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.
1,269
questions with no upvoted or accepted answers
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Vector valued 2-forms which satisfy Jacobi Identity
Motivated by this MO question we ask the following two questions:
1)What is an example of a compact manifold $M$ which does not admit any smooth (1,2) tensor $\omega$ which restriction to each ...
10
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0
answers
2k
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Linear independence of tensor product basis $\{ v_i \otimes w_j\}$ for $\{v_i\}$ and $\{w_j\}$ linearly independent.
Show that the set $\{v_i \otimes w_j\}$ is a linear independent subset of $V\otimes W$ when $\{v_i\}$ and $\{w_j\}$ are independent subsets of V and W respectively.
I want to find an error in a proof ...
7
votes
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What is the difference between tensor calculus and exterior derivative type concepts?
I am trying to clarify terms in order to help me figure out what I'd like to study.
I understand that $p$-forms and $p$-vectors are used with things like wedge products, exterior algebras, and a ...
7
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260
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Mnemonic device for relationships between Hom and Tensor
Probably this is a stupid question, but nevertheless...
Let $A$, $B$, $C$ and $D$ be rings, and $M$, $N$ and $K$ be appropriate bimodules between them. There are extremely well-known canonical ...
6
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What matrices $M$ satisfy $M_{il} M_{jm} M_{kn} \delta_{lmn} = \delta_{ijk}$?
I'm trying to better understand matrices that are defined by preserving some other tensor. I recently asked a more involved question, and I realized it might be better to learn simpler examples first.
...
6
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'Tensor Calculus' by J.L. Synge and A. Schild (1979 Dover publication) . Exercise 12 page 110.
I'm solving all the exercises of 'Tensor Calculus' by J.L. Synge and A. Schild (1979 Dover publication) . Till now everything went smooth, but now I'm stuck at exercise 12. page 110 of the third ...
6
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162
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How to express the covariant derivative on $TM$ by the covariant derivative on $M\times M$?
Let $(M,g,\Gamma)$ be a Riemannian manifold with the Levi-Civita connection. $M\times M$ and $TM$ have natural metric structrures inherited from $(M,g,\Gamma)$. Let $\phi:TM \rightarrow M \times M$ be ...
6
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616
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Integration by parts for covariant tensor fields
Let $(M,g)$ be a compact Riemannian manifold with boundary, and suppose $N$ is the outward unit normal vector field along $\partial M$. I am trying to prove the following integration by parts formula: ...
6
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What is the definition of the nuclear norm (aka trace norm, Ky-Fan n-norm) of a tensor?
What is the direct definition for the trace norm of a tensor?
By direct I mean without matricization.
Edit 1: By tensor, I mean a multi-way array, a generalization of vectors and matrices. (The ...
6
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680
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How do Riemann and Ricci tensors represent curvature?
I have started an attempt to self-study Riemannian Geometry.
I well understand all the algebraic properties of the Riemann tensor (With a symmetric connection), and how it gradually becomes less and ...
6
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319
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What are spinor fields?
For example, a tensor field $T=T_{a,b}^{\ \ \ \ c}$ on a manifold $M$ should be thought as
$$
T_{a,b}^{\ \ \ \ c}dx^{a}\otimes dx^{b}\otimes \frac{\partial}{\partial x^{c}}
$$
with local coordinate $...
6
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0
answers
398
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Extending Tensor Fields defined on Manifolds to Ambient Space
I am currently reading about tensor fields on manifolds, and I came across two comments that sound contradictory to me.
The first comment is made in the book by James Munkres "Analysis on Manifolds", ...
5
votes
1
answer
210
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Symmetric Rank-1 Decomposition for Density Matrices
Let $(H,\langle\cdot,\cdot\rangle)$ be an $n$-dimensional complex Hilbert space. For concreteness, you can just take $H=\mathbb{C}^n$ with standard inner product. Note that we will use the physicist's ...
5
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1
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How to show that extension of linear connection commutes with contraction.
I am reading John Lee's Riemannian Manifolds, and am having trouble with exercise 4.3.
In order to solve the exercise, you must show that the following definition of a connection on a tensor bundle ...
5
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Is "$:$" so-called Frobenius inner product?
I see some notation like
\begin{align*}
\int \nabla \mathbf{u} : \nabla \mathbf{v} \; dx
\end{align*}
Here I think the two vectors $\mathbf{u}$ and $\mathbf{v}$ should be column vectors, i.e. $\...
5
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Trace in Riemannian geometry
The first time I met the definition of the trace of the Ricci curvature $Ric$ on a Riemannian manifold (M^n,g), it was formulated thanks to local orthonormal coordinates: if $(x_{1}, \cdots, x_{n})$ ...
5
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0
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483
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Eigendecomposition and eigenvalue scaling for tensor networks
I've recently been interested in studying tensors and networks of tensors. Consider the following quantity:
$$Tr(A^N)$$
Where $A$ is a square matrix and $N$ is large. This is extremely easy - all we ...
