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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 ...
Jack's user avatar
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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 ...
Awe Kumar Jha's user avatar
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1 answer
40 views

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 ...
liyiontheway's user avatar
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12 views

Kronecker tensor product preserving operations

Suppose I have two tensors, of order $m$ $$ A = \sum_{i=1}^p u_i \otimes \cdots \otimes u_i, \qquad B = \sum_{i=1}^p v_i\otimes \cdots \otimes v_i. $$ What are the class of transformations $T(u_i) = ...
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Changing coordinates of $2$nd order partial operators

Let's work in $\mathbf R^n$. If we want to change coordinates $\mathbf x\to\mathbf r$, with them related like $$ \mathbf x=\mathbf x(\mathbf r) $$ Then, the second order generic partial operator in ...
Conreu's user avatar
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1 vote
1 answer
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Coordinate transformations of a scalar field

I'm very confused about coordinate transformations and the statement that scalar fields remain unchanged under coordinate transformations. Consider a coordinate transformation $$ x \rightarrow x' $$ ...
bennnn's user avatar
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1 answer
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Exploring the relationship between Tensorial Techniques and Coordinate-Independence?

I am studying Differential Geometry and Riemannian Geometry.I need some help from my stack exchange community members in this regard. Recently I happened to get a lecture series on Riemannian Geometry ...
Kishalay Sarkar's user avatar
1 vote
3 answers
48 views

endomorphism and tensors

Can someone explain to me how an endomorphism can be related to a tensor $T_{1,1}$? I don't understand how a tensor, which a priori takes a linear functional and a vector and produces a scalar, can be ...
JL14's user avatar
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0 answers
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Gradient of Lagrangian function (tensor calculus)

I have the following lagrangian function: $\mathscr{L} = \frac{1}{2} \sum_{ij} u_i K_{ij} u_j - F_i w_i + \frac{1}{2} \sum_{ijk} E_k~ \hat{g}_{kij}~ \partial_i u_j + \frac{1}{2} \sum_{ijk} E_i~ \hat{...
user134613's user avatar
3 votes
2 answers
91 views

Dual basis of a matrix space

I am considering the following basis $B$ of $S_2$: $$B=\left\{\begin{pmatrix}1&0\\0&0\end{pmatrix},\begin{pmatrix}0&1\\1&0\end{pmatrix},\begin{pmatrix}0&0\\0&1\end{pmatrix}\...
JL14's user avatar
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$(p, q)$ tensors and multidimensional arrays

I am trying to understand connections between different interpretations of tensors. In many contexts, tensors are treated simply as multidimensional arrays. Let us consider the following example. Let $...
mathslover's user avatar
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Simplify $\nabla_{\nabla_{[W,X]}Y-\nabla_W\nabla_XY+\nabla_X\nabla_WY}Z[f]$

I'm trying to simplify the formula as in the title. I know that $$R(W,X,Y,Z)=\langle\nabla_{[W,X]}Y-\nabla_W\nabla_XY+\nabla_X\nabla_WY,Z\rangle$$ by the definition. Then I may conclude that $$\left(\...
一団和気's user avatar
2 votes
0 answers
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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 $M$ ...
ProphetX's user avatar
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0 answers
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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 ...
pfloutch's user avatar
2 votes
3 answers
477 views

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: ...
Christian S.'s user avatar
1 vote
2 answers
45 views

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 ...
Unknown x's user avatar
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0 answers
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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 ...
amateur's user avatar
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0 answers
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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 ...
Aidan Beecher's user avatar
2 votes
0 answers
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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 ...
Raoní Cabral Ponciano's user avatar
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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 ...
Bremen000's user avatar
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0 votes
1 answer
55 views

Constructing a basis for a tensor product

Let $V$ and $W$ be vector spaces of a field $\mathbb{K}$. Let $\left\{v_i : i \in \mathcal{I}\right\}$ be a basis of $V$, and $\left\{w_j : j \in \mathcal{J}\right\}$ be a basis of $W$. Let $V \otimes ...
mathslover's user avatar
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0 answers
26 views

Derivative with respect to a derivative -- is covariant or contravariant?

