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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65 views

Rotation of axes transformation as definition of vectors

Given a three-axes coordinate system ${1,2,3} $ by the right-hand rule, and a new coordinate system ${1',2',3'}$ , I know that one can define a vector $\vec{x}$ to be something that obeys the ...
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1answer
53 views

Tensors and Transformations

In Griffiths E&M book, he says that a second rank tensor transforms with two factors of some transformational tensor on each of its nine components-I'm not sure why that is. I thought a second ...
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74 views

The trace of a wedge product of matrices

I'm trying understand a computation on Besse's book (p. 371). I already know the curvature operator $R:\bigwedge^2\to\bigwedge^2$ may be written in block diagonal form relative to the direct sum ...
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2answers
57 views

Is every tensor an element of a vector space?

As, the tensor product of two vector spaces $V$ and $W$ over a field $K$ is another vector space over $K$, is it true to say that every tensor is an element of a vector space ? (if we do not consider ...
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181 views

Number of isotropic tensors of rank N

Could someone please help me out with this problem? I would like to know what the number of distinct isotropic tensors of rank N is. I am new to the field and got confused by apparently contradictory ...
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1answer
65 views

Partial derivative with respect to $ \left( \frac{dx^m}{ds} \right) $

I don't understand how the following $$ 2g_{ml} \frac{dx^l}{ds} $$ partial derivative was obtained below. It is supposedly the partial derivative of the value between the parenthesis. $$L = ...
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1answer
299 views

Inverse of a matrix

I am looking for a way to derive that the inverse of a matrix using Levi-Civita. I know that the final result looks like this for a $3 \times 3$ matrix: $$(A^{-1})_{ij} = \frac{1}{2!}\frac{1}{\det A} ...
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1answer
67 views

Why must metric tensor be invertible?

The metric can be written as a matrix, but why must this matrix be invertible? At the points where the matrix is singular, why is the metric not defined?
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1answer
274 views

Transforming a matrix from cartesian to spherical coordinates

Consider a variable matrix $$\left[\begin{array}{ccc}a_{11}(x,y,z) \quad a_{12}(x,y,z) \quad a_{13}(x,y,z)\\ a_{21}(x,y,z) \quad a_{22}(x,y,z) \quad a_{23}(x,y,z)\\ ...
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0answers
63 views

Curvature tensor on the sphere

While reading R. Hamilton paper "Three-manifolds with positive Ricci curvature" I came across the sentence: For the sphere we have $$R(u,v,u,v)=R_{ijkl}u^{i}v^{j}u^{k}v^{l}>0,$$ which is the ...
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1answer
47 views

Calculating a certain tensor

Consider the $\mathbb{Z}$ module $\mathbb{Z}/n\mathbb{Z}$. What is $\mathbb{Z}/n\mathbb{Z} \otimes_{\mathbb Z} \mathbb{Z}/n\mathbb{Z}$?
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1answer
107 views

Tensor Einstein summation notation

I have two tensors $A^i$ and $B_j$ with components $(2,3,4)$ and $(1,2,3)$ respectively. What is the difference between $A^i B_i$ and $A^i B_j$? Is it just: $A^i B_i = 2+6+12 = 20$ $A^i B_j =$ $ ...
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1answer
79 views

Symmetry of the Riemannian curvature tensor

The Riemannian curvature tensor, in local coordinates, $R_{ijkl}$, has the following symmetries: $$R_{ijkl}+R_{jikl}=0;$$ $$R_{ijkl}+R_{ijlk}=0;$$ $$R_{ijkl}=R_{klij};$$ ...
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5answers
637 views

What, Exactly, Is a Tensor?

I've repeatedly read things that reference tensors, and despite reading the wiki page and other answers here on stackexchange I still don't know what a tensor is. I'm fine with hearing things in the ...
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1answer
33 views

Levi-Civita symbol identity: $\epsilon_{ijs}x_s L_i = -\epsilon_{ijs}x_i L_s$

I have the following identity I want to use but don't know whether it's correct or not (and if it is, why so): $$\epsilon_{ijs}x_s L_i = -\epsilon_{ijs}x_i L_s$$ Is this correct? How to arrive at ...
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0answers
29 views

Prove $\frac{1}{N!}\varepsilon_{i_1\dots i_N}\varepsilon_{j_1\dots j_N}A_{i_1 j_1}\dots A_{i_N j_N} = \det A$

Is there any simple way to prove the following: $$\frac{1}{N!}\varepsilon_{i_1\dots i_N}\varepsilon_{j_1\dots j_N}A_{i_1 j_1}\dots A_{i_N j_N} = \det A. \tag{$1$} $$
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2answers
87 views

What is the value of the 2-form $X(u(Y))-Y(u(X))-u([X,Y])$, is it $2du(X,Y)$?

