} c^i_rc^j_sc^k_t $$ by the definition of ...
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2answers
75 views

Looking For a Coordinate Free Way to Prove This Linear Algebra 'Fact'

$$\newcommand{\mc}{\mathcal}$$ Let $V$ be an $n$-dimensional vector space over a field $F$. (We use $\mc L(V)$ to denote $End(V)$). For each $v\in V$, define $\Theta_v:\mc L(V)\to V$ as ...