Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [tensors]

For questions about tensor, tensor computation and specific tensors (e.g.curvature tensor, stress tensor).

1
vote
1answer
58 views

Questions on Cartan's magic formula $\mathcal{L}_X=i_X \circ d + d\circ i_X$

Algebra $A$ is called graded algebra if it has a direct sum decomposition $A=\bigoplus_{k\in\Bbb Z} A^k$ s.t. product satisfies $(A^k)(A^l)\subseteq(A^{k+l}) \text{ for each } k, l.$ A ...
0
votes
0answers
30 views

When do derivatives cancel inside integrals when working with tensors?

While doing a problem recently I realised I'm not clear about when derivatives inside integrals will cancel when working with tensors. For example, I have come across integrals such as: $\int \...
0
votes
0answers
41 views

Why are all physical quantities tensors? [on hold]

I have been struggling for years to understand what a tensor is. I know all physically meaningful quantities are to be described as tensors but why? What is the link between a "p+q-linear form on ...
0
votes
1answer
37 views

If the derivative of tensors are not generally tensors, why does vector calculus work?

There's this chart on Wikipedia (source: https://en.wikipedia.org/wiki/Matrix_calculus) Suppose I have the function $$f(x,y) = x^2y^3$$ and I compute the gradient $$\nabla f(x,y) = (2xy^3,3x^2y^...
1
vote
2answers
44 views

divergence of gradient of scalar function in tensor form

I found simple expression in tensor notation for a divergence of product vector and gradient of scalar function: $$\operatorname{div}(\mathbf{j}) = 0 \text{, where } \mathbf{j} = \mathbf{m}\times \...
4
votes
1answer
73 views

Use abstract index notation to prove Leibniz rule for exterior derivative

I want to use abstract index notation to prove Leibniz rule for exterior derivative of wedge product: For $\omega\in \Omega^k(U),\eta\in\Omega^l(U)$, d$(\omega\wedge\eta)=\text{d}\omega\wedge\eta +(...
1
vote
1answer
19 views

Symmetrization is the unique $k$-tensor

$\newcommand{\Sym}[1]{\operatorname{Sym}{#1}}$ Let $V$ be a $n$-dim real vector space with dual space $V^*$. Let $\alpha$ be a covariant $k$-tensor, i.e., $\alpha \in T^k(V^*) \equiv (V^*)^{\otimes k}...
-1
votes
0answers
33 views

differential 'df' as a tensor.

In my text book there is question which says: If f is smooth real valued function on a manifold M then df is a) linear map on b)a co-vector on M c) (0,2) tensor d)(1,0) tensor. Since df at a point is ...
0
votes
0answers
18 views

If antisymmetric tensors are differential forms, what are symmetric tensors?

Let $\mathbf{T}$ be (for example) a rank-2 antisymmetric covariant tensor, with components $T_{ij}$. In the language of differential forms, we can represent $\mathbf{T}$ as $$ \mathbf{T}=\sum_{i,j}\...
1
vote
2answers
42 views

Prove that $2 \otimes 1 \neq 0$ in $2\Bbb Z \otimes {\Bbb Z/2\Bbb Z}$ the tensor is over $\Bbb Z$.

Prove that $2 \otimes 1 $ is zero in $\Bbb Z \otimes {\Bbb Z/2\Bbb Z}$ but not a zero in $2\Bbb Z \otimes {\Bbb Z/2\Bbb Z}$ the tensor is over $\Bbb Z$. It is easy to show the first part that $2 \...
0
votes
0answers
23 views

What is the derivative of Cauchy-Green tensor with respect to deformation gradient

The Right Cauchy-Green tensor is $ \mathbf {C=F^TF}$. Its derivative with deformation gradient in index notation is ${\partial {F_{iI} F_{iJ}}}/{\partial F_{ mk}}$ $\\ = \delta_{Ik} \delta_{im} {F_{...
0
votes
0answers
22 views

What operation makes a multi-row matrix from a single-row matrix?

