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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2answers
117 views

Curl, $\vec\nabla \times (\hat{a}\times \vec{b})$

EDIT: FIXED TYPOS & Deleted most of my wrong work pointed out by others. Calculate the curl of $f(\vec r,t)$ where the function is given by $$ f(\vec r,t)=- (\hat{a}\times \vec{b}) \frac{e^{i(c ...
3
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1answer
122 views

The Curvature Tensor

I present three different ways I've seen the Riemann curvature written: $R(X,Y)Z=D_XD_YZ-D_YD_XZ-D_{[X,Y]}Z$ $R(e_c,e_d)e_b=D_{e_c}D_{e_d}e_b-D_{e_d}D_{e_c}e_b-D_{[e_c,e_d]}e_b$. $R^{\rho}_{\space ...
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0answers
28 views

Is there a name for this type of tensor rank?

Let $A\in\mathbb{R}^{n_1\times n_2\times n_3 \times n_4}$ be a tensor. Suppose that $k$ is the minimum integer there exist matrices $X_1,\ldots,X_j\in\mathbb{R}^{i_1\times i_2}$ and ...
1
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1answer
41 views

Proving that the Moment Tensor is super-symmetric

The Carathéodory theorem in the image bellow is the one about convex hull, isn't it? Would you please explain why can the tensor F be rewritten as that sum? From that representation the author ...
1
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1answer
803 views

Trace of tensor product vs Tensor contraction

I have come across various sources that talk about traces of tensors. How does that work? In particular, there seem to be such an equality: $$ \text{Tr}(T_1\otimes ...
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0answers
33 views

Tensor notation and the “Zero-Value Theorem”

In the following picture: taken from Martin Sadd's book on elasticity, I am having trouble understanding the "zero-value theorem". I can't understand why this theorem is true. For example, when ...
7
votes
1answer
107 views

Tensor fields and vector bundles

Let $M$ be a differentiable manifold, $TM$ and $T^*M$ a tangent and cotangent bundle of $M$ and let $\Gamma (TM),\ \Gamma (T^*M)$ be spaces of smooth sections of $TM$ and $T^*M$. Let $T_s^r (M)$ ...
1
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1answer
82 views

tensor notation surprise

I'm trying to study tensors from several textbooks. One early example completely confuses me: Islam, Tensors and their Applications, in the "Preliminaries" chapter, gives this example (page 3, using ...
2
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0answers
76 views

Covariant vectors and Dual spaces

There are contravariant vectors and their duals called covariant vectors. But the duals are defined only once the contravectors are set up because they are the maps from the space of contravariant ...
0
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1answer
35 views

Is tensor product commutative on orthonormal basis?

In general the tensor product $\varphi\otimes\psi$ is not commutative, but I was thinking that if I have tensor product on two orthonormal bases of Hilbert spaces are they commutative i.e is ...
2
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2answers
56 views

Cartesian & Tensor Product

What is the difference between a cartesian product and tensor product of two vector spaces $V_1$ and $V_2$ defined over same field $F$ ?
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0answers
32 views

how to make geometric figure with tensor, are tensors covariant or contravariant

Heloł dear Colegues! Im curious how to make some geometric figure ie. cube, simple hull of ship like half of fish or half of cylinder, with tensor, or in more mathematical language: how to make ...
2
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0answers
118 views

Prove the Curvature Tensor is a Tensor

For an affine connection $\nabla$, prove the curvature R $R(X,Y,Z,\alpha)=\alpha(\nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z -\nabla_{[X,Y]}Z)$ with $X,Y,Z$ vector fields and $\alpha$ a co-vector, is ...
1
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1answer
36 views

derivation of rank of tensor from the product of two tensors

If $A^p$ is a first rank tensor and $A^pK^{qrs}$ is a 4th rank tensor we have to prove that $K^{qrs}$ is a tensor of rank three?.we can check here clearly that $A^p$ is first rank tensor and $K^{qrs}$ ...
2
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1answer
134 views

using the kronecker product and vec operators to write the following least squares problem in standard matrix form

I have a least squares problem with the following form: $$ \min_\mathbf{X} ~ \left\| \sum_{i=1}^n \mathbf{u}_i^\top \mathbf{X} \mathbf{v}_i - b_i \right\|^2 $$ where $\{\mathbf{u}_i\}_{i=1}^n$ and ...
1
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0answers
32 views

Isomorphism between $T^k_{l+1}(V)$ and $\mathcal{L}\left( (V^*)^k \times (V)^l\right)$.

V is a real $n$-dimensional vector space. I'm using the notation as in John Lee's Riemannian Manifolds: an Introduction to Curvature for the set of $\binom{k}{l}$ tensors and the set of multilinear ...
0
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0answers
25 views

degrees of freedom (df) of a third order tensor

Does it make sense to simply unfold the tensor into a matrix and apply the df metric used for matrices? That is, a $\ n_1\times n_2 $ matrix of rank $\ r$ has df =$\ n_1r + (n_2 - r)r$ and so, a ...
0
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0answers
64 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 ...
2
votes
1answer
51 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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0answers
70 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 ...
-2
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1answer
56 views

Is it true to say that every tensor is an element of a monoid?

If we consider that: by definition, a tensor is an element of the tensor product of two algebraic structures the most abstract algrebraic structure on which the tensor product is defined are ...
0
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2answers
55 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 ...
5
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0answers
155 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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0answers
22 views

Alternative operator is a homomorphism?

Let $V$ be a real vector space of dimension $n$, for a (real valued) tensor $f$ of order $r$, define the alternative operator $A$ by $$(Af)(v_1,\cdots, v_r)=\frac{1}{r!}\sum_{\sigma\in ...
1
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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 = ...
3
votes
1answer
226 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} ...
0
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1answer
55 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?
3
votes
1answer
208 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)\\ ...
1
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0answers
56 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 ...
2
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1answer
43 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
101 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: $A^i B_i = 2+6+12 = 20$ $A^i B_j =$ $ ...
4
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1answer
71 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
519 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 ...
0
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1answer
31 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 ...
1
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0answers
28 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: $$\frac{1}{N!}\varepsilon_{i_1\dots i_N}\varepsilon_{j_1\dots j_N}A_{i_1 j_1}\dots A_{i_N j_N} = \det A. \tag{$1$} $$
3
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2answers
86 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 ...
1
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1answer
80 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 ...
0
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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, ...
1
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1answer
39 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. ...
1
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1answer
77 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
276 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 ...
7
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1answer
595 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 ...
1
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2answers
30 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
51 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
104 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
95 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 ...
1
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2answers
351 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}^{\ ...
2
votes
1answer
69 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
52 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
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1answer
598 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 ...