Use this tag for questions about specific tensors (curvature tensor, stress tensor), or questions regarding tensor computations as they appear in multivariable calculus and differential/Riemannian geometry (specifically, when it is amenable to be treated as objects with multiple indices that ...

learn more… | top users | synonyms

0
votes
1answer
17 views

Deriving the Geodesic Equation

I found a derivation of the geodesic equation that includes this step as I write it: $$ \frac{d (g_{ab}\dot{x}^b)}{dt}=\frac{1}{2}\partial_ag_{bc}\dot{x}^b\dot{x}^c \Rightarrow \\ \\ ...
1
vote
0answers
21 views

Comparing metric tensors of the Poincare and Klein disk models

I was trying to compare the metric tensor at the wikipedia pages of the Beltrami Klein model https://en.wikipedia.org/wiki/Klein_disk_model and the metric tensor of the Poincare disk model at ...
1
vote
0answers
27 views

How does a 4-tensor act as a linear trasformation of 2-tensors?

I'm trying to understand tensors by looking at this table and thinking about the various types of transformations the tensors represent. From the linked table, I tried looking up some of the less ...
1
vote
1answer
19 views

Locality of tensors part of definition?

I am wondering whether linearity with respect to scalar functions $f \in C^{\infty}(M, \mathbb{R})$ is part of the definition of a tensor? Let me explain it by referring to the Riemann curvature ...
3
votes
1answer
25 views

Why should the metrical groundform on a variety be a quadratic form?

I'm learning General Relativity and I can't understand why the distance function on space time is a quadratic form $$\textrm{d}s^2=g_{\mu\nu}\textrm{d}x^{\mu}\textrm{d}x^{\nu}$$ I explain it through ...
3
votes
1answer
49 views

Show that $\mathbb{C} \otimes_\mathbb{Z} \mathbb{C} \cong \mathbb{C} \otimes_\mathbb{Q} \mathbb{C}$

Show that $\mathbb{C} \otimes_\mathbb{Z} \mathbb{C} \cong \mathbb{C} \otimes_\mathbb{Q} \mathbb{C}$ This is not homework, it is part of an answer of Show that $\mathbb{A}_\mathbb{C}^2 \ncong ...
1
vote
0answers
15 views

The space $V^{0}_{p}$ of p times covariant tensors and canonical isomorphisms

I have been studying tensor calculus by myself, but I have found the following claim in my book: The space $V^{0}_{p}=V^{*} \otimes \cdots \otimes V^{*}$ of $p$ times covariant tensors is ...
1
vote
1answer
16 views

Showing that two tensors are coaxial

Given two second-order tensors $\mathbf{C} = C_{ij}\mathbf{e}_i\otimes\mathbf{e}_j$ and $\mathbf{U} = U_{ij}\mathbf{e}_i\otimes\mathbf{e}_j$ with the following relation between $\mathbf{C}$ and ...
4
votes
2answers
24 views

A question about irreducible representation of symmetric group (permutation group) in tensor space and tensor contraction

In chapter 13 of the textbook of Group Theory in Physics by Wu-Ki Tung, Lemma 2 discusses the equivalence of two irreducible representations of GL(m) on ${T^i}_j$. In its proof, it simply mentioned ...
2
votes
2answers
37 views

Is there a way to factor out the middle tensor product?

Given a state $(|1\rangle \otimes |0\rangle \otimes |1\rangle) + (|0\rangle \otimes |0\rangle \otimes |1\rangle)$, is it possible to factorise out the $|0\rangle$ in the middle of both of them? ...
1
vote
0answers
19 views

Is this proof of the product of determinants in tensor notation correct?