I use the common physics notation "$,\nu$" to mean $\partial/\partial x^\nu$. Since $$\frac{\partial}{\partial x^{\nu'}} = \frac{\partial}{\partial x^\mu} \frac{\partial x^\mu}{\partial x^{\...
Khun Chang's user avatar
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0 answers
11 views

derivative of the inverse transpos of a second order tensor with itself

consider $\boldsymbol{F}$ is a second order tensor, and $\boldsymbol{F}^{-T}$ the invers traspos of the same tensor. Is it correct to take the derivative of the $\boldsymbol{F}^{-T}$ with respect to ...
Khoder Alshaar's user avatar
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0 answers
57 views

Confusion over tensor definition of exterior power of a vector space and exterior algebra

I am new to and currently learning about Tensor Algebra and Exterior Algebra. I am confused about the definition of the exterior power of a vector space $V$, $\textstyle \bigwedge^k (V)$, and the ...
Christian S.'s user avatar
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0 answers
36 views

Determining the Signature of a Metric Tensor Given a Set of $g-$null Vectors

I am currently working through the following worksheet on metric manifolds and am having trouble understanding the solution for part of the following question: Let $g$ be a symmetric $(0,2)-$tensor ...
Taylor Rendon's user avatar
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0 answers
19 views

derivation of a third-order normal component tensor of a fourth-order tensor

I have the following constitutive material fourth-order tensor $\boldsymbol{\mathcal{C}}$ \begin{equation} \boldsymbol{\mathcal{C}}= \frac{\partial \boldsymbol{P}}{\partial\boldsymbol{F}}, \quad ...
Khoder Alshaar's user avatar
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1 answer
39 views

Double Trace of the tensor product of the metric tensor with vector fields.

So I am currently preparing for an exam on General Relativity and while reading the notes I stumbled upon this: $$ tr[tr[g \otimes X \otimes Y]]= g(X,Y) $$ Where $$ g=g_{ij} dx^{i}\otimes dx^{j} $$ is ...
Geotrael's user avatar
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Werner Greub's formulation of the Universal Property of the Tensor Product

$\def\id{\operatorname{id}}$ $\def\Im{\operatorname{Im}} $In Section 1.4 of Multilinear Algebra Werner Greub starts with a bilinear map $\otimes: E \times F \rightarrow T$ where $E,F,T$ are vector ...
Ted Black's user avatar
  • 896
2 votes
1 answer
60 views

Mechanics of a contraction on the Kronecker product matrix

As a follow-up question to this one: According to Wikipedia is the bases vectors are fixed the two tensors $A$ and $B$ result in the tensor product given by the Kronecker multiplication: $$A = \begin{...
JAP's user avatar
  • 597
1 vote
1 answer
51 views

Apparent dimensional problem when calculating a tensor contraction with matrices

According to Wikipedia is the bases vectors are fixed the two tensors $A$ and $B$ result in the tensor product given by the Kronecker multiplication: $$A = \begin{bmatrix} a_{1,1} & a_{1,2} \\ a_{...
JAP's user avatar
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1 vote
0 answers
63 views

Express a diagonal $3 \times 3$ matrix using the (powers) of traces

I have a $3 \times 3$ diagonal matrix which can be written as \begin{equation} \bf{A} = \begin{bmatrix} A_1 & 0 & 0\\ 0 & A_2 & 0\\ 0 & 0 & A_3 \end{bmatrix} \end{equation} I ...
Maozhu Peng's user avatar
0 votes
0 answers
34 views

Equivalent definitions of tensor power of a vector space

I have two definitions of the tensor power $T^nV$ of a vector space $V\in \bf{kVect}$. For every $n$-multilinear map $f:V\times ...\times V\rightarrow W$, there exists a unique $\bar{f}:T^nV\...
Wyatt Kuehster's user avatar
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0 answers
25 views

Divergence and rotational calculation for cylindrical coordinates with tensor calculus

Hi I'm asked to use the following relations (which I have demostrated before) in order to calculate div and rot in cylindrical coordinates: (where X is a vector field) $$div X = (-1)^{d-1}\nabla_\...
Guillermo Fuentes Morales's user avatar
0 votes
1 answer
36 views

Tensor chain rule

Say if there is a function $f$ that maps a real number $x\in\mathbb{R}$ (some parameter) to a tensor $\mathbf{y}\in\mathbb{R}^{3\times256\times256}$ (some rgb image data). Then there is another ...
Scanners's user avatar
  • 310
4 votes
1 answer
286 views

Linearity of differential forms

I put my hands on "Linear Algebra" by Serge Lang, second edition, and I noticed that it contains some sections that were later removed in the following third one. In one of the removed parts ...
Andreas Compagnoni's user avatar
0 votes
1 answer
93 views

Abstract index notation - can't understand identity $(dx^{\mu})_aT^b\partial_bv^a=T^b\partial_b[(dx^{\mu})_av^a]$

