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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1k views

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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237 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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268 views

Transformation law for the Ricci curvature

In 4 dimensions, for a conformal change of metric $g=e^{2u}g_0$ the Ricci curvature tensor $\operatorname{Ric}$ satisfies the transformation law \begin{equation}\tag{1} \operatorname{Ric}_g = \...
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144 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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202 views

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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765 views

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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262 views

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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207 views

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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157 views

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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198 views

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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267 views

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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428 views

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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476 views

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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735 views

Tensors as mutlilinear maps

I am aware that many books on differential geometry define tensors as multilinear maps. Namely $$ V\otimes W := L_2(V^*\times W^*,\Bbb F) $$ I am also aware that this space is isomorphic to the tensor ...
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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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51 views

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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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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475 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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202 views

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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94 views

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
161 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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967 views

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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256 views

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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1answer
238 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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2k views

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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dot product between vector and matrix

In my book on fluid mechanics there is an expression $$ \boldsymbol{\nabla}\cdot \boldsymbol{\tau}_{ij} $$ where $\boldsymbol{\tau}_{ij}$ is a rank-2 tensor (=matrix). Given that $\boldsymbol\nabla=(\...
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295 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?
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222 views

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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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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427 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 ...
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316 views

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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88 views

Is there a specific mathematical terminology for matrix/tensor with holes(missing elements)?

Consider a matrix-with-hole like below: $$ \begin{bmatrix} None & x \\ y & 0 \end{bmatrix} $$ We can define two variants of means for the above object: one is to calculate ...
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Find $T\left(\frac{\partial}{\partial x^k}, dx^l\right)$ for a tensor of type ${1}\choose{1}$ on $T_p(\mathbb{R}^n)$

So say I have the tensor of type ${1}\choose{1}$, with $T \in T_p(\mathbb{R}^n) \otimes T_p^*(\mathbb{R}^n)$ where $$T = T^{a}_{b} \frac{\partial}{\partial x^a} \otimes dx^b$$ summed over all $a, b$. ...
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99 views

Pulling back symplectic structure to $TQ$ - why is a certain term (not) zero?

Let $(Q,g)$ be a pseudo-Riemannian manifold, $(q^1,\ldots, q^n)$ be local coordinates in $Q$ and $(q^1,\ldots, q^n, v^1,\ldots, v^n)$ the induced tangent coordinates in $TQ$. I wanted to check that $$\...
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1answer
200 views

What is the name for the third-order tensor of third-order partial derivatives?

Given a sufficiently nice function $$f:\mathbb R^n\rightarrow\mathbb R$$ one can define a first-order tensor of all first-order partial derivatives in the standard way to obtain the gradient, and the ...
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57 views

Papers on Tensor factorization

I started reading a few papers on Tensors. Right now I am reading a paper by Chi and Kolda called “On Tensors, Sparsity, and Nonnegative Factorizations”. I passed linear algebra and calculus courses. ...
3
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1answer
170 views

How to think of the trace of a linear map as connecting its output back to its own input

In both string diagrams (i.e. Penrose graphical notation) and tensor index notation, the trace of a linear map has a nice representation, either as drawing a string connecting a tensor's output back ...
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1answer
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Demonstration of relation between geodesics and FLRW metric

I am reading a book of General Relativity and I am stuck on a demonstration. If I consider the FLRW metric as : $\text{d}\tau^2=\text{d}t^2-a(t)^2\bigg[\dfrac{\text{d}r^2}{1-kr^2}+r^2(\text{d}\theta^...
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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 ...
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244 views

Strain rate and Euclidian norm of the deformation tensor

I'm very confused with the the following terms. In fluid mechanics, the gradient of velocity can be written as a $3\times 3$ matrix, which can be split into the sum of two matrices, i.e., the ...
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Career advice in Math and Physics

So, I am currently doing a bachelor in mathematics, as I have found out that math is one of my great passions. However, as time has passed after graduating from upper secondary school (~High school) I ...
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229 views

Outer product of cross product vector with itself

I'm wondering if there is another, possibly more efficient, way to get to the $3 \times 3 $ symmetric matrix $\mathbf{D}$ below from 3-vectors $\mathbf{a}$ and $\mathbf{b}$ then the straight forward ...
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100 views

Deduce Geodesics equation from Euler equations

I am using from the following Euler equations : $$\dfrac{\partial f}{\partial u^{i}}-\dfrac{\text{d}}{\text{d}s}\bigg(\dfrac{\partial f}{\partial u'^{i}}\bigg) =0$$ with function $f$ is equal to : $...
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245 views

Summation convention, partial derivatives.

I've started intro to tensor calculus and I'm pracicing Einstein's summation convention. My taks is to derive the expression for third-order partial derivatives. There is a function $f(u) = F(a(u))$ ...
3
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136 views

How is Tensor Decomposition (Factorization) related to Topological Data Analysis?

I have been researching modern exploratory data analysis techniques, and came across two promising approaches: Topological Data Analysis (TDA) and Tensor Decomposition/Factorization (TF). I am ...
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185 views

Relation between the curl of a vector field and the divergence of a tensor

The following seemingly-simple problem came up when working on a problem in the fluid theory of plasmas. Given a vector field $\mathbf{A}$, find a symmetric tensor $\mathbf{P}$ such that $\...
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194 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 ...
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46 views

Apparent (minor) error in Cauchy's article on pressure or tension in a solid body

In his article De la pression ou tension dans un corps solide [On the pressure or tension in a solid body], Cauchy introduces a theory that allows to define Cauchy stress tensor. It looks like he ...
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51 views

Need help calculating entropy using huge tensors

I am no expert in tensor algebra. I am stuck with computing the following tensor product (you will recognize an entropy-like equation): $H(\mathbf{T}) = \mathbf{T}^t \cdot \log_2(\mathbf{T})$ where $...