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 ...

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39 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 ...
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1answer
37 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 ...
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0answers
23 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 ...
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3answers
190 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} ...
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1answer
37 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 ...
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1answer
47 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 ...
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47 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} + ...
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1answer
34 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 ...
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0answers
26 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 ...
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1answer
55 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, ...
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1answer
35 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} & ...
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1answer
53 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
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1answer
93 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
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1answer
22 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
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3answers
88 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
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1answer
32 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
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2answers
58 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
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2answers
60 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
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0answers
91 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 ...
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0answers
22 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
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1answer
46 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 ...
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2answers
79 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 ...
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0answers
49 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
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0answers
87 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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2answers
126 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
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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 ...
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1answer
63 views

derivative of tensor

Hi I am trying to simplify $$ A=\frac{1}{2}\left(\partial_j u_i+\partial_i ...
5
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3answers
217 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
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1answer
57 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
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0answers
43 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 ...
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0answers
46 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 ...
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1answer
43 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
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1answer
74 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 ...
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0answers
88 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. ...
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2answers
33 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?
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0answers
27 views

Linear transformation and riemannian metric tensor

When is it true that for a linear transformation $A$ and a symmetric bi-linear form $g$ (i.e. (0,2) tensor) on a riemannian manifold: $$ g( AV, W)= g(V, AW)?$$
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0answers
18 views

Symmetry and Skew-Symmetry in covariant derivatives

Okay,so, E$_i$$_j$ is a rank 2 symmetric tensor. So, E$_i$$_j$= E$_j$$_i$ So, when there is, E$_i$$_j$,$_k$$_l$$_t$ where the comma indicates covariant differentiation with respect to the indices ...
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3answers
672 views

Qualitatively, what is the difference between a matrix and a tensor?

Qualitatively (or mathematically "light"), could someone describe the difference between a matrix and a tensor? I have only seen them used in the context of an undergraduate, upper level classical ...
2
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3answers
90 views

What are the dual basis vectors?

What exactly are dual basis vectors such as those which arise in non-orthogonal co-ordinate systems? What is their physical interpretation. Please note, I don't know much tensor calculus yet. I am ...
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1answer
68 views

How do you prove this tensor identity?

Suppose we have a tensor field ${\Gamma_{ab}}^{c}$ on a smooth manifold $M$ such that for all vector fields $X^a$, we have $X^aX^b{\Gamma_{ab}}^{[c}X^{d]} = 0$. How do you show that there exists a ...
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1answer
92 views

Curiosity about the wedge product and the Levi-Civita symbol

Let us use the definition of the wedge product of two vectors: $$\vec{u}\wedge\vec{v} = \vec{u}\otimes\vec{v} - \vec{v}\otimes\vec{u}$$ writing $\vec{u}$ and $\vec{v}$ in dyadic form as $\vec{u} = ...
0
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1answer
39 views

Fixing some notations about tensor calculus

I'm studing some articles about Ricci Flow and Ricci Solitons. I realize the first thing to do is choose a notation, thus I'm using the Huai Dong Cao's notation (which is the same Ricgard Hamilton's ...
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0answers
19 views

Integrating a tensor over the whole of $R^3$

I want to find $\int_{R^3}r^{-3}\exp^{-\lambda r^2}x_ix_jdV$ where $r = x_kx_k$. I had the idea to translate that into finding $\delta_{ij}\int_{R^3}r^{-1}\exp^{-\lambda r^2}dV$ However, when I put ...
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1answer
23 views

Integrating the alternating tensor over a sphere

I'm having some trouble working out how to evaluate $\int_S\epsilon_{ijk}x_kdS$, where S is the unit sphere. My thoughts are that $x_kdS$ is the surface element, but the combination of that with ...
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0answers
47 views

Integrating a particular tensor

I have a tensor defined by $T_{ij} = \delta_{ij} + \epsilon_{ijk}x_k$, and I want to find $\int_ST_{ij}dS$, where S is the surface of the unit sphere. I'm having some problems because I really don't ...
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12 views

Simpliication of Tensor operations

Is it possible to obtain $F(n,E)$ such that $$(n_k E_{kj} n_j ) n_i u_i + n_k E_{ki} u_i = F(n,E) n_i u_i$$ where $n$ is a unit vector in an unknown direction, $E$ is a symmetric tensor (specifically ...
0
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1answer
23 views

Tensor algebra problem from the derivation of covariant derivative

I was reading a way of deriving the covariant derivative (using (1,1)-tensor) and found it hard to understand one specific operation. $${\frac{\partial a^{k}_{m}}{\partial y^{p}}=\frac{\partial ...
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0answers
36 views

LeviCivita identity and the vector cross product

How can I verify the identity $\sum_{k} \epsilon _{ijk} \epsilon _{mnk} = \delta _{im}\delta _{jn}-\delta _{in}\delta _{jm}$ ? How can I use this identity to calculate $(\vec{A}\times \vec{B})\cdot ...
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2answers
38 views

Are four vectors in Special Relativity considered to be tensors?

In particular, I would like to know if the four velocity and the four acceleration are tensors.
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57 views

Lowering and raising tensor indices without using metric tensor

Is it possible to lower and raise tensor indices using non-metric tensor? For example if we define non-singular tensor ${a_{ij}}$ such that ${a_{ij}a^{jk}=\delta^{k}_{i}}$ and it is not metric tensor. ...