I'll start with the matrix C which is the product of the matrices A and B. $$c^i_k = a^i_jb^j_k$$ The determinant of C is $$\frac{1}{3!}\delta_{ijk}^{rst} c^i_rc^j_sc^k_t $$ by the definition of ...
1
vote
2answers
68 views

Looking For a Coordinate Free Way to Prove This Linear Algebra 'Fact'

$$\newcommand{\mc}{\mathcal}$$ Let $V$ be an $n$-dimensional vector space over a field $F$. (We use $\mc L(V)$ to denote $End(V)$). For each $v\in V$, define $\Theta_v:\mc L(V)\to V$ as ...
1
vote
1answer
24 views

How to simplify this expression using tensor notaion?

$\nabla^2 (\phi A)-A \nabla^2 \phi -2(\nabla \phi \cdot\nabla)A$ Where $A,\phi$ are any sufficiently smooth vector and scalar fields respectively.
5
votes
2answers
55 views

Curvature tensors and bivectors

At the beginning of the paper "The curvature of 4-dimensional Einstein spaces", by Singer and Thorpe, the authors define the space $\mathcal{R}$ of curvature tensors of the vector space $V$ as the set ...
1
vote
0answers
42 views

Matrix transponse in tensor notation

In this paper http://www.ita.uni-heidelberg.de/~dullemond/lectures/tensor/tensor.pdf at the end of chapeter 2 the author says that in index notation a matrix is written as $A^\mu_{\;\;\nu}$ and its ...
4
votes
3answers
84 views

On Learning Tensor Calculus

I am highly intrigued in knowing what tensors are, but I don't really know where to start with respect to initiative and looking for an appropriate textbook. I have taken differential equations, ...
2
votes
1answer
39 views

When is shear useful?

I'd never heard of the shear of a vector field until reading this article. Shear is the symmetric, tracefree part of the gradient of a vector field. If you were to decompose the gradient of a vector ...
0
votes
0answers
20 views

Finding the basis one forms (covectors) corresponding to a particular formulation of basis vectors

This formulation of the basis may be wrong, or I may be missing something, but I can't see a way to formulate the covectors this particular basis: \begin{align} \vec{e}_0 &= \vec{x} + \vec{y} ...
1
vote
2answers
64 views

Dual tensor for partial derivative, if it has any meaning

I'm trying to find out some details about tensors, so my question maybe isn't quite correct. What if $\omega$ is volume form in $(x,y,z)$ coordinates, then how to understand that ...
1
vote
2answers
33 views

Contraction as Adjoint of Wedging

Let $V$ be an $n$-dimensional vector space. Given $\phi^1\wedge \cdots \wedge \phi^k\in \bigwedge ^k(V^*)$ and $v_1\wedge\cdots\wedge v_k\in \bigwedge^k(V)$, we write $$ \langle \phi^1\wedge ...
0
votes
0answers
17 views

Find all components of the tensors $T_{ij}=u_iv_j+v_iw_j$ and $S_{ijk}=u_iv_jw_k-v_iu_jw_k+v_iw_ju_k-w_iv_ju_k+w_iu_jv_k-u_iw_jv_k$.

Given three vectors, $\vec u=(u_1,u_2,u_3)$, $\vec v=(v_1,v_2,v_3)$ and $\vec w=(w_1,w_2,w_3)$. Find all components of the tensors $T_{ij}=u_iv_j+v_iw_j$ and ...
0
votes
0answers
8 views

Tensors which are symmetric and antisymmetric in overlapping groups

Say I have the following tensor $T_{abc}$ such that $$ T_{(a[b)c]} $$ Ergo, it is symmetric in indices $a$ and $b$ and antisymmetric in $b$ and $c$. Keeping in mind the various properties that ...
2
votes
0answers
28 views

$T(M) \cong R<x_1,\ldots,x_n>$ isomorphism question

I have that $T(M) = \bigoplus_{i =1}^{\infty} T^i(M)$ where $T^k(M) = \bigotimes_{i =1}^{k} M$. In a paper i have that to prove such isomorphism, we define: $$\Phi:R<x_1, \ldots, x_n> ...
0
votes
1answer
39 views

Trace in Einstein notation

I know quite well what the trace of a matrix is; however, I am not quite sure I understand the meaning of the 'trace' concept when applied to tensors. I would be very grateful to you if: 1) You could ...
0
votes
0answers
54 views

Forming a (1,1)-tensor field from a (0,2)- and (2,0)-tensor field

Let $A$ and $B$ be 2-tensor fields on a manifold, contravariant and covariant respectively. Prove that there exists a smooth (1,1)-tensor field $C$ with components defined by $$C^i_j = ...
3
votes
1answer
47 views

Is there an “internal” definition of the tensor product?

