Questions tagged [tensors]

For questions about tensor, 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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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 ...
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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 ...
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242 views

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 ...
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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 ...
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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 ...
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753 views

Is there a method of knowing which of the Christoffel symbols of second kind survive and vanish for a given metric?

I'm trying to solve a problem and the given $g_{ij}$ metric is $$\left[\begin{array}{cc}1&0&0\\0&(x^1)^2&0\\0&0&(x^1 \sin x^2)^2\end{array}\right]$$ The non-zero Christoffel ...
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147 views

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 ...
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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: ...
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1answer
172 views

Intuition of divergence and curl

There is the well known expression for the divergence of a vector field $V$ as the limit of smaller and smaller surfaces of the flux of a surface. However it occurred to me that there is another way ...
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829 views

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 $(x_{1}, \cdots, x_{n})$ ...
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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 ...
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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 ...
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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 ...
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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/...
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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 $...
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Intuition about the Riemann and Ricci tensors

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 ...
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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 ...
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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} $ ...
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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 $...
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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", ...
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Mean Gaussian Curvature using non-unit vectors.

Pg.248 of "Textbook in Tensor Calculus and Differential Geometry" by Prasun Nayak. Let us suppose that $\lambda_{h|}^i$ is not a unit vector and therefore, the mean curvature $M_h$ in this ...
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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 ...
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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 ) \...
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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 ...
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381 views

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 ...
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1answer
155 views

Isotropic tensor functions that map antisymmetric tensors to zero (Navier-Stokes derivation)

I'm going through Chorin and Marsden's derivation of the Navier-Stokes equations in A Mathematical Introduction to Fluid Mechanics. There are three assumptions made about the Cauchy stress tensor $\...
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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 $\nabla\mathbf{u}$ ...
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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) = \...
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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 ...
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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{\...
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Proving a vector identity using Einstein summation

I need to prove $$\nabla \times [(u\cdot \nabla)u]=(u\cdot \nabla)(\nabla \times u)-[(\nabla \times u)\cdot \nabla]u+(\nabla\cdot u)(\nabla \times u)$$ So the ith component of each side using ...
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1answer
232 views

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\...
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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 ...
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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} {...
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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 (...
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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\...
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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?
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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 ...
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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 ...
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An svd-like tensor decomposition splitting a tensor into two lower-dimensional tensors and a singular vector

I have also posted this question on mathoverflow, but it seems there are a lot of questions related to SVDs here and a tag "tensor-decomposition", so I will give it a shot. I am looking for ...
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1answer
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Are $(1, 0)$ tensors always vectors? (resolved)

An $(r, s)$ tensor $T$ is defined to be an element of the tensor product of a vector space and its dual: $$T \in T^r_sV := V^{\otimes r}\otimes V^{* \otimes s}.$$ However, when $V$ is finite ...
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Naturality/when diagrams commute in Commutative Algebra

As I work my way through commutative algebra (Atiyah and Macdonald), I am often confronted with the following sort of scenario. One has an exact sequence $$ 0 \to M' \to M \to M'' \to 0 $$ of modules ...
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Taking Discrete Convolution of a tensor with respect to another tensor

I have an equation $$y_\mu y_\nu=By^2\delta_{\mu\nu}+Cy^2{p_\mu p_\nu \over p^2}$$. Can I take convolution of this equation with respect to ${p_\mu p_\nu} \over p^2 $? On doing so, I get $\Sigma_{\mu \...
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Mistake in computing the evolution of $\frac{\partial}{\partial t} R_{ik}$ - where did the term $-2g^{jp}g^{lq}R_{pq}R_{ijkl}$ go? Why did it vanish?

I'm trying to prove that under the Ricci flow, the Ricci tensor evolves by the following equation: $$\frac{\partial}{\partial t} R_{i k}=\Delta R_{i k}+2 g^{p q} g^{r s} R_{p i k r} R_{q s}-2 g^{p q} ...
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1answer
158 views

Riemann Curvature of SO(3)

The problem (from Misner, Thorne, and Wheeler's Gravitation, exercise 11.12) is as follows: Calculate the components of the Riemann curvature tensor for the rotation group's manifold SO(3); use the ...
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Approximation in canonical format by rank 1 tensors

Let $I$ be a finite nonempty set and $H_i$ be a pre-$\mathbb R$-Hilbert space and $H:=\bigotimes_{i\in I}H_i$. Let $v\in H$. I would like to show that there is a $u\in H$ with $\operatorname{rank}u=1$...
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1answer
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Question on the differences in the definitions of what a tensor is

Below are the common definitions of tensor. a. "a tensor is a quantity which transforms according to a definite law under the change of the coordinate system". b. "a tensor is a multilinear function ...
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410 views

Divergence in cylindrical coordinates - Tensor Calculus way

I am currently taking a course in Electrodynamics - really beautiful physics, utterly mathematical. Because of that, I am trying to have a few mathematical barriers as possible. A big focus on the ...
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137 views

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 (...
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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} + ...

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