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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-2
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1answer
40 views

Questions related dual vector and covector through a example.

I am finding dual vector very confusing to understand. I would like to understand it using example: Let $V=\Bbb{R}^3$, $v=\begin{bmatrix}2\\3\\5\\ \end{bmatrix} ∈ V$ and $f_1, f_2, f_3 ∈ V^*$ are ...
2
votes
1answer
69 views

Exterior derivative commutes with postcomposition by symmetric multilinear functionals?

Let $\frak{g}$ be a finite-dimensional real Lie algebra, $\varphi: \bigotimes^l \frak{g} \to \mathbb{R}$ a symmetric multilinear functional, and $\psi \in \Omega^k(M; \bigotimes^l \frak{g})$ a ...
0
votes
0answers
82 views

What is dual vector and covector?

I tried to learn dual vectors online but failed to exactly understand it, I know that it could be understood using change of basis. Below is a example for change of basis, kindly help me with this. ...
1
vote
2answers
31 views

Varying Order of Covariant Differentiation

Does the order of covariant differentiation matter? Will E$_i$$_j$,$_k$$_l$$_t$ =E$_i$$_j$,$_l$$_t$$_k$ ? Does it matter if the tensor E$_i$$_j$ is continuously differentiable?
0
votes
0answers
25 views

Linear transformation and riemannian metric tensor

When is it true that for a linear transformation $A$ and a symmetric bi-linear form $g$ (i.e. (0,2) tensor) on a riemannian manifold: $$ g( AV, W)= g(V, AW)?$$
0
votes
0answers
17 views

Symmetry and Skew-Symmetry in covariant derivatives

Okay,so, E$_i$$_j$ is a rank 2 symmetric tensor. So, E$_i$$_j$= E$_j$$_i$ So, when there is, E$_i$$_j$,$_k$$_l$$_t$ where the comma indicates covariant differentiation with respect to the indices ...
10
votes
3answers
633 views

Qualitatively, what is the difference between a matrix and a tensor?

Qualitatively (or mathematically "light"), could someone describe the difference between a matrix and a tensor? I have only seen them used in the context of an undergraduate, upper level classical ...
2
votes
3answers
74 views

What are the dual basis vectors?

What exactly are dual basis vectors such as those which arise in non-orthogonal co-ordinate systems? What is their physical interpretation. Please note, I don't know much tensor calculus yet. I am ...
1
vote
1answer
65 views

How do you prove this tensor identity?

Suppose we have a tensor field ${\Gamma_{ab}}^{c}$ on a smooth manifold $M$ such that for all vector fields $X^a$, we have $X^aX^b{\Gamma_{ab}}^{[c}X^{d]} = 0$. How do you show that there exists a ...
0
votes
1answer
75 views

Curiosity about the wedge product and the Levi-Civita symbol

Let us use the definition of the wedge product of two vectors: $$\vec{u}\wedge\vec{v} = \vec{u}\otimes\vec{v} - \vec{v}\otimes\vec{u}$$ writing $\vec{u}$ and $\vec{v}$ in dyadic form as $\vec{u} = ...
0
votes
1answer
38 views

Fixing some notations about tensor calculus

I'm studing some articles about Ricci Flow and Ricci Solitons. I realize the first thing to do is choose a notation, thus I'm using the Huai Dong Cao's notation (which is the same Ricgard Hamilton's ...
1
vote
0answers
16 views

Integrating a tensor over the whole of $R^3$

I want to find $\int_{R^3}r^{-3}\exp^{-\lambda r^2}x_ix_jdV$ where $r = x_kx_k$. I had the idea to translate that into finding $\delta_{ij}\int_{R^3}r^{-1}\exp^{-\lambda r^2}dV$ However, when I put ...
0
votes
1answer
22 views

Integrating the alternating tensor over a sphere

I'm having some trouble working out how to evaluate $\int_S\epsilon_{ijk}x_kdS$, where S is the unit sphere. My thoughts are that $x_kdS$ is the surface element, but the combination of that with ...
0
votes
0answers
33 views

Integrating a particular tensor

I have a tensor defined by $T_{ij} = \delta_{ij} + \epsilon_{ijk}x_k$, and I want to find $\int_ST_{ij}dS$, where S is the surface of the unit sphere. I'm having some problems because I really don't ...
0
votes
0answers
12 views

Simpliication of Tensor operations

Is it possible to obtain $F(n,E)$ such that $$(n_k E_{kj} n_j ) n_i u_i + n_k E_{ki} u_i = F(n,E) n_i u_i$$ where $n$ is a unit vector in an unknown direction, $E$ is a symmetric tensor (specifically ...
0
votes
1answer
22 views

Tensor algebra problem from the derivation of covariant derivative

I was reading a way of deriving the covariant derivative (using (1,1)-tensor) and found it hard to understand one specific operation. $${\frac{\partial a^{k}_{m}}{\partial y^{p}}=\frac{\partial ...
0
votes
0answers
36 views

LeviCivita identity and the vector cross product

How can I verify the identity $\sum_{k} \epsilon _{ijk} \epsilon _{mnk} = \delta _{im}\delta _{jn}-\delta _{in}\delta _{jm}$ ? How can I use this identity to calculate $(\vec{A}\times \vec{B})\cdot ...
2
votes
2answers
38 views

Are four vectors in Special Relativity considered to be tensors?