5
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0
answers
241
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Examples of geometric structures on real manifold that lead to almost complex structure
I have a $M^{2n}$ real manifold. I am interested, if there are any well known geometric structures on manifold that lead in some natural manner to almost complex structure on $M.$
I read on wiki ...
5
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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 $\mathrm{d}s^2 = (\sin x^\varphi)^2 (\mathrm{d} {x^\theta})^2 + (\mathrm{d} {...
5
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Decomposition of order-$n$ tensors
If $V$ is a finite-dimensional vector space, then $V\otimes V\cong\mathbf{S}^2(V)\oplus\bigwedge^2(V)$.
The first summand on the right is the symmetric part of $V\otimes V$ and the second summand is ...
5
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295
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Elastic wave equation in curvilinear coordinates: how do you perform a coordinate change?
The essence of this question is that I don't know how to convert an equation from Cartesian coordinates into curvilinear coordinates, and would like to know how, preferably using the language of co/...
5
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461
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Is the identification between symmetric tensors and homogeneous polynomials useful?
The general question:
Given an $n$-dimensional vector space $V$ over a field $k$, there exists an identification
$$\mathrm{Sym}^d(V) \sim k[x_1, \dots, x_n]_d$$
between the space of symmetric order $...
5
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0
answers
506
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Multilinear or Tensor Regression?
Given input data $x_t\in \mathbb{R}^n$ and output data $y_t\in\mathbb{R}^m$, the closed form solution to $\min_A \sum_t \|y_t - Ax_t\|^2_2$ is given by $A = (XX^T)^{-1}XY^T$ where $x_t$ form the ...
5
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279
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Use of graph theory to determine tensor contraction ordering
I am considering using a computer program to execute tensor contractions like the following:
$\displaystyle \sum_{ij}^{o} \sum_{ab}^{v} \sum_{KL}^{X} B_{ia}^{K} B_{ia}^{L} B_{jb}^{K} B_{jb}^{L} $
...
4
votes
0
answers
90
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Combinations of simple tensors
This is a follow-up question to https://math.stackexchange.com/a/4872343
Let $X$ and $Y$ two Banach spaces and let $X\otimes Y$ their tensor product. Let $A(u)$ be the collection of all finite sets of ...
4
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0
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116
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Compute the gradient of polar basis vectors in tensor calculus
$(1)$ Compute the gradient of polar basis vectors,
\begin{equation}
\tilde{\nabla} e_\rho=\frac{1}{\rho} \widetilde{e}_\rho \otimes e_\theta \text { and } \tilde{\nabla} e_\theta=\frac{1}{\rho} \tilde{...
4
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63
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Dynamical degrees of freedom of the gravitational field
In their book on the nonlinear stability of Minkowski spacetime, Christodoulou and Klainerman state (p. 15, in the print version): "We recall that the space of dynamical degrees of freedom of the ...
4
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0
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266
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Proof that Nijenhuis Tensor is a Tensor
I have spent probably more time than I should attempting to verify this fact which is a question in Da Silva's "Lectures on Symplectic Geometry." I attempted to show it is $C^{\infty}(M)$-...
4
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0
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Is $\langle \nabla \omega, \eta \rangle \mathrm{dVol}_g=\nabla \omega \wedge \star \eta$ meaningful? How?
I know that $ \omega\wedge \star \eta=\langle \omega, \eta \rangle\mathrm{dVol}_g$. I want to know
Q1: why the following make sense? $$\langle \nabla \omega, \eta \rangle \mathrm{dVol}_g=\nabla \...
4
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0
answers
81
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Curvature of a circle using tensors
Using the tensor methods in polar coordinates, find the curvature of the circle:
$x^1=b$
$x^2=t$
with $t$ a free parameter and $b$ a constant.
I know how to obtain the curvature of any parametric ...
4
votes
0
answers
415
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is a Block matrix a Tensor?
Currently I am starting to study tensor calculus and I came across the definition of the tensor product, and more specifically the definition of tensor rank (ex. a tensor product of 2 rank 1 tensors (...
4
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0
answers
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Exterior power of symmetric product of GL(2,R) tensor representations
A paper I am reading uses an identification of $GL(2,\mathbb{R})$ representations:
\begin{align}
\Lambda^2(\text{Sym}^3\mathbb{R}^2) = (\text{Sym}^4\mathbb{R}^2 \otimes \Lambda^2\mathbb{R}^2 ) \...
4
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0
answers
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Is there a coordinate-free proof of this Lie derivative identity?
Wikipedia mentions (here and here) that the Lie derivative has the following appealing commutator: $$[\mathcal{L}_X,\iota_Y]=\iota_{[X,Y]}$$ The only way I know to demonstrate this identity relies on ...
