# 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.

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### Decomposing multidimensional matrices into contraction of 2-dimentional matrices using the singular value decomposition.

My question is easy to state, but I will state it fist for 3-dimensions because I think it is more intuitive, then n-dimensions, and then the 2-dimensional case which I have solved. I feel like there ...
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### Visualization of skew-symmetric rank-2 tensor fields

Background I was reading Einstein's The Meaning of Relativity in which he points out that axial vectors are usually used in place of rank $2$ tensors for the sake of geometrical picturisation, as in ...
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### Standard form of positive semidefinite polynomial

The standard form of a positive semidefinite quadratic polynomial of $n$ variables is a sum of n squares. Or in the language of linear algebra, a positive semidefinite symmetric matrix can be ...
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### (Necessary and sufficient) Conditions for the Ricci tensor of an affine connection to be symmetric.

Let $\nabla$ be an affine connection on a smooth manifold $M$. It is widely known, that if $\nabla$ is torsion-free, then its Ricci tensor is symmetric iff there exists a volume form $\omega$ on $M$ ...
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### Variational formulation of the vector Laplace equation in cylindrical coordinates

I want to solve the vector Laplace equation $\nabla^2 \mathbf{v}=\mathbf{f}$ in arbitrary coordinate systems using finite-elements. The usual way to derive the variational form necessary for the ...
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### Two definitions of antisymmetrization of a tensor?

I am currently learning about tensors and the exterior product, and I have found some contradictory information. I have seen some sources define the antisymmetrization of a tensor as the following: ...
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### Prove that $\det(A)=\frac{1}{3!}\varepsilon_{IJK}\varepsilon_{ijk}A_{Ii}A_{Jj}A_{Kk}?$

How do I prove for the matrix $A=(A_{ij}) \implies$ $\det(A)=\frac{1}{3!}\varepsilon_{IJK}\varepsilon_{ijk}A_{Ii}A_{Jj}A_{Kk}?$ where $\varepsilon_{ijk}$ is the Levi-Civita symbol. My attempt: I could ...
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### What is this Tensor operation?

I am trying to identify an operation that takes two matrices of dimensions $n^2$ and output a matrix of dimensions $n^2$. Every element in the output matrix incorporates all elements from the input ...
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### Confusion about integration in manifolds

I'm reading through Sean M. Carroll's text on General Relativity. In section 2.10, he explains that the volume element $d^nx$ transforms as a tensor density. Consequently, to get a "tensorial ...
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### Equivalence of Sobolev spaces Definitions on Riemannian manifold via $k$-Covariant Derivative and $k$-Gradient Derivative

I am studying Sobolev spaces on Riemannian manifolds and have encountered two different definitions in the literature. In the references: E. Hebey. Nonlinear analysis on manifolds: Sobolev spaces and ...
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### Dual of the sum of dual metrics

Suppose I have a (smooth, connected, compact) Riemannian manifold $M$ and two smooth metric tensors $g_1$ and $g_2$. I can define the metric tensor $g:= (g_1^\star+g_2^\star)^\star$ where for a tensor ...
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### permutation of indices in tensor notation

I have probably a stupid question. Somehow tensor notation often confuses me. Let's say we have the following equality: $$f_{ijk}=g_{ijml}\epsilon_{mlk}.$$ Does this imply $g_{ijml}=-g_{ijlm}$? ...
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### Degrees of freedom of an $r$-ranked tensor?

I'm trying to determine the degrees of freedom for parameters of a tensor in shape $(J_1,\dots,J_D)$ and with rank $r$, where "rank" refers to the smallest number of rank-1 tensors whose sum ...
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### Proving a property of an antisymmetric tensor $\textbf{T}$, where $\textbf{a}\cdot \textbf{T} \cdot \textbf{a}=0$.

I would like to prove that $\textbf{a}\cdot \textbf{T} \cdot \textbf{a}=0$, where $\textbf{T}$ is a second order antisymmetric tensor and $\textbf{a}$ is a vector. I know I can prove this by proving ...
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Standard CP/PARAFAC decomposition: A tensor $\mathcal{T}$ in shape $(I_1,\dots,I_N)$ is produced by $N$ matrices $\mathbf{A}^{(1)}, \dots, \mathbf{A}^{(N)}$ where each $\mathbf{A}^{(n)}$ is in shape \$(...