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

Understanding contra/covariance

From Hehl and Obukhov's Foundations of Classical Electrodynamics: A transformation from a basis $e_\alpha$ of $V$ to another one $e_{\alpha '} = (e_1', \dots, e_n')$ is described by a matrix $L:= ...
5
votes
0answers
140 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 ...
2
votes
1answer
18 views

Definition of brackets in index notation

I've come across the notation $B_{[ij]}$ and $\partial_{[k}B_{ij]}$. What do these notations mean? The text says that $[ij] := (ij-ij)/2$, but doesn't the $ij$ just specify the which component of ...
1
vote
1answer
44 views

If $\mathfrak{g}\subseteq\mathfrak{so}(V)$, then $H^{0,2}(\mathfrak{g})=(\mathfrak{so}(V)/\mathfrak{g})\otimes V^*$

Problem I'm learning about $G$-structures and was assigned this exercise (Cartan for Beginners Exercise 8.3.6.1): Let $$ H^{0,2}(\mathfrak{g})=(V\otimes\Lambda^2V^*)/\delta (\mathfrak{g}\otimes ...
0
votes
1answer
34 views

How to show that the isotropic tensor of order n is a multiple of the kronecker delta

I have already found this question here but with the property of invariant under rotation. However I don't have this property and I want to prove that $T_{ij} = \alpha \delta _{ij}$ where $T_{ij}$ ...
2
votes
1answer
72 views

Is this 3 Qubit state Entangled?

Is this 3-Qubit state entangled? http://prntscr.com/8vsg0b $|X\rangle=\frac{1}{\sqrt 2}~~|000\rangle+ \frac{i}{\sqrt 2}|111\rangle$ I've worked with 2-Qubit states and you can turn them into ...
0
votes
0answers
28 views

Covariant derivative of a covariant vector

$$\mathbf{A} = A^{1}\mathbf{e_{1}}+A^{2}\mathbf{e_{2}}$$ $$A= \sum_{i=1}^{n}A^{i}\mathbf{e_{i}}$$ Taking the derivative wrt the tangent basis vector and dropping the summation by Einstein convention: ...
0
votes
0answers
16 views

How would you express $(\underline{a} \times \underline{b}) \times (\underline{a} \times \underline{c})$ in index notation?

At a guess, I would have said that the answer is $\varepsilon_{ijk}\varepsilon_{jlm}a_{l}b_{m}\varepsilon_{kpq}a_{p}c_{q}$, but I'm guessing that this is incorrect. What is the corrdct expression? ...
0
votes
0answers
23 views

Proving a matrix transform like a tensor

If a Matrix C is given and asked to determine if it transforms as a tensor, what are the essential conditions to be determined to prove it as a tensor? I can do it with e vector and see if it ...
0
votes
2answers
61 views

What does it mean for a vector to be a derivative?

I'm reading on Killing vectors and Killing vector fields, and one notion that keeps coming up is a derivative being a vector. For example, it's put here(Eq. 5.46) that in 2-dimensional Euclidean ...
1
vote
1answer
36 views

Does the set of null vectors of an indefinite scalar product determine the product up to scale?

If $V$ is a vector space with index 1, let $g$ and $\hat{g}$ be scalar products, and denote \begin{align}\Lambda &= \{v \in V \mid g(v,v)=0\} \\ \hat{\Lambda} &=\{v \in V \mid ...
1
vote
1answer
35 views

Rank one decomposition or elementary tensor decomposition of matrices over commutative rings

I'm facing the following problem: Let $A$ be an $m\times n$ matrix over a commutative ring $R$ (to begin with, a finite field would be sufficient too) and want to compute a decomposition in terms ...
0
votes
0answers
28 views

The explicit expression of symmetric tensors, and the Symmetrize gradient of symmetric tensors

This question has been asked here long time ago but I am still confused... Let me repeat some definitions. In general we define on $\mathbb R^2$ that $$ \mathcal T^k(\mathbb R^2):=\{\xi:\,\mathbb ...
0
votes
1answer
29 views

Corresponding matrix field basis

Hi people, I'm reviewing my notes for an exams and this is a question which I was unable to wrap my head around for many months. It should be fairly simple but I might be lacking a crucial piece of ...
1
vote
0answers
14 views

Is there any clue for the supremum of projective norm in an unit ball of injective tensor product space

Is there any information about the supremum, that is in a d-order tensor space $\mathcal{T}=\mathcal{R}^{n_1}\times\mathcal{R}^{n_2}\times\cdots\times\mathcal{R}^{n_d}$, $x\in \mathcal{T}$ what is ...
1
vote
0answers
36 views

definition of tensors and its connection to examples of tensors

Tensors are often introduced using tensor products or multilinear maps. I think I understand how they hang together. The examples given (see https://en.wikipedia.org/wiki/Tensor) are then a bit ...
1
vote
1answer
33 views

Question about the metric tensor

From wikipedia: The metric can be written in the form $g=g_{ij}dx^i \otimes dx^j$. The metric is thus a linear combination of tensor products of one form gradients of coordinates. If we denote the ...
1
vote
0answers
22 views

Simple (?) tensor index notation; When do the indices mean inner product and in what order?