In particular, I would like to know if the four velocity and the four acceleration are tensors.
1
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0answers
54 views

Lowering and raising tensor indices without using metric tensor

Is it possible to lower and raise tensor indices using non-metric tensor? For example if we define non-singular tensor ${a_{ij}}$ such that ${a_{ij}a^{jk}=\delta^{k}_{i}}$ and it is not metric tensor. ...
3
votes
1answer
58 views

Abstract formulation of the Riemann-tensor in index notation

Let $M$ be a smooth manifold and let $g$ be a Riemannian or pseudo-Riemannian metric on it, with $\nabla$ being the associated Levi-Civita connection. The indices $a,b,c,d...$ denote abstract indices ...
0
votes
0answers
37 views

How to prove a symmetric tensor is indeed a tensor?

Our professor defined a rank (k,l) tensor as something that transforms like a tensor as follows: $$T^{\mu_1' \mu_2'...\mu_k'}{}_{\nu_1'\nu_2'...\nu_l'} ~=~ ...
1
vote
1answer
42 views

Tensors and suffix notation

I'm just looking for an explanation as to why $R_{ip}R_{iq} = \delta_{ij}$ $\\ $ Here R is a rotation, and is orthogonal, and $det(R) = 0$. One of the explanations I've seen is that $R_{ip}R_{jp} = ...
2
votes
0answers
65 views

Integration is only possible for forms and not for general tensors?

Integration is only possible for forms and not for general tensors? What is the true reason for this? Or can integration of $k$-forms be extended in some natural way to arbitrary $(k,l)$ tensors;if so ...
5
votes
2answers
74 views

Tensor product.vector space,equivalent definitions

Let $V$ be a real vector space. There are 2 definitions of $V \otimes V$. One is the set of all multilinear maps $L(V^*,V^*,R)$,and the other is the qutionet group $G/H$,where $G$ is free abelian ...
1
vote
1answer
50 views

The proof of a Riemannian metric as a $(0,2)$ tensor

I was told that the Riemannian metric is a $(0,2)$ tensor. I have trouble understand this. I know very little geometry, I learnt that the Weingarten map of a hypersurface is a linear map from ...
1
vote
0answers
44 views

What exactly is the Tensor Property? And how can i check if something is a Tensor?

Dear StackExchange users, i have heard that there is something like a Tensor Property, but what exactly is that and how can i check if something is a Tensor or not? For example my book on fluid ...
4
votes
1answer
52 views

Does bounded scalar curvature imply bounded Ricci curvature?

Does bounded scalar curvature imply bounded Ricci curvature? It is trivial to show the converse, but I do not know whether the above is true. Inspired by a vaguely similar question, I am thinking ...
2
votes
0answers
80 views

Identities involving covariant derivatives

Is there an identity that says for a tensor of rank $4$, if we cycle the indices, including the index with respect to which its covariant derivative is taken, will the sum of all those quantities ...
0
votes
1answer
55 views

Tensor notation problems

I don't understand this thing: let's assume I have this equation: $$\partial_{\rho} C_{\rho\rho} = 0$$ My professor said: "There is an index of too: two out of the three indexes do contract ...
3
votes
1answer
135 views

Vectors, Forms, Multivectors, and Tensors

In researching some of the ways that vectors (and vector fields) generalize I find that there are apparently many different objects that generalize them -- matrices, differential forms/ covectors, ...
0
votes
1answer
38 views

Curl of Curl of Rank 2 Covariant Tensor

Can anyone please tell me the expression for the curl of curl of a rank 2 covariant tensor? I've been going through a lot of books and sources and have not found an exact expression.
0
votes
0answers
35 views

Bijection between tensors and permutations (in linear $O(n)$ time)

The number of permutations of the set $S=\{1, \dots, n\}$ is $n!$, or in other words the permutation group $S_n$ has $n!$ elements The number of tensor components of a tensor in $n$ dimensions ...
0
votes
0answers
17 views

Covariant Differentiation

Does it mean anything when you take a covariant derivative of a rank 2 tensor twice? I know that the first covariant derivative of a rank 2 tensor shows the rate of change of the tensor in a changing ...
1
vote
0answers
51 views

Identity regarding the components of a dual basis

This problem is from Robert Wald's "General Relativity." The problem is 4(b) from chapter 2. Let $Y_1\cdots Y_n$ be smooth vector fields on an $n$-dimensional manifold $M$ such that at each $p\in M$ ...
3
votes
1answer
38 views

