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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variation metric tensor

$\newcommand{\Tr}{\operatorname{Tr}}$(I asked this on physicsexchange but no reply) I have the metric tensor $g_{\mu\nu}$. I want to make the variation of $\sqrt{-g}$ where $g=\det g_{\mu\nu}$. Can ...
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36 views

Contraction of $(2k,2l)$tensor

In picture below ,how to get the equality in red box?
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1answer
47 views

Can someone help me understand the Euclidean metric?

A Euclidean metric is defined as: $g_{euclid} = g_{ij} = \delta_{ij} dx^i \times dx^j = dx^1dx^1 + \ldots + dx^ndx^n$ Can someone explain the following: why do we use $dx^i$ instead of $x^i$ which ...
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27 views

Higher order derivatives than Riemann Tensor

Does anyone know of any meaningful tensors that are related to the derivative of the riemann tensor? i.e. in the following picture we can consider Given arbitrary Pseudoriemannian manifold and metric ...
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22 views

Infinitesimal displacement- The inf. rotation tensor.

I have a question related to the infinitesimal rotation part- the skew-sym. part- of the infinitesimal displacement gradient matrix. Let the infinitesimal displacement gradient, say D, and E and W ...
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1answer
51 views

How to understand acting one tensor on another tensor to obtain a third tensor?

I've already known the definition of the tensor that a tensor T of type $(k,l)$ is a multilinear map from a collection of dual vectors and vectors to $\mathbf{R}$: $$ T: ...
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3answers
64 views

Tensors as geometric objects

Wikipedia's article on tensors starts with: "Tensors are geometric objects..." https://en.wikipedia.org/wiki/Tensor However there is no definition of "geometric object" in Wikipedia. To my amateur ...
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1answer
36 views

Dual Spaces vs Dual Bases

I'm trying to wrap my head around differentiable manifolds and tensors. I partially worked through a question which asked me to use the metric tensor and the line element in spherical polar ...
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18 views

Positive semidefiniteness of higher order tensor

I am trying to understand how positive semidefiniteness of hessians extends to higher dimensions. Specifically I am considering a case where I have a fourth order Hessian tensor such that each ...
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1answer
55 views

raising/ lowering indices

Here is my understanding of tensors: There is more than one way to think about tensors. One way is be thinking about tensors as objects with components which obey some transformation laws. For ...
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1answer
43 views

The formula for the differential of a vector-valued function

If we have a vector, $\,U=U\left(x_1,x_2,x_3\right)$, in the coordinate axis $\left(x_1,x_2,x_3\right)$, then why does the following differential relation hold? $$ dU= \left(\frac{\partial ...
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2answers
84 views

Einstein Tensor Notation: Addition inside a function

Main Question Can I represent addition of multi-dimensional variables in this linear function in Einstein Summation Convention? $$ f(\mathbf{x} + \mathbf{v}) $$ This didn't seem right ...
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1answer
19 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:= ...
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151 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 ...
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1answer
20 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 ...
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1answer
45 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 ...
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1answer
39 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 $T_{ij}$ ...
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1answer
78 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 ...
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33 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: ...
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0answers
17 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? ...
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2answers
63 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 ...
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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 ...
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1answer
39 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 ...
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0answers
32 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 ...
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1answer
34 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 ...
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17 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 ...
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0answers
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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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
38 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
54 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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48 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
37 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
28 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
60 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
59 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 ...
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1answer
98 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. ...
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1answer
24 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 ...
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3answers
91 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 ...
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1answer
33 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 ...
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2answers
60 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 ...
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2answers
61 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 ...
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92 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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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 ...
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1answer
47 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
89 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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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 ...
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101 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 ...