In index notation, does the term $σ_{ik}x_{j}n_{k}$ mean $\bf{σx}\cdot\bf{n}$ or $\bf{xσ}\cdot\bf{n}$? Here $σ$ is a second-order tensor and $x,n$ are vectors. On the same note, is ...
6
votes
3answers
186 views

How to simplify $\frac{\partial^m}{\partial y_i^m}\mathrm{div }(A\nabla u({\bf x}({\bf y})))$

Let $\bf{x}\in\mathbb{R}^n$ (interesting in $n\in\{2,3\}$) and let $A=A_{n\times n}=\mathrm{diag}(a_1({\bf x}),\dots,a_n({\bf x}))$, that is $$A_{2\times2}=\begin{bmatrix} ...
1
vote
1answer
28 views

Deriving the Epsilon-Tensor (Levi-Civita Symbol)

Given $$\hat{e}_i \times \hat{e}_j=\sum_{k=1}^3\epsilon_{ijk}\hat{e}_k$$ and $\hat{e}_1, \hat{e}_2, \hat{e}_3$ are the unit vectors in a right handed cartesian coordinate system ...
0
votes
1answer
42 views

Show that $(2, 0)$ tensor is not a product of two vectors

The problem I am trying to solve is: Show that a general $(2, 0)$ tensor $K$, in $n$ dimensions, cannot be written as a direct product of two vectors, $A$ and $B$, but can be expressed as a sum of ...
0
votes
0answers
40 views

Jacobi Identity for Tensors

I am trying to derive the Maxwell's equations from the electromagnetic field tensor $F_{\alpha\beta}$ by using Jacobi identity: $$\partial_\gamma F_{\alpha\beta} + \partial_\alpha F_{\beta\gamma} + ...
0
votes
1answer
30 views

Cartesian tensors/change of basis/rotation - Dot product geometric issue

I was hoping someone could help with the following: [Working in index notation] I understand that we can write a new basis e'_i = (a_ij)(e_j) , where e'_i is the new basis, a_ij is the rotation ...
1
vote
0answers
23 views

Proof of a theorem on the topic of alternating tensors

I need help to prove this theorem about alternating tensors and Kronecker Deltas $\varepsilon_{ijk}\varepsilon_{imn}$ = $\delta_{jm}\delta_{kn} - \delta_{jn}\delta_{km}$ I have no idea on how to go ...
1
vote
1answer
47 views

Tensor Contraction producing a trace.

I was reading a meta-paper and came across this notation. Given V as a vector space, then there exists a dual space V*. V*⊗V is isomorphic to the set of linear transformation that map V into V, ...
0
votes
1answer
33 views

Tensor with two indices

A tensor with two indices can be represented by a $3\times3$ matrix. \begin{equation} A= \left( \begin{array}{ccc} A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & ...
1
vote
1answer
43 views

Basis of a Tensor Product

I was wondering if anyone could explain why the following proof of linear independence is not valid. Choose the identity function, $i: V \times W \to V \times W$. This function is clearly bilinear so ...
1
vote
1answer
85 views

Tensor notation for 3-D matrix expression

I have the expression $y_i = \displaystyle\sum_j x_j \!\cdot\!a_{ij} \!\cdot\! \exp \Big(\sum_k b_{ijk} \!\cdot\! x_k \Big)$ which I want to shorten without introducing more notation than necessary. ...
1
vote
1answer
20 views

Tensor transformations in curvilinear coordinates

According to tensor notation, to transform a vector from coordinates $x^\mu$ to $x^{\mu'}$, one applies the rule $$ V^{\mu'} = \frac{\partial x^{\mu'}}{\partial x^\mu} V^\mu $$ However, for example ...
4
votes
3answers
79 views

Determinant of transpose intuitive proof

We are using Artin's Algebra book for our Linear Algebra course. In Artin, det(A^T) = det(A) is proved using elementary matrices and invertibility. All of us feel that there should be a 'deeper' or a ...
0
votes
1answer
31 views

Quick Question about summing $(n^{\alpha}_{\ \ \alpha})^{2}$

This is a pretty dumb question, but regardless: Suppose $n$ is a rank-2 tensor with components $n_{\alpha \beta}$, where $\alpha$ runs from $1$ to $2$. How would I evaluate the quantity ...
3
votes
2answers
55 views

Repeated application of the gradient on a Riemannian manifold

While reading about Sobolev spaces on manifolds, I encountered the following notation regarding the norm in $H^k$: $\| \nabla ^k f \|_{L^2}$. There are two questions here: 1) What kind of object id ...
3
votes
2answers
55 views

Problems with Leibniz rule in calculating the covariant derivative of a $(1,1)-$ tensor. Where is my mistake?

