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

1
vote
1answer
18 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
57 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
42 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
20 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
76 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
26 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
67 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
50 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
120 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
23 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
66 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
41 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
29 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
44 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
59 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
48 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
53 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
49 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
33 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
25 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
29 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
71 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
19 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
85 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
19 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
44 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
30 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
23 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
48 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
15 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
81 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
36 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
125 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 \ ...
2
votes
1answer
31 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
47 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
43 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
61 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 ...
0
votes
1answer
52 views

Notation in symmetric and alternating products of forms

In Lee's Riemannian Manifolds book, we see that a Riemannian metric $g$ can be expressed locally in coordinates $(x^1, \ldots, x^n)$ by $g = g_{ij} dx^i \otimes dx^j$. Introducing the symmetric ...
1
vote
1answer
32 views

How to prove the skew symmetry of the Riemannian tensor?

I parallel transported a vector around a parallelogram formed by another two vectors to get the Riemannian tensor: $$R^m{}_{lkj} = \left( \partial_k \Gamma^m{}_{jl} + \Gamma^m{}_{kn} \Gamma^n{}_{jl} ...
2
votes
1answer
30 views

Are there any 3rd order tensors satisfying $e_{ijk} e_{lmk} = \delta_{il} \delta_{jm}$ in dimensions higher than three?

My question is simply wrote on the title. (I'm using Einstein's contraction rule.) In the case of three dimensions, I can construct the Levi-Civita-like tensor as follows. \begin{align} e_{ijk} = ...
2
votes
1answer
70 views

Minimize Product of Sums of Squared Distances

The Question Given two sets of vectors $S_1$ and $S_2$,we want to find a unit vector $s$ such that $$\{\sum_{u\in S_1}(\|u\|^2-\langle u, s \rangle^2)\} \cdot \{\sum_{v\in S_2}(\|v\|^2 - \langle v, ...
7
votes
0answers
77 views

Determinant of a tensor

Is there such a thing as the determinant of a tensor of rank $\gt 2$? I'm tried to think how it might be defined -- potentially like, the determinant of the tensor $A=a_{ijk}$ is ...
3
votes
0answers
50 views

Difference of two connections is a tensor

I am currently reading through Jost's Riemannian Geometry and Geometric Analysis and am seeking clarification of the statement in the title. Jost abstractly defines a connection on a vector bundle ...
1
vote
0answers
32 views

Why the following is not a tensor?

Say we have an arbitrary coordinate system, in which a position vector is represented by: $$\vec V=Z^i\vec Z_i$$ Where $\vec Z_i$ is a covariant basis. Now $\partial {\vec Z_i}/\partial Z^j$ term has ...
4
votes
2answers
54 views

Proof of a tensor identity involved in the derivation of the Einstein field equations?

On the wikipedia page for the einstein-hilbert action, the section for the derivation of the einstein field equations cites this identity: $$ \sqrt{g} \nabla_\mu A^\mu = \partial_\mu (\sqrt{g} A^\mu) ...