We have the following "internal" definition of the direct sum: A vector space $V$ with subspaces $S,T$ is said to be the direct sum of $S$ and $T$ if $S + T = V$ and $S \cap T = \{0\}$. (Of course ...
2
votes
2answers
52 views

Is $a_i\mathbf e^i$ always equal to $a^i\mathbf e_i$?

The way that the covariant basis was described to me was that we could represent any vector $\mathbf a$ as either $\mathbf a=a_i\mathbf e^i$ or $\mathbf a = a^i\mathbf e_i$ (with the Einstein ...
1
vote
1answer
46 views

Resources on exterior algebra, wedge product, geometric product and tensors

I would like to learn exterior algebra, wedge product and geometric product along with their applications in physics. Is there a good source you can recommend? Should I study differential geometry in ...
1
vote
1answer
21 views

Tensor Notation Upper and Lower Indices

I want to ask what the difference between the tensors $T_i^{\; j}$ , $T_j^{\; i}$ , $T_{\; i}^{ j}$ , and $T_{\;i}^{j}$ are. In particular I am asking about the matrix representations of these tensors ...
0
votes
0answers
24 views

derivative of a tensor function

If e is a second order tensor and it's symmetrical, assuming abs() is a tensor function such that returns all the components of e be positive, I am interested the derivatives of the function abs(e) ...
1
vote
0answers
23 views

Pullbacks of symmetric tensors commute with products

The problem: Show that $$F*(AB)=(F*A)(F*B)$$ where F is a smooth map from a smooth manifold M to another smooth manifold N, A and B are symmetric tensor fields on N, and $F*$ denotes the pullback ...
2
votes
1answer
37 views

Matrix inversion via Levi-Civita symbols

Using Cramer's formula for the inverse of a matrix $M_{ij}$, is it possible to express the entries $(M^{-1})_{ij}$ in terms of the entries $M_{ij}$ using the Levi-Civita symbol and Kronecker deltas? ...
1
vote
0answers
14 views

Normalize tensors ($3$-by-$3$ matrix) so that the largest eigenvalue is 1?

I am trying to "normalize" a tensor $T$ (a $3$-by-$3$ matrix). The paper says ... the normalization of a tensor scales all eigenvalues so that the largest one equals to $1$. I am confused. ...
3
votes
2answers
79 views

How are these definitions of the inertia tensor the same?

I'm looking for some help in understanding the inertia tensor (not the physics, just the math). I'm trying to figure out how to convert between the wedge product and tensor product definitions. ...
2
votes
0answers
18 views

An extension of change of variables in double (and $n$-?) integrals - second-order Jacobian?

I'm aware that there are many, many questions regarding changing variables in double and triple integrals. The equation that typically pops up in textbooks is \begin{align} ...
2
votes
1answer
30 views

Proving vector Identities (Using the Permutation Tensor and Kroenecker Delta)

Prove the following vector identities by using permutation tensor and kroenecker delta. $$(\vec{A} \times \vec{B}) \times (\vec{C} \times \vec{D}) = (\vec{A} \cdot (\vec{B} \times \vec{D})) \cdot ...
-2
votes
1answer
27 views

derivative of a linear mapping

What is the derivative of a linear mapping A: R^n -> R^n? I assume it must be a tensor. In particular, if I have a linear function of a vector x, A(x), what is DA(x)?
0
votes
1answer
19 views

Little mistake with Levi-Civita symbol property

I have this equation $ \varepsilon_{ijk}B_k = \partial_iA_j - \partial_jA_i $ and I was asked to prove $\mathbf{B}=\nabla\times\mathbf{A}$, where $\mathbf{B}=B^i\mathbf{e}_i$ and ...
0
votes
1answer
22 views

How to convert $V \otimes W^*$ to a matrix space?