Suppose we have the matrix: $$m = \begin{bmatrix}-1 \\0 \\1\end{bmatrix}$$ What standard matrix operations on $m$ would give $m_2$ and $m_3$ below? $$ m_2 = \begin{bmatrix}-1&\vert\\0&m \\1&...
0
votes
1answer
58 views

Prove derivative of contravariant tensor of rank 1 is a mixed tensor of rank 2

$A^\alpha$ is a given contravariant vector (when $\alpha\in {0,1,2,3}$ a $4$-vector in Minkowski space) I need to show that the derivative $\frac{\partial A^\alpha}{\partial x^\beta}$ is a mixed ...
0
votes
0answers
34 views

Why if a tensor vanishes in some coordinates system it vanishes in all coordinates systems?

If we take how the components of a tensor on a pseudo riemannian manifold transform $$T'^{\mu\nu}(x')=\frac{\partial x'^\mu}{\partial x^\alpha}\frac{\partial x'^\nu}{\partial x^\beta}T^{\alpha\beta}(...
0
votes
1answer
30 views

Confusion about tensor fields taking values in vector fields rather than functions

I'm beginning with tensorial calculus and I have some questions. Let $(M,g)$ a riemannian manifold with $\nabla$ his Levi Civita connection. The curvature tensor $R$ is defined as \begin{align*} R : \...
-1
votes
0answers
25 views

Tensors and group theory

How tensors transform under SO(3) , Can we need to pre and post multiply tensor with an SO(3) element and its transposed conjugate ? , is it tensors exist in complex vector space those transform under ...
0
votes
0answers
36 views

Geodesic equations in polar coordinates (Euclidean space)

I tried to derive the geodesics in polar coordinate system (which should be a straight line since the metric is still Euclidean), and arrived the same equations as in this question: How to calculate ...
0
votes
1answer
51 views

What is a linear isomorphism between $\underset{n\times m}{\times} \mathbb{R}$ and $\mathbb{R}^n\otimes\mathbb{R}^m$? [closed]

What is a linear isomorphism between $\underset{n\times m}{\times} \mathbb{R}$ and $\mathbb{R}^n\otimes\mathbb{R}^m$? Where $n\times m :=\{(i,j):0\le n-1,0\le j \le m-1\}$. Since $\underset{n\times ...
2
votes
1answer
79 views

Interpretation of $\nabla$ operator in an expression

Let a tensor (3x3) be of the form $U = \mathbf{u}\mathbf{v}$ ($\mathbf{u}$ and $\mathbf{v}$) being two fluid velocity vectors (of dimension 1x3). In my analysis, for such a tensor $U$, following ...
1
vote
0answers
20 views

What are the components of the vectors $\mathbf{Z}_i$ with respect to the covariant basis $\mathbf{Z}_j$?

I am studying the book Introduction to tensor analysis and the calculus of moving surfaces, where the covariant basis is defined as the collection of vectors $\mathbf{Z}_i$ obtained from a position ...
0
votes
0answers
45 views

Tensor products $ f^∗(S\otimes T) = (f^∗S)\otimes (f^∗T) $ .

Let $V$,$W$ be finite dimensional vector spaces and let $f$ be a linear map, define $f^*$ to be the adjoint of $f$. If $S$,$T$ are tensors on $W$ show that $f^∗(S\otimes T) = (f^∗S)\otimes (f^∗T)$ ...
0
votes
1answer
26 views

The gradient of a scalar function

I found this definition of gradient of scalar function $\Phi$: $\nabla \Phi = (g^{ij}\partial_{j}) \vec{g_{i}}$ And I know: Metric tensor of spherical coordinates $g_{11} = 1$ $g_{22} = r^2$ $g_{...
1
vote
0answers
13 views

Representing position vector

I know the position vector in spherical coordinates: $\vec r = rsin\theta cos\phi \hat r + rsin \theta sin\phi \hat \theta + rcos \theta \hat\phi$ But I do not know the derivation of it. How is the ...
3
votes
1answer
123 views

Derivation of Vector Laplacian in Cylindrical Coordinates through Tensor Analysis

I'm currently trying to derive the Navier-Stokes equations in cylindrical coordinates through tensor analysis. I am only struggling with the last term on the right side, which is a vector Laplacian: $...
0
votes
0answers
12 views

