Questions tagged [tensors]
For questions about tensors, tensor computation and specific tensors (e.g.curvature tensor, stress tensor). Tensor calculus is a technique that can be regarded as a follow-up on linear algebra. It is a generalisation of classical linear algebra. In classical linear algebra one deals with vectors and matrices.
3,874 questions
3
votes
2
answers
86
views
How to derive Levi-civita identity: $\varepsilon_{kj}\delta_{il}-\delta_{ij}\varepsilon_{kl}=-\varepsilon_{jl}\delta_{ik}$ in 2D
I came across the following identity in 2D (and I believe in 2D only):
$$\varepsilon_{kj}\delta_{il}-\delta_{ij}\varepsilon_{kl}=-\varepsilon_{jl}\delta_{ik},$$
with $\varepsilon$ the Levi-Civita ...
- 426
0
votes
1
answer
93
views
Is it valid to set up differential geometry by starting with the structure group $GL(n,R)$?
This is the construction I have in mind : We start witha. Topological manifold $M$. We define a Principal $GL(n,R)$ bundle on it using the fibre bundle construction theorem, by specifying the ...
- 1,679
0
votes
2
answers
94
views
Problem in understanding tensor equality
I read in the book about vectors and tensors and I have a problem in understanding.
Given:
$V$ is scalar field (scalar function of $x_1$,$x_2$,$x_3$).
$x_i$ is coordinate in $XYZ$ coordinate system.
$...
- 1,161
0
votes
0
answers
28
views
How the energy-momentum tensor comes from the action in kalb-ramond gravity? [closed]
The action
\begin{align}\label{action}
S=\int d^4x\sqrt{-g}\bigg[\frac{1}{2\kappa}\bigg(R-\varepsilon\, B^{\mu\lambda}B^\nu\, _\lambda R_{\mu\nu}\bigg)-\frac{1}{12}H_{\lambda\mu\nu}H^{\lambda\mu\nu}-V(...
- 1
0
votes
0
answers
39
views
Fourier transform and the acoustic tensor
Let $C$ the fourth-order tensor of elastic constants which can be seen as as a linear transformation from Sym into Sym (matrices). We denote the action of $C$ on a symmetric matrix $A$ as $C[A]$. ...
- 8,158
1
vote
1
answer
72
views
Does the tensor $\Phi$ take this form?
We will use Einstein summation convention. I apologize if this question is too easy—it has been years since I've had to to work with some of the mentioned material.
Have the the set of all linear ...
- 1,881
2
votes
1
answer
210
views
Proving Tensor Product of Vector Spaces is a Vector Space Using Only Bilinearity and Axioms
In the arXiv paper "An introduction to tensors for path signatures" by Jack Beda, Gonçalo dos Reis, and Nikolas Tapia, they motivate the construction of direct sums and tensor products as ...
- 33
0
votes
1
answer
120
views
Cofactor of a tensor
In the book that I’m actually using for tensor algebra (second order tensors in $\mathbb{R}^3$), the author defines the cofactor of a tensor as the tensor that transforms the area vector, that is, ...
1
vote
1
answer
67
views
a problem with pushforward and pullback and Lie derivative
I have a problem with these notations:
let $\varphi:M\to N$ be a difeomorphism between the $C^\infty$ manifolds $M$ and $N$. Given $X\in{\mathfrak X}(M)$, the vector field image $\varphi\cdot X$ of $X$...
2
votes
0
answers
69
views
Covariant derivatives of tensor densities
The answer to this question proves the following:
\begin{equation}
\partial_\sigma \sqrt{-\det{\mathrm{g}}} = \frac12 \sqrt{-\det{\mathrm{g}}} \;g^{\alpha\beta}\partial_\sigma g_{\alpha\beta} \qquad (...
- 21
5
votes
1
answer
226
views
Looking for a Basis-Free Definition of a Tensor Operator in Quantum Mechanics
I have learnt in linear algebra that a tensor is defined via a multi-linear map from a vector space $V$ (its dual space $V^*$) onto the field, usually $\mathbb{R}$ or $\mathbb{C}$. In classical ...
- 133
3
votes
1
answer
51
views
Powers and roots of 2-nd order tensors in Euclidean space
I'm currently studying tensor algebra and analysis in $\mathbb{R}^3$ because I need it for continuum mechanics purposes and I'm currently focusing on 2-nd order tensors. The vector space is Euclidean ...
0
votes
0
answers
42
views
Is there any (1,3)-type tensor, whose contraction becomes the Bakry-Emery Ricci curvature $Ric_f$?
The contraction (trace) of Riemann-Christoffel curvature $R$ is the Ricci curvature $Ric$. Again, the Bakry-Emery Ricci curvature $Ric_f$ is a generalization of Ricci curvature $Ric$, which is defined ...
3
votes
1
answer
104
views
Do the Riemann tensors lie in $\text{Young}(13,24)$?
In the Wikipedia article on the Riemann curvature tensor, it is stated that:
“The algebraic symmetries are also equivalent to saying that $R$ belongs to the image of the Young symmetrizer ...
- 695
1
vote
0
answers
35
views
4th Order Tensor multiplication Rules for Sparse Regression analysis
I am working on a problem which involves working with stress and deformation tensors of the order 4. I have a set of data at different time steps for 20 cases and each element stress is 3x3 matrix, so ...