Namely let's say we have chosen basises $e_1, e_2, ... e_k$ for $V$ and $j_1, j_2, ... j_n$ for $W$. Now, since we can always just convert them separately, and then add the matrixes, how we represent ...
1
vote
2answers
47 views

maximizing a coordinate of $x^T A^T A x \leq r^2$

Given a vector $\mathbf{x} \in \mathbb{R}^n$, a scalar $r\gt 0$ and an invertible matrix $\mathbf{A} \in \mathbb{R}^{n\times n}$, I'd like to maximize one of the components $x_\alpha$ constrained by ...
0
votes
0answers
11 views

Why the gradient of the r vector is the identity map, geometrically speaking?

When doing some simple quantum mechanics problem involving commutators, I forgot the result of this expression $$\left[\vec{r} ,\hat{p}\right]$$ Thus I then brute force it using the definition of ...
4
votes
2answers
79 views

How do I understand constraints on high order derivatives of the Gauss Map?

I'm trying to understand the constraints resulting from differentiating an unit normal field $N$ on a surface $S$ in $\mathbb{R}^3$. If I write the unit-length constraint at a point $p \in S$, I ...
0
votes
0answers
33 views

How do you get the curvature tensor of the Schwarzschild Solution?

So, on the Wikipedia page on the derivation of the Schwärzschild solution , I get everything up to the part about the Ricci tensor. What were the components of the tensor that were used? Could ...
0
votes
1answer
40 views

What is the prerequisiste to study Tensors for application in signal processing?

I want to study Blind Source separation in signal processing for this I need to study Tensors and have a basic idea about rank, border rank and other concepts. Right now I am studying from ...
6
votes
2answers
114 views

Intuition about $v\otimes w$

In Physics and Differential Geometry usually tensors of type $(k,l)$ on a vector space $V$ over $\mathbb{F}$ are defined as multilinear functions $$f : \underbrace{V\times\cdots\times V}_{k \ ...
0
votes
0answers
19 views

Tensor as multi-dimensional array

I have a finite (but unknown) dimensional discrete object that I would like to represent as tensor (as a multi-dimensional matrix) with some basis. Can someone guide me on matrix-transformations that ...
2
votes
1answer
27 views

Showing $T$ equivalent to linear map

Let $T$ be a $(1,1)$ tensor over a vector space $V$. Let $\left\{e_a\right\}$ be a basis for $V$ and $\left\{f^a\right \}$ be its dual basis. Show that $T$ is equivalent to a linear map $V^* ...
2
votes
1answer
40 views

Relationsip between two definitions of the christoffel symbol?

When I first started learing about tensor calculus, the professor defined the Christoffel symbol as $$\Gamma ^a _{bc} = Z^a \cdot \partial_b Z_c $$ Where $Z^a$ is a contravariant basis vector and ...
0
votes
1answer
37 views

What is meant by “trace on any pair of indices”?

I am reading the book Riemannian Manifolds, written by John M. Lee. On page $53$, the author defines a connection on the tensor bundle $\text{T}_l^k(M)$ and says it satisfies (d) $\nabla$ commutes ...
1
vote
1answer
54 views

How to prove that below quantity is a Third Rank Tensor

$F^{ik}$ is an antisymmetric tensor. I want to prove that below quantity is a Third Rank Tensor. $$\dfrac{\partial F_{ik}}{\partial x^{l}} + \dfrac{\partial F_{kl}}{\partial x^{i}} + \dfrac{\partial ...