Inverse of the second-order metric in GR

How do we calculate the second-order metric tensor? Given a metric tensor which includes a second order perturbation around a background metric $\bar{g}_{ab}$ $g_{ab} = \bar{g}_{ab} + \epsilon h_{ab}...
0
votes
1answer
30 views

Tensor chain rule reference request

I am a Maths major student. Question: Given a function $f:\mathbb{R}^2\to\mathbb{R},$ $g:\mathbb{R}^2\to\mathbb{R}^2$ and $(a_1,a_2)\in \mathbb{N}^2.$ Assume that $f$ and $g$ are infinitely ...
0
votes
1answer
34 views

Material Derivative in Cylindrical Coordinates

I'm testing myself on my knowledge from this book by taking the material derivative of velocity in cylindrical coordinates: $$\frac{D\mathbf u}{Dt}=\mathbf u\cdot \nabla \mathbf u$$ Which, in tensor ...
0
votes
0answers
36 views

Why are $T^k(V)$ and $V^* \otimes… \otimes V^*$ just isomorphic?

In Lee, p.178, it is said that $T^k(V)$ is isomorphic to $V^* \otimes... \otimes V^*$. But is this not stronger? Isn't this an equality? I thought if $v_1,..., v_n$ is a basis of $V$ and $\epsilon^1,.....
1
vote
0answers
33 views

Cannot simplify expression with rotor and nabla with index notation

I need to handle simple operation which needs some skill in tensor algebra. I have to take $\mathrm{rot}$ from $ (\vec u \cdot\nabla)\vec u $. I am not very good at tensors operations, but I know ...
0
votes
0answers
24 views

Relation between vector gradient (double-dotted) and divergence

I've been working on a fluid mechanics problem and have come across the following conundrum. I am asked to show two constants must satisfy $a>0$ and $(b+\frac{2}{3}a)>0$ given that $$2a\mathbf{...
0
votes
1answer
44 views

Excluding basis in tensor notation

My question is: is there anywhere in tensors that we lose something by dropping the basis, or where it makes something more difficult? Like by saying the a tensor $T^{ij}e_i\otimes e_j$ is represented ...
0
votes
0answers
21 views

Finding “determinant” for rank-4 tensor that's not for linear algebra

Please pardon me if I'm not using the right terminology. I'm using a rank-4 tensor T ($n \times n \times n \times n$) to store connections among n nodes through tetrodes (things connecting to 4 nodes ...
0
votes
1answer
14 views

Scalars in tensor notation

I know that usually the number of indices on a tensor indicates it’s rank, so how do you represent a scalar/rank zero tensor. I’ve had trouble making this work and it seems at times like there’s ...
0
votes
1answer
18 views

How to prove an expression to be a tensor?

How to prove that the expression $\varphi_{,ij}:=\frac{\partial^2\varphi}{\partial x_i\partial x_j}=\nabla\nabla\varphi$ is a tensor of second order where $\varphi$ is a scalar? Furthermore, how to ...
1
vote
1answer
49 views

Interpretation of Einstein notation for matrix multiplication

Consider the matrix product $C = AB$ where $A \in \mathbb{R}^{m \times n}, B \in \mathbb{R}^{n \times p}$. The Einstein summation notation for this is $$ c_{ik} = a_{ij}b_{jk}. $$ Is there any ...
0
votes
1answer
60 views

Why is the Kronecker sum defined for square matrices?

Background From Wikipedia, if A is an $m\times n$ matrix and B is a $p\times q$ matrix, the the Kronecker product $\mathbf A\otimes \mathbf B$ is the $mn\times nq$ block matrix $$\mathbf A\otimes \...
0
votes
0answers
19 views

Laplacian Identity in Tensor Notation

I am working through a problem in this book where it asks for the proof of the following identity: $\nabla^2f=g^{ij}\nabla_if_{,j}$ This is what I came up with: $\nabla^2f=\nabla\cdot\nabla f$ $\...
1
vote
2answers
38 views

Covariant derivative of a scalar field

I am working on a problem that asks to use the following identity to compute the Laplacian in different coordinate systems: $\nabla^2 f = g^{ij} \nabla_i f_{,j}$ In the cylindrical coordinate system,...
0
votes
0answers
23 views

The Kronecker product w.r.t. addition

On two occasions I have come upon problems requiring the differences between each point in two matrices or vectors. One such example is to find the extremum points in two arbitrary curves. The ...
2
votes
1answer
62 views

Inconsistent vector notation in physics?