Determining the value of A given $Z_4=Z_8\oplus Z_2/A$

Let Z define the integers and $Z_a$ define the integer group modulo a. I want to determine what A is. Given $Z_4\cong Z_8\oplus Z_2/A$, where $A\subset Z_8\oplus Z_2$, am I able to just say that ...
1
vote
1answer
66 views

Metric tensor for n-sphere in ambient coordinates

Let $S^n$ be the unit n-sphere embedded in $\mathbb{R}^{n+1}$: $$ S^n = \{ a \in \mathbb{R}^{n+1} \mid a \cdot a = 1 \} $$ What is the induced metric tensor for the sphere, in $\mathbb{R}^{n+1}$ ...
6
votes
1answer
62 views

Ricci curvature along Killing vector field

If $V$ is a Killing vector field, I need to prove that $$V^{m}\nabla_{m}R = 0$$ where $R$ is the Ricci scalar $R = g^{mn}R_{mn}$. I´m having some trouble with this, I already showed that ...
0
votes
0answers
26 views

Tensors, indices and matrix notation - is there a common convention?

For a tensor named T with two indices, there are four possibilities: $T_{ij}$ , $T_i^{\ j}$, $T^i{\ _j}$ and $T^{ij}$. Is there a common convention as to how these tensors would be represented as ...
1
vote
1answer
161 views

Christoffel Symbols in Flat Space-Time

Does it make sense to talk about Christoffel symbols in flat space time? Do they have non-zero values? I understand that the Christoffel symbols appear as an indication of curvature in space. So, are ...
2
votes
0answers
32 views

Border rank of tensors

Can anyone help me find the rank and border rank of the following tensor: \begin{align} T=a_{11}\otimes b_{11}\otimes c_{11}+a_{12}\otimes b_{21}\otimes c_{11}+a_{11}\otimes b_{12}\otimes c_{12}\\ ...
0
votes
1answer
50 views

Is every tensor the gradient of a vector?

I guess no, since every vector is not the gradient of a scalar - I assume ! Please confirm or guide in this regard .
0
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0answers
48 views

Strassen's Laser Method Technique AND Tensors in matrix multiplication algorithms

I understand the first algorithm presented by Strassen in 1968, for fast matrix multiplication. This was the first improvement to the naive approach of multiplying matrices. Thereafter, he went on to ...
1
vote
0answers
17 views

Ricci Contraction

Is Ricci Contraction different from ordinary Contraction or are they the same? I understand that in ordinary tensor contraction, you contract( equate two indices) one index of the tensor with respect ...
1
vote
1answer
41 views

In the formula for Ricci curvature, do the terms $\Gamma^{\ell}_{ij}\Gamma^m_{\ell m} - \Gamma^m_{i\ell}\Gamma^{\ell}_{jm}$ cancel each other out?

$$\large R_{ij} = R^{\ell}_{i\ell j} = g^{\ell m}R_{i\ell jm} = g^{\ell m} R_{\ell imj} = \frac{\partial\Gamma^{\ell}_{ij}}{\partial x^{\ell}} - \frac{\partial ...
2
votes
1answer
89 views

Matrix representations of tensors

I've been trying to teach myself general relativity, and I always get stuck at the same point: I don't really understand what the metric tensor is. Unless I'm incorrect, and please correct me if I'm ...
2
votes
2answers
62 views

Index notation interpretation for matrices

I want to understand the how to interpret the matrices which are represented by index notation. Here is my matrix $𝜎_{𝑖𝑗}+𝜎_{𝑖𝑘}𝑤_{𝑘𝑗}−𝑤_{𝑖𝑘} 𝜎_{𝑘𝑗}$ All the matrices in the equation ...
2
votes
2answers
31 views

Matrix tensor indices

Suppose I have an orthonormal basis $$B = \left \{ u_{i} \right \}_{i=1}^{\infty}$$ Then for a matrix $K$, do I represent it as $$K = \sum_{j,k=0}^{\infty}k_{jk}\left ( u_{i}\bigotimes u_{j} ...
0
votes
0answers
85 views

Exponential map on a sphere in spherical coordinates

Let $M = \{ (x^\varphi, x^\theta) : x^\varphi \in [0, \pi), \thinspace x^\theta \in [0, 2\pi) \}$ be a manifold with metric $\mathrm{d}s^2 = (\sin x^\varphi)^2 (\mathrm{d} {x^\theta})^2 + (\mathrm{d} ...
2
votes
1answer
56 views

Tensors in matrix multiplication algorithms

Fast matrix multiplication algorithms, be it the Winograd and Coppersmith algorithm or any further improvement of it, extensively use tensors. In fact, the entire construction is based on tensor ...
2
votes
1answer
41 views

The Differential Geometry of a 2-D Surface

I'm currently self-studying the differential geometry of embedded surfaces. My question is, how am I to chose the appropriate coordinates and derive the covariant basis for the surface I'm interested ...