Let $X,Y$ be two vector fields on a Riemannian manifold $(M,g,\nabla)$ where $\nabla $ is the symmetric metric connection on $M$ and let $u$ be a 1-form. I want to find the value or a simple form or ...
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1answer
88 views

Question about tensor product of homomorphisms

I've come to think about this problem when reading a proof in Commutative Algebra by N. Bourbaki. Say, let $R$ be a commutative ring, given 3 $R-$modules $A$, $B$, $C$, and the $R$-homomorphism $f:B ...
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1answer
29 views

Vanishing of 1-form

If $\theta \in \frak{X}^* \mathrm{(M)}$ and $\theta (X) = 0$ $\forall X \in \frak{X} \mathrm{(M)}$ then $\theta = 0$. How do I prove this statement? Consider a manifold $M$ with chart $x^1, \dots, ...
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1answer
45 views

Significance of bump function in the proof.

In the book Semi-Riemannian Geometry with applications to Relativity by Barrett O'Neill, in Chapter 2 (Tensors), he stated that: However, I don't get why he had to use a bump function in the proof. ...
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1answer
84 views

Why is this true: $\nabla \cdot (S\cdot \vec v )=S:(\nabla \otimes \vec v)+\vec v \cdot (\nabla \cdot S)$?

I am doing fluid mechanics and I don't understand a particular step that is being used. It is the following step which I don't understand: $\nabla \cdot (S\cdot \vec v )=S:(\nabla \otimes \vec ...
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3answers
279 views

Resources for properly developing a modern understanding tensors

I am currently learning about tensors as they come up in the mathematics behind continuum mechanics. I was fairly disappointed with my initial foray into tensors, as presented in the book Classical ...
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1answer
608 views

I feel that (physics) notation for tensor calculus is awful. Are there any alternative notations worth looking into?

I am reading through Fung and Tong's "Classical and Computational Solid Mechanics", and feel that the Einstein summation convention saves a few symbols, at the expense of a lot of clarity. Along with ...
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2answers
32 views

Multilinearity of the exterior derivative of a one-form.

I wish to show that the exterior derivative $d \theta$ of a one-form $\theta$ is $\frak{F} \mathrm{(M)}$-multilinear, therefore, a tensor. Let $X, Y, V, W \in \frak{X} \mathrm{(M)}$ and $f, g, h, k ...
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0answers
52 views

Metric on an open subset of $\mathbb{R}^d$ and Christoffel symbol of the second kind under diffeomorphisms

This is a follow-up question to this question I proved the following identity. $\Omega \subset \mathbb{R}^d$ is open and $g$ is a metric field on $\Omega$. Further $\Gamma_{\,kl}^j$ denotes the ...
2
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0answers
108 views

Stress Tensor and the ubiquitous cube: a misrepresentation?

So I'm just learning about Stress Tensors and have had an on going confusion and I'm wondering if the cube diagrams may to be blame. Usually when a stress tensor is introduced they show a cube and ...
3
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0answers
109 views

Intuition about the Riemann and Ricci tensors

I have started an attempt to self-study Riemannian Geometry. I well understand all the algebraic properties of the Riemann tensor (With a symmetric connection), and how it gradually becomes less and ...
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2answers
380 views

Multiplication of 3 matrices - Index vs. Matrix notation

i'm having a problem multiplicating 3 matrices in index notation. I know this should be trivial but i just can't figure it out. Is there any formula like $\ A'_{\mu\nu} = M_{\mu}^{\ ...
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1answer
73 views

Tensor Product definition (help with certain step)

I'm going over some notes I took from the blackboard, and reached a slight hitch. I thought that maybe someone could help. Let $E,F$ be vector spaces over a field $\mathbb K$. A tensor product of $E$ ...
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1answer
54 views

Why is it true that ${(\nabla Y)'^i}_j = {Y'^i}_{,j} + {\Gamma'^i}_{jk}Y'^k $?

I do not understand why this equation transforms as it does : ${(\nabla Y)'^i}_j = {Y'^i}_{,j} + {\Gamma'^i}_{jk}Y'^k $ Could someone give me a detailed explanation of why this is true please? I ...
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1answer
795 views

Christoffel Symbols and the change of transformation law.

I have seen it written that the change of co-ordinate form is given by the following: $$ \tilde \Gamma^{i}_{jk} = {\partial \tilde x^i \over \partial x^\alpha} \left [ \Gamma^\alpha_{\beta ...
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1answer
71 views

Metric tensors. Have I got the correct understanding?

My course is covering metric tensors in a slap-dash way, so I want to ensure I have understood correctly how they are described. I hope you can confirm this! So, I believe that a metric tensor is a ...
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1answer
126 views

Derivation for affine connection formulas on differentiable manifolds (General tensors)

Let $p\in U\subseteq M$ be a point in some neighborhood of a finite-dimensional differentiable manifold, $\{x^i\}$ a set of local coordinates with respect to $U$, and ...
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1answer
702 views

Riemann tensor in terms of the metric tensor

The Riemann tensor is a function of the metric tensor when the Levi-Civita connection is used. Most tensor notation based texts give the Riemann tensor in terms of the Christoffel symbols, which are ...
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1answer
109 views

Tensor product and valence of a tensor?