I'm trying to get familiar with abstract index notation. Came across this identity: $$(dx^{\mu})_aT^b\partial_bv^a=T^b\partial_b[(dx^{\mu})_av^a]=T^b\partial_bv^{\mu}$$ where $T$ and $v$ are vector ...
Shirish's user avatar
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0 votes
0 answers
27 views

The almost product structure of the cartesian square of a smooth manifold

I suspect that the cartesian square of a manifold $M$ admits a "canonical" almost product structure, that is, a smooth $(1,1)$-tensor on $M\times M$ whose square is the identity. But I need ...
Parco Macelli's user avatar
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0 answers
24 views

Show that the conformally complete Schwarzschild spacetime is asymptotically flat at null infinity

I am trying to show that the conformal factor used to conformally complete the Schwarzschild spacetime renders it asymptotically flat at null infinity (according to the mathematical definition given ...
darkside's user avatar
  • 589
0 votes
1 answer
44 views

trace of a matrix squared:a formula

Can someone explain to me why $$\operatorname{Tr}K^2=\sum_{\alpha,\beta} K_{\alpha\beta}K^{\alpha\beta}$$
user122424's user avatar
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0 votes
0 answers
24 views

Can the wedge product acting on continuous functions be written differently?

I have $M=\mathbb{R}^2$, $N=\mathbb{R}^3$ $$F(\theta,\phi)=\big((\cos\phi+2)\cos\theta,(\cos\phi+2)\sin\theta,\sin\phi\big)$$ $$\omega=y\text{d}z\wedge\text{d}x$$ Calculating: $$F^*\omega=F^*(y\text{d}...
Superunknown's user avatar
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1 vote
0 answers
49 views

Compute d$\omega$ in Cartestian coordinates for a given $\omega$

Define a $2$-form $\omega$ on $\mathbb{R}^3$ by $$\omega=x\text{d}y\wedge\text{d}z+y\text{d}z\wedge\text{d}x+z\text{d}x\wedge\text{d}y$$ Compute d$\omega$. Using the formula for $$\text{d}\omega=\text{...
Superunknown's user avatar
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0 votes
0 answers
39 views

Sign error in the computations of $R_{\theta\phi\theta\phi}$ for the Schwarzschild metric

I am computing the components of the Riemann tensor for the Schwarzschild metric using the following formula $R_{\alpha\beta\mu\nu}$=$(\partial_\alpha\Gamma^l_{\beta\mu}-\partial_\beta\Gamma^l_{\...
darkside's user avatar
  • 589
3 votes
1 answer
41 views

Proving a proposition on Lie derivatives of differential forms.

Suppose $M$ is a smooth manifold, $V\in \mathfrak{X}(M)$, and $\omega$, $\eta \in \Omega^*(M)$. Then, $$\mathscr{L}_V(\omega\wedge\eta)=(\mathscr{L}_V\omega)\wedge \eta +\omega\wedge (\mathscr{L}_V\...
Superunknown's user avatar
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1 vote
0 answers
59 views

What is the symbol $\mathfrak{X}$ for smooth manifolds describing?

In Lee's "Introduction to Smooth Manifolds" the symbol $\mathfrak{X}$ arises after chapter 12, particularly on page 318: d) If $X_1,\dots, X_k\in \mathfrak{X}M$, then the function $A(X_1,\...
Superunknown's user avatar
  • 2,973
6 votes
2 answers
210 views

Derivatives of the determinant of a singular matrix w.r.t the matrix

I have a $3 \times 3$ matrix $\bf{A}$, and want to find the second order derivative of its determinant w.r.t the matrix itself $\frac{\partial ^2 \det({\bf{A}})}{\partial {\bf{A}} ^ 2}$. Everything I ...
Maozhu Peng's user avatar
0 votes
1 answer
50 views

General way to find derivative of tensor operation.

Suppose a domain $D_S\subseteq \mathbb {N}^n$ of dimension $n$ and of shape $S\in\mathbb {N} ^n$ is the index set $$ D_S=\{(d_0,\ldots,d_{n-1}) \in \mathbb {N_1} \times \ldots \times \mathbb {N_n} \ | ...
Iain's user avatar
  • 191
0 votes
1 answer
60 views

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 $g_{ijml}=-g_{ijlm}$? ...
BitterDecoction's user avatar
2 votes
0 answers
19 views

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 ...
graphitump's user avatar
1 vote
0 answers
41 views

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 ...
Patrick's user avatar
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0 votes
0 answers
14 views

What if a CP/PARAFAC tensor decomposition can be further decomposed, "recursively"?

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 $(...
graphitump's user avatar

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