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 ...

learn more… | top users | synonyms

2
votes
1answer
66 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 $E$ ...
2
votes
1answer
51 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 ...
0
votes
1answer
530 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 ...
1
vote
1answer
64 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 ...
0
votes
1answer
98 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 ...
1
vote
1answer
508 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 ...
0
votes
1answer
94 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 ...
3
votes
2answers
199 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 ...
1
vote
0answers
39 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
votes
1answer
111 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 ...
4
votes
1answer
161 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 ...
0
votes
1answer
32 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 ...
0
votes
0answers
59 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 ...
2
votes
2answers
95 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 ...
3
votes
0answers
69 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 ...
5
votes
1answer
208 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
votes
1answer
27 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 ...
8
votes
1answer
587 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 ...
4
votes
2answers
107 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 ...
2
votes
1answer
156 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 ...
1
vote
1answer
245 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 ...
1
vote
1answer
110 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 ...
1
vote
3answers
59 views

Tensor notation about $A^Tx$

I can express $x=x^ie_i$ and $x^T$ by $x_ie^i$. But how to express $A^Tx$ where $A=a^i_je_i\otimes e^j$? I don't think I can write as $a^j_ix^i$ or $a^j_ix^j$.
1
vote
1answer
63 views

Why is this combination of a covariant derivative and vector field a (1,1)-tensor?

I have a question regarding something Penrose says in section 14.3 of The Road to Reality. It says '...when $\nabla$ acts on a vector field $\xi$, the resulting quantity $\nabla \xi$ is a ...
1
vote
0answers
50 views

Derivative of a tensor

I have a rank-2 tensor given by $$ P_{\alpha \beta} = p\delta_{\alpha \beta} + (u_1^2, u_1u_2 ; u_1u_2, u_2^2) $$ whose derivative with respect to $x_{\alpha}$ I would like to find. According to my ...
1
vote
1answer
43 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 ...
1
vote
0answers
19 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 ...
1
vote
2answers
119 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 ...
1
vote
2answers
82 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 $\nabla _a\nabla ^af=\nabla ^c\epsilon _{cb}\nabla ^bh$ ...
0
votes
2answers
82 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 ...
1
vote
1answer
72 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 ...
4
votes
1answer
103 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 ...
2
votes
0answers
63 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 $A\circ M$ ...
0
votes
1answer
65 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} = ...
2
votes
1answer
220 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} ...
0
votes
1answer
43 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
votes
1answer
75 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?
2
votes
1answer
169 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 ...
1
vote
1answer
146 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 ...
1
vote
1answer
218 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 ...
1
vote
1answer
52 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 ...
0
votes
3answers
147 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 ...
0
votes
2answers
107 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 ...
1
vote
1answer
49 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?
0
votes
2answers
139 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 ...
1
vote
3answers
137 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?
1
vote
0answers
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 ...
0
votes
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.
0
votes
1answer
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 ...
0
votes
1answer
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 ...