Given a vector space $V$, its dual vector space $V^{*}$ and a tensor $\mathbf{T}$: $\mathbf{T} \in \underbrace{V \otimes\dots\otimes V}_{n\text{ copies}}\otimes \underbrace{V^{*}\otimes\dots\otimes ...
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2answers
239 views

Using the Levi-Civita alternating tensor and suffix notation to concisely write the vector product rule.

I am reading through a section on vector calculus in an electromagnetism book and it has started to use suffix notation and the Levi-Civita alternating tensor in order to prove some identities. Some ...
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0answers
44 views

A tricky tensor

There's this question from Schaum's Outlines-Tensor Calculus: If the $a_{ij}$ are constants, calculate the partial derivative $\partial\over\partial x_k$$(a_{ij}x_ix_j)$. We use the product rule and ...
3
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1answer
119 views

Why the space of skew-symmetric tensors $\Lambda^{n}V$ is a one dimensional if $dim(V)=n$

While reading Liviu Nicolaescu Lectures on the geometry of manifolds, I came accross the notion of "determinant line": Definition: Lev $V$ be an n-dimensional R-vector space. The one dimensional ...
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1answer
244 views

Tensor basis change

I have a question regarding tensors and basis changes, however upon searching the web I've found an infinity of definitions for tensors so I'll have to give the one I know first: Given a vector space ...
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1answer
38 views

choosing indices in tensor notation

I have the following operator, where $\rho$ is a scalar and $u$ is a vector: $$ \nabla (\rho u) - (\nabla \rho)u - u(\nabla \rho) $$ My book writes this in index notation as $$ \partial_\alpha(\rho ...
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0answers
67 views

writing tensor in index notation

I have the following 3D tensor $$ T = \nabla \cdot \sum_{i}{c_ic_ic_i} $$ I would like to write this using index notation. According to my book it becomes $$ T_{ab} = \partial_y ...
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2answers
102 views

Square vs non-square tensors?

In mathematics, tensors are objects that operates on vector space. In physics or engineering, tensors usually operates on one vector space and its dual space: $V^{*} \times V^{*} \times V^{*} \times ...
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0answers
73 views

dot product between vector and matrix

In my book on fluid mechanics there is an expression $$ \boldsymbol{\nabla}\cdot \boldsymbol{\tau}_{ij} $$ where $\boldsymbol{\tau}_{ij}$ is a rank-2 tensor (=matrix). Given that ...
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1answer
234 views

Metric on an open subset of $\mathbb{R}^d$ and Christoffel symbol of the second kind

I need to prove the following identity. $\Omega \subset \mathbb{R}^d$ is open and $g$ is a metric field on $\Omega$. Further $\Gamma_{\,kl}^j$ denotes the Christoffel symbol of second kind. $g^{ij}$ ...
0
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1answer
30 views

What's the name of these transformations.

I was self-studying Spivak's Calculus on Manifolds and on page 89, two transformations $f_*$ and $f^{*}$ are defined as the following. Given a differentiable function ...
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1answer
881 views

Vectors, Basis, Dual Vectors, Dual Basis and Tensors

I'm trying to understand tensors and I know they have something to do with the basis and the dual basis of a vector space and a dual space. First I will give a concrete example to make clear what I ...
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2answers
129 views

How to generalize symmetry for higher-dimensional arrays?

@BrianM.Scott 's answer to this question Q: 3-dimensional array suggests that there is no standard concept of symmetry for 3-, 4-, N-dimensional arrays, in constrast to the case for 2-D arrays, as in ...
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1answer
184 views

Putting Maxwell's Equations in Tensor Form. (Carroll Chapter 1 Question 11)

Simply put, if you look at https://en.wikipedia.org/wiki/Electromagnetic_tensor#Significance it says you can go from the traditional four "vector calculus" maxwell equations to two tensor Maxwell ...
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1answer
292 views

Trace of tensor product

I am currently struggeling with the following problem: Imagine that you have 4 vectors $v_1,v_2,v_3,v_4$ in $\mathbb{R}^2$ and then you want to calculate $\hbox{trace}(v_4 \otimes v_3 \cdot ...
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1answer
122 views

Tensor Property

Show $$ (Q^3)_{ij}-\frac{1}{2}Tr(Q^2)Q_{ij}-\det(Q)\delta_{ij}=0 $$ where Q is a real symmetric traceless tensor. $δ_{ij}$ is Kronecker delta symbol which is 1 if $i=j$ or $0$ otherwise. We can ...