Let be $$R=\sum _{\alpha, \beta} R^\alpha_\beta \frac{\partial}{\partial x^\alpha} \otimes dx^\beta. $$ I want to calculate $\nabla_\gamma(R)=\nabla_{\frac{\partial}{\partial x^\gamma}}(R).$ My book ...
3
votes
0answers
88 views

Tensor notation generally

I'm pretty new to tensors in differential geometry and I have a basic question about the notation used. In general a vector field $X$ can be expressed as $$X=\sum_{i=1}^n X^i \partial_i,$$ where ...
1
vote
0answers
20 views

Hessian and Ricci Curvature [duplicate]

I just came across a term called the hessian and read that it represents the local curvature of a function at a point. So, if it represents local curvature, then is there any way the Hessian can be ...
1
vote
1answer
43 views

Natural Isomorphism between $T_1^1(V)$ and End$(V)$

I'm a little stuck on showing that there is a natural isomorphism between the $\mathbb{R}$ vector space of $(1,1)$ tensors, and the $\mathbb{R}$ space of of linear maps $T:V\to V$. The hint is define ...
0
votes
2answers
59 views

Help understanding the proof of a vector identity using Levi-Civita and Kronecker notation.

I am looking at the proof of the following identity: a x (b x c) = (a.c)b - (a.b)c I have only just been introduced to Levi-Civita notation and the Kronecker delta, so could you please break down ...
0
votes
0answers
46 views

Raising and lowering indices - any chance of checking my work?

I'm trying to teach myself some basic differential geometry in relation to general relativity. Would anyone be willing to check my work showing the steps involved in multiplying ...
2
votes
0answers
59 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 ...
1
vote
1answer
109 views

Basic tensor derivation

Let $D$ be a tensor derivation on a mnaifold $M$. I have to show that if $D(\partial_i)=\sum F_i^j \partial_j$, then $D(dx^j)=-\sum F_i^j dx^i$. Any help on how to do this? Thanks in advance.
2
votes
0answers
55 views

Connecting physical tensors to mathematical tensors

I feel like I (maybe) understand the mathematical /algebraic perspective on tensors, for example as described in Wikipedia/Monoidal Category. You need a (simultaneously left & right) unit and an ...
1
vote
1answer
57 views

derivative of tensor

Hi I am trying to simplify $$ A=\frac{1}{2}\left(\partial_j u_i+\partial_i ...
4
votes
3answers
135 views

How to intuitively understand parallel transport

In the article I've referenced below, and many other articles for that matter, the notion of parallel transport along a line of latitude $\theta=\theta_0$ on the unit 2-sphere is spoken about. What I ...
1
vote
1answer
55 views

Tensor Algebra for Riemannian Geometry

I'm trying to learn a little bit about Riemannian geometry, but the books that I'm looking through seem to assume that the reader is familiar with topics such as contracting tensors, raising and ...
2
votes
0answers
42 views

Tensor varieties?

I read somewhere that the space of rank-one tensors, known as the Segre variety, defined by $$ Seg: \mathbb{P}V_1 \times \cdots \times \mathbb{P}V_n \rightarrow \mathbb{P}(V_1 \otimes \cdots \otimes ...
1
vote
0answers
40 views

Levi-Civita and Matrix Determinant

I have the following relations: $$\epsilon_{lmn}\det{A} = \epsilon_{ijk}a_{li}a_{mj}a_{nk} $$ and $$\epsilon_{ijk}\det{A} = \epsilon_{lmn}a_{li}a_{mj}a_{nk} $$ I can see from the logic why these ...
-2
votes
1answer
40 views

Questions related dual vector and covector through a example.

I am finding dual vector very confusing to understand. I would like to understand it using example: Let $V=\Bbb{R}^3$, $v=\begin{bmatrix}2\\3\\5\\ \end{bmatrix} ∈ V$ and $f_1, f_2, f_3 ∈ V^*$ are ...
2
votes
1answer
69 views

Exterior derivative commutes with postcomposition by symmetric multilinear functionals?

Let $\frak{g}$ be a finite-dimensional real Lie algebra, $\varphi: \bigotimes^l \frak{g} \to \mathbb{R}$ a symmetric multilinear functional, and $\psi \in \Omega^k(M; \bigotimes^l \frak{g})$ a ...
0
votes
0answers
82 views

What is dual vector and covector?

I tried to learn dual vectors online but failed to exactly understand it, I know that it could be understood using change of basis. Below is a example for change of basis, kindly help me with this. ...
1
vote
2answers
31 views

Varying Order of Covariant Differentiation

Does the order of covariant differentiation matter? Will E$_i$$_j$,$_k$$_l$$_t$ =E$_i$$_j$,$_l$$_t$$_k$ ? Does it matter if the tensor E$_i$$_j$ is continuously differentiable?