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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1answer
19 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
28 views

the gradient of the product of a scalar by a vector [on hold]

I have the following expression (gradient of scalar multiplied by a vector) $$\vec{\nabla}\left(a\cdot \vec v \right) = ?$$ where $a$ is a scalar.
3
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1answer
36 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
36 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 $\mathbb{R}^{n+1}$ ...
5
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1answer
38 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 ...
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0answers
15 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
24 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
22 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
25 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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0answers
21 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
9 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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0answers
18 views

Ricci Curvature in Terms of Christoffel Symbols

$$\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
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1answer
25 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
40 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
18 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
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0answers
29 views

Help explain “3d algebra”

The following is part of my lecture note, but I get lost after the first paragraph. I know what is "even" permutation and "odd permutation" which I learned from my abstract algebra course, and figured ...
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0answers
42 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
31 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
37 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 ...
1
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1answer
31 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
38 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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0answers
41 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 ...
0
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2answers
33 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
13 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
27 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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0answers
19 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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0answers
25 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 ...
0
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1answer
63 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 ...
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0answers
30 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? $${g^{mr}g_{rn}=\delta^{m}_{n}}$$
4
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1answer
79 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 ...
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0answers
37 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 ...
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0answers
20 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 ...
2
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1answer
62 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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0answers
44 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 ...
0
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1answer
23 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 ...
2
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0answers
28 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
32 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 ...
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0answers
29 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 ...
1
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1answer
27 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} ...
5
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1answer
68 views

Coordinate-Free Definition of Trace.

$\DeclareMathOperator{\tr}{trace}$ I am reading the wikipedia article on the trace operator. The section titled Coordinate-Free Definition defines the trace as follows. Let $V$ be a finite ...
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2answers
19 views

The Riemannian Curvature in a solid sphere

Is the Riemannian Curvature at the centre of a solid sphere zero?
2
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1answer
32 views

Couple stress tensor reference.

Can someone give me a good mathematical reference for couple stress tensor in its most basic form. Thank you.
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0answers
24 views

Natural bilinear map $B\colon Alt^p(E^*)\times Alt^p(E)\rightarrow\mathbb R$

$Alt^P(E^*):=\{ u\colon \overbrace{E^*\times\cdots\times E^*}^{p- times}\rightarrow \mathbb R\ \ , u \text{ is alternating multilinear map}\}$ $Alt^P(E):=\{ \alpha\colon ...
2
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1answer
38 views

Defining a partial derivative with respect to an antisymmetric tensor/matrix

I'm looking at some nonlinear electrodynamics, and have been following a textbook which contains a primer on some of the stuff I'm interested in following up. However, I seem to have fallen at the ...
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1answer
46 views

Components of a vector product as an antisymmetrical rank 2 tensor

So I'm reading Landau and Lifshitz' Theory of Elasticity (https://archive.org/details/TheoryOfElasticity) and they have done, among (many) other things, something I simply don't understand. On page ...
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0answers
36 views

The space of alternating multilinear maps and existence of a bilinear map [duplicate]

Before, I ask similar this. But here I change question settings since it was incomplete. I hope receive good ideas. Let $E$ be a finite dimensional vector space over field $\mathbb R$ with $E^*$ as ...
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2answers
60 views

Differential at a point and differential (Differential Geometry)

Given $f\in C^\infty(U)$, $U$ open set of $\mathbb{R}^n$, we define the differential of $f$ at $p$ $$ (df)_p:T_p\mathbb{R}^n\to\mathbb{R}\\ (df)_p(v):=v(f) $$ and the differential of $f$ $$ df:U\to ...
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1answer
29 views

Some calculations with skew forms and wedge product

I have some problems with the language of multilinear forms. I have to prove that if $dim(V)\le 3$, then every $\omega\in\Lambda^q(V^\ast)$ is such that $\omega\wedge\omega=0$. I consider the case ...
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0answers
27 views

$G$-structure defined by a tensor

Let $M$ be an $n$ dimensional manifold with its bundle of linear frames $\pi:L(M)\to M$. Suppose $T_0$ is a tensor on $\mathbb R^n$ and $u\in L(M)$. We may view $u$ as a linear map $u:\mathbb R^n\to ...
1
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1answer
36 views

How can point group symmetry operations be used to reduce the number of independent crystal properties?

How can point group lattice symmetry operations be applied to reduce the full second-rank elasticity tensor (in Voigt notation) from: to, for example, in the cubic case, this: A reference would ...