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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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'} ~=~ \Lambda^{\mu_1'}{}_{\mu_1}...\Lambda^{\mu_k'...
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1answer
48 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} = (...
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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 ...
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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 ...
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1answer
59 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 $T_x\...
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0answers
81 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 ...
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1answer
64 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 ...
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83 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 ...
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1answer
58 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 ...
4
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1answer
155 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, ...
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1answer
46 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.
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0answers
39 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 $(d_1=...
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19 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 ...
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0answers
55 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 $p\in M$ ...
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1answer
40 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
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1answer
90 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 $\mathbb{R}^{n+1}$ ...
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1answer
70 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 $R = g^{mn}R_{mn}$. Iยดm having some trouble with this, I already showed that $$\nabla_{...
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29 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 ...
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1answer
362 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
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0answers
39 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}\\ ...
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1answer
82 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 .
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1answer
64 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 ...
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0answers
23 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 ...
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1answer
42 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 \Gamma^{\ell}_{i\...
2
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1answer
122 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
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2answers
66 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 $๐œŽ_{๐‘–๐‘—}+๐œŽ_{๐‘–๐‘˜}๐‘ค_{๐‘˜๐‘—}โˆ’๐‘ค_{๐‘–๐‘˜} ๐œŽ_{๐‘˜๐‘—}$ All the matrices in the equation ...
2
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2answers
33 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} \...
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0answers
114 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
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1answer
72 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
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1answer
46 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 ...
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1answer
42 views

What do tensors of second order map to?

On page 15 of James G. Simmonds book "A brief on Tensor Analysis" (chapter 1 of the first published edition), a second order tensor is described as an operator that sends vectors into vectors. On ...
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1answer
79 views

Equivalent conditions for a linear connection $\nabla$ to be compatible with Riemannian metric $g$

I am reading John M. Lee's Riemannian Manifolds: An Introduction to Curvature. In Lemma $5.2$, it is said that the following conditions are equivalent for a linear connection $\nabla$ on a Riemannian ...
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1answer
56 views

Converting tensor product from one coordinate to another

This is a long multi-steps question and I'm stuck at the last leg. I believe my question to be trivial but after 3 hrs of staring and trying all sort of methods (ridiculous ones even) I'm not getting ...
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2answers
75 views

contraction of the Riemann-Christoffel tensor

I'm attempting to prove that a particular contraction of the Riemann-Christoffel tensor is zero. I know that when the top and second of the bottom indices are contracted we get the Ricci tensor. But ...
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0answers
15 views

Basis and dual basis relationship to tensor

I'm having a very tough time understanding the solution. Are the entries of the matrix vectors instead of integers? I would really appreciate it if someone could show me how the matrix ...
0
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1answer
28 views

How to compute the following relationship with tensor notation?

I am really struggling understanding the following equalities, particularly the last one. I think the first one is using the delta function, however I do not understand why the negative sign ...
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49 views

How can obtain curvature from the gradient of a rotation tensor?

I have proper rotation function $R$ over $\mathbb{R}^3$ that yields a $3 \times 3$ tensor $R(x,y,z)$ for every point (x,y,z) in space. If I differentiate this tensor with respect to position (x,y,z), ...
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58 views

Symmetry of the second derivative

For the purposes of this question, a function $f$ is differentiable at $x\in \mathbb{R}^d$ iff (i) the directional derivative $\mathrm{D}_vf(x)$ exists for all $v\in \mathrm{T}_x(\mathbb{R}^d)$ and (...
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1answer
90 views

How to express the second fundamental form in terms of deformation second gradient

Suppose we have a surface $\Omega$ with prescribed principal curvatures, $\kappa_1$, $\kappa_2$, say. An isometric deformation ${\bf r}:\Omega\rightarrow\mathbb{R}^3$ maps the surface into $\mathbb{R}^...
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0answers
55 views

Connection between covariant and contravariant components o tensor

What is the general proof of the relation between covariant and contravariant components of a tensor using the metric tensor? $${g^{mr}g_{rn}=\delta^{m}_{n}}$$
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1answer
101 views

$Alt(T)=0$ if $T$ is a symmetric tensor

Question is to prove that $Alt(T)=0$ if $T$ is a symmetric tensor. We have $$Alt(T)=\sum_{\sigma}sgn(\sigma)T^{\sigma}$$ As $T$ is symmetric we have $T^{\sigma}=T$ for all $\sigma$. So, we have $$Alt(...
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0answers
47 views

Transformation laws for tensors on general manifolds

I was interested in tensor products rather from the mathematical, abstract point of view, so topics like various tensor products in the context of, for example, Banach spaces, $C^*$-algebras and so on....
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0answers
54 views

derivative of a tensor A with respect to transpose(A)*A?

What is the derivative of $\partial A/\partial ({A^T}A)$ ? Where $A$ is a 3x3 tensor. (in index notation, I want to find explicit components of ${D_{ijpq}} = \partial {A_{ij}}/\partial ({A_{kp}}{A_{...
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1answer
74 views

$d \times d \times d$ tensor rank vs $d \times d$ tensor rank

I am trying to understand rank of a $d \times d \times d$ tensor. The way that I understand the $d \times d$ case is that a rank $r$, $d \times d$ tensor is a tensor that can be written as the sum of ...
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45 views

Example of two modules M, N where the set of the $m\otimes n$ is not a submodule

If I remember my Algebra class correctly, this is in general not true because, for example, the addition isn't closed. So if I take: $B:\mathbb{R}^2 \times \mathbb{R}^2\to \mathbb{R}^{2^2}\cong \...
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1answer
45 views

Solution to tensor/matrix equation

I need to find a real, symmetric matrix, $A$, that satisfies: $\sum_{i,j} c_i c_j A_{ki}A_{jl} = A_{lk}$ I believe this is an equation of the form: $c^T B c = A$, where $c$ is $\mathbb{R}^{N \times ...
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0answers
34 views

Does $\mathfrak T^r(\Bbb R^m)$ count as an vector space?

Here $\mathfrak T^r (\Bbb R^m)$ denotes all the $r$-th tensors (multi-linear functions) acting upon the elements $(u_1,\cdots,u_r)$ from the product space $\displaystyle \prod^r \Bbb R^m$. And the ...
2
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1answer
59 views

Understanding Symmetric tensor field

I am reading an article in which author calls some basic tensor analysis result. He states in general we define on $\mathbb R^N$ that $$ \mathcal T^k(\mathbb R^N):=\{\xi:\,\mathbb R^N\times\cdots\...
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40 views

Jaumann deviatoric stress rate.

Being a bit cheeky as I asked this question over on Physics but didn't get a response. http://physics.stackexchange.com/questions/196393/jaumann-deviatoric-stress-rate Background about terms in this ...
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1answer
38 views

How to denote a tensor in terms of matrices product?

How to write a tensor in terms of a product of matrices? For example, I have $a \times b$ matrix $F$, and I want to create a 3D $a \times a \times a$ tensor $T$, where $T_{i,j,k} = \sum_{m=1}^{b} F_{...