I'm physics student, so please forgive my abuse of notation. I know some of my notation are not rigorous, like $\nabla$ is not a really vector. It's not wholly my fault, because I learned that way ...
0
votes
1answer
17 views

$\mathbb{P}_{n,m} \sim \mathbb{P}_n \otimes \mathbb{P}_m$…

let $\mathbb{P}_{n,m}$ be a set of polynomials $P(x,s)$ with complex coefficients such that $P(x,s) = 0$ or $deg(P(x,1)) \leq n-1 $ and $deg(P(1,s)) \leq m-1$ show that $\phi: \mathbb{P}_n \otimes \...
1
vote
0answers
23 views

How to perform these two conversions from a symmetric metric

Suppose in Cartesian coordinate system a Minkowski metric for flat spacetime can be written as : $$ ds^2 = a^2 (t)[-(1 + 2ψ(t,x,y,z))dt^2 - 2B_idx^idt + (1 - 2ψ(t,x,y,z))dx^{(i)2}] $$ This is a ...
2
votes
1answer
68 views

Integration by parts for tensor fields on Riemannian manifold

I'm working on the following exercise in my Riemannian manifolds book: Suppose $M$ is a compact, oriented Riemannian manifold with boundary. Show that if $\omega$ is any $k$-tensor field and $\eta$...
0
votes
0answers
13 views

Tensor computation and testing

I am currently testing several tensor-libraries in C++ for speed and performance. My tests are comprised of increasing size, dimensionality and complexity of the computation. But i would also like ...
3
votes
1answer
47 views

How to transfer a metric to the orthonormal coordinate?

Suppose in Cartesian coordinate system a Minkowski metric for flat spacetime can be written as : $$ ds^2 = -[1 + 2ψ(t,x)]dt^2 + a^2(t) [1 - 2ψ(t,x)]dx^2 $$ This is a diagonal metric. How can I ...
5
votes
1answer
60 views

Levi-Civita & Kronecker delta identity

I am trying to prove the following identity: $\epsilon^{ijk}\epsilon_{pqk}=\delta_p^i\delta_q^j-\delta_p^j\delta_q^i$ Starting from the following identity: $\epsilon^{ijk}\epsilon_{pqr}=\begin{...
1
vote
1answer
35 views

Pushforward of covariant and contravariant tensor.

Le $F : M \rightarrow N$ be a map between manifolds. What is the pushforward of a covariant or contravariant tensor? I think that for a covariant tensor $T : T_pM \times...\times T_pM \rightarrow \...
1
vote
2answers
69 views

If $V=k[x]$ then $V \otimes_k V \simeq k[x,y]$

Let $V=k[x]$. Show that $V \otimes_k V \simeq k[x,y]$. I consider the function $\phi : V \otimes_k V \to k[x,y]$ given by $\phi(f(x) \otimes g(y)) = f(x)g(y)$. I could show that $\phi$ is injective, ...
0
votes
0answers
86 views

What is the meaning of $A^T A$?

Given are an $m$-dimensional vector-space $V$ and a base $\mathscr{B}$ of $m$ vectors $\mathbf{e}_i \in V$. Given $m$ vectors $\mathbf{a}_j$ of an $n$-dimensional vector-space $W$, one can construct ...
1
vote
0answers
66 views

In what sense is the cross-product not a tensor?

Using the definition of tensors as tensor products, let $\operatorname{CROSS}$ be a type $(1,2)$ tensor defined by $$\operatorname{CROSS} =\sum_i\sum_j\sum_k (e_k \cdot (e_i \times e_j))\,\,e_i^*\...