4
votes
0
answers
461
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Hessian squared and Laplacian on a Riemannian manifold
I have been trying to understand more about the norm squared of the Hessian on a Riemannian manifold, that is
$$
|\nabla^2f|^2
$$
This quantity shows up in the Bochner formula, for instance. On $\...
4
votes
0
answers
361
views
Verify Nijenhuis tensor is a (1,2) tensor
I've been working the following problem:
If $T_i^j$ is a type (1,1) tensor field show that $$ H_{ij}^k = T_i^r
\dfrac{\partial T_j^k}{\partial x^r} - T_j^r \dfrac{\partial
T_i^k}{\partial x^r} + ...
4
votes
1
answer
393
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Geometric meaning of second Covariant Derivative
This other question exists, but it doesn't answer my question: Geometric interpretation of the second covariant derivative
I know the Riemann Tensor can be written as the commutator of the second ...
4
votes
0
answers
100
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What is a neat way to solve $\nabla\mathbf{u}+\nabla\mathbf{u}^T=\mathbf{\mathbf{C}}$?
Let $\mathbf{u}:\mathbb{R}^3\to\mathbb{R}^3$ be a smooth enough vector field that satisfies the following equation
$$\nabla\mathbf{u}+\nabla\mathbf{u}^T=\mathbf{C},\tag{1}$$
where $\nabla\mathbf{u}$ ...
4
votes
1
answer
357
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Musical Isomorphisms
I'm studying from Fecko's Differential Geometry and Lie Groups for Physicists, and in the part introducing metric tensors, Fecko introduces the musical isomorphisms between the tangent and cotangent ...
4
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What is the physical OR geometrical interpretation of the exterior derivative?
The exterior derivative of a $C^{\infty} 0-$ forms on an open set $U$ of $\mathbb R^n$ look like the total derivative of $f$ in the calculus.
My doubt-
Why did an exterior derivative of $C^{\infty} ...
4
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0
answers
805
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Showing curl curl of symmetric gradient is 0
I am trying to show that $curl\,curl\,(\mathcal E)=0$, where $\mathcal E$ represents the symmetric gradient, i.e., for vector-valued function $u(x_1,x_2)=(u_1(x_1,x_2),u_2(x_1,x_2))$
$$
\mathcal E(u)=
...
4
votes
0
answers
459
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Is this a valid way to think of decomposable $k$-forms?
Let $V$ be a finite dimensional vector space and $\eta \in \Lambda^k(V^\ast)$ be decomposable and non-zero. Then there exists covectors $\omega^1,\dots,\omega^k$ such that
$$\eta(v_1, \dots, v_k) = \...
4
votes
0
answers
389
views
Historical motivation for the tensor definition
I would like to understand more about the tensors product/tensors and apart from the definition, I think it would be useful to understand first historical use or motivation of a tensors. For example ...
4
votes
0
answers
254
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Derivative of product tensor (or matrix product )
if I define a "product tensor" of two second-order tensors as the second-order tensor:
$\boldsymbol{C}=\boldsymbol{AB}$ such that $C_{ij}=A_{ik}B_{kj}$ does the product rule applies to $\dfrac{\...
4
votes
1
answer
288
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Divergence operator of higher order and intrinsic point of view
Let $\underline{u}$ be a $1$ - order tensor (say a column vector) I want to prove that :
$\underline{\operatorname{div}} \left( (\underline{\underline{\operatorname{grad}}} \, \underline{u})^T\...
4
votes
0
answers
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Are quaternions used in tensor analysis?
At the moment I am learning tensor analysis. In the books I study (civil engineering/mechanical books on tensors) there is a lot written about rotation matrices/tensors, however quaternions are not ...
4
votes
0
answers
463
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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 (...
4
votes
0
answers
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Tensor derivatives of a second order tensor function with respect to itself
In a continuum mechanical context, let $\mathbf{A}$ be a second order tensor function.
We consider our operations in an orthonormal context
$$\mathbf{A} = A_{ij}\, \mathbf{e}_i\otimes\mathbf{e}_j\...
4
votes
0
answers
333
views
What is tensor, really?
How can one understands the definition of tensor from the purely formal point of view?
To what abstract structure this concept can be generalized?
4
votes
0
answers
92
views
Is $H^{\alpha}_{ij,k}=H^{\alpha}_{ik,j}$ for submanifolds in $R^n$?
Consider $\Sigma^n\subset {\bf R}^{n+p}$ a submanifold and $\{e_i, e_\alpha\}$ an orthonormal frame where the Greek indexes stand for the normal vectors.
Then define
$H^{\alpha}_{ij,k}=e_k\langle ...
4
votes
0
answers
565
views
Constant tensors and covariant derivatives
I seem to have trouble with an elementary computation, and figure it may help others if faced with a similar situation. The basic question is as follows: if I have a tensor field $T$ on some ...