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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1answer
56 views

How to extend the parallelepiped volume formula to higher dimensions?

The volume of a parallelepiped $(V)$ is given by the triple scalar product: $$V=\mathbf{c}\cdot{}(\mathbf{a}\times\mathbf{b})$$ where $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ are the vectors ...
0
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1answer
82 views

Book recommendation for rigorous multilinear algebra , tensor analysis, manifolds.

I am looking for recommendation on books about multilinear algebra, tensor analysis, manifolds theory, basically everything to be able to understand basic concepts of general relativity. I am ...
0
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1answer
59 views

Contravariant vector example with polar coordinates

My book gives me this definition for contravariant vector: Let an n-tuple of real numbers $a^1,a^2, \dots, a^n$ be associated with a point P of an n-dimensional Riemannian space with coordinates ...
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0answers
31 views

Solving PDE's in Tensor form

Is there a straight forward method to solving PDE's in tensor form? How do boundary conditions work? For example, I may get the wave equation, $$ \partial_\mu \partial_\nu \eta^{\mu \nu} \psi = 0 $$ ...
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1answer
42 views

Confusion regarding notation of a dual transformation

I'm reading Spivak's Calculus on Manifolds and in Chapter 4 he defines the dual transformation (although he doesn't call it that) as follows: If $f:V \rightarrow W$ is a linear transformation, a ...
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1answer
39 views

Changing variables: partial derivatives of a tensor

Given is the tensor $T$ in Cartesian coordinates $T=\operatorname{diag}\{T_{xx},T_{yy},T_{zz}\}$ in cylindrical coordinates $T=\operatorname{diag}\{T_{rr},T_{\theta\theta},T_{zz}\}$ How does one ...
2
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1answer
32 views

Show that $R_{\mu\nu}=fg_{\mu\nu}$ (Ricci and metric tensors) and $\dim(M)\geq 3$ then $f$ is constant

I need to prove that given the Ricci and metric tensors $R_{\mu\nu}=fg_{\mu\nu}$ and $\dim(M)\geq 3$ then $f$ is constant. I tried to use some identities but I end up with some sort of a proof ...
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0answers
21 views

Cylindrical coordinate derivative of a vector field.

Considering the following identity transformation in cylindrical coordinate: $$\mathbf{v}(R,\theta,z)=R\;\mathbf{e}_{R}+\theta\;\mathbf{e}_{\theta}+z\;\mathbf{e}_{z} $$ Taking its derivative ...
3
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1answer
53 views

Sectional curvature in a paraboloid is always positive.

I'm working on Lee's book ''Riemaniann Manifolds an Introduction to Curvature''. One exercise (11.1) is about to see that the paraboloid given by the equation $y=x_1^2+...+x_n^2$ has positive ...
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0answers
23 views

Variation of a tensor $\delta T$.

Let a change of coordinates be given by $x^{\mu}\to x^{\mu '}=x^{\mu}+\varepsilon \xi^{\mu}(x)$ with epsilon a small quantity. Given a tensor $T$ we define $\delta T:=T'(x)-T(x)$. I guess this means ...
2
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1answer
55 views

Decomposing a tensor product space into direct sums

I'm trying to understand how to decompose certain symmetric and anti-symmetric tensor products of vector spaces into direct summands. Let $V$ be a complex finite dimensional vector space and denote ...
3
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1answer
48 views

The Relation Between Kronecker's Delta and the Permutation Symbol

The Kronecker's Delta is defined as $$\delta_{ij}= \begin{cases} 1 & i=j \\ 0 & i \ne j \end{cases}$$ Also, the Permuation Symbol known as Levi Cevita's Symbol is introduced as ...
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1answer
29 views

Treatments of Lie theory and Noether theorems in tensor notation

I am looking for a conceptual treatment of Lie theory and Noether theorems that uses tensors calculus rather than exterior calculus. I know that tensor calculus is not optimal for these subjects, ...
1
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1answer
52 views

Invert tensor expression involving Levi-civita symbol

I have to prove that $$ \omega_{\mu \nu} = \epsilon_{\mu \nu \lambda \kappa} \omega^\lambda u^\kappa $$ given the relations: $$ u_k u^k = -1 $$ $$ \omega_{\mu \nu} u^\nu = 0 $$ $$ \omega^\mu = ...
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0answers
11 views

Adjoint of $SU(2)$ from tensor product of fundamental 2-dimensional representation

Given elements of the fundamental 2-dimensional representation of $SU(2)$, for example, $a=(1,0)$ and $b=(0,1)$, how can I multiply them correctly to yield an element of the adjoint? $$ a \otimes b ...
1
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1answer
20 views

Index notation of tensors and mnemonics

I've been trying to learn to manipulate tensors but I've got probably too comfortable with all the matrices in my Linear Algebra course, that it gets really difficult beyond rank-3 tensors. So, ...
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0answers
23 views

Tensor products over Ring of Integers

I am having trouble computing the following tensor product(s): $$\mathcal {O}_{Q} \otimes _\mathbb {Z} \mathbb {C} = $$ Where in this case we use the ring of integers of the field of rationals, or ...
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0answers
61 views

Comodule induced from a comodule over a graded coalgebra

Let $G$ be a group with $1$ be the identity element of $G$. Let $(C,\bigtriangleup,\epsilon)$ be a coalgebra over a field $K$. $C$ is called $G$-graded if $C$ admits a decomposition as a direct sum of ...
2
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0answers
48 views

Why is the transformation law for tangent space coordinate basis vectors “easily seen” using the chain rule?

In Spacetime & Geometry, Carrol immediately posits that, given that you have some tangent space vector (in the coordinate basis representation): ...
1
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1answer
48 views

Is a tensor always identifiable as some symbol with some amount of indices?

Say I have a (1,1) tensor $T^{\mu}_{\;\nu}$, I can represent it in some basis as follows (using Sean Carrol's notation): $T= T^{\mu}_{\;\nu} \; \hat{e}_{(\mu)} \otimes \hat{\theta}^{(\nu)}$ If I let ...
0
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1answer
34 views

Interchange symmetry of a tensor of type $(0,4)$

Suppose $A$ is a tensor of type $(0,4)$ on any manifold having the following symmetries $$A_{abcd}=-A_{bacd}=-A_{abdc}$$ and $$A_{abdc}+A_{acbd}+A_{adcb}=0.$$ How can I show that ...
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0answers
42 views

Christoffel symbols for the Poincaré ball model

The metric tensor $g_{ij}$ of the Poincaré ball model is $$ g_{ij} = \frac{\delta_{ij}}{(1 - x_k x^k)^2} $$ where $\delta_{ij}$ is the Kronecker delta and $x^k$ are the ambient Cartesian ...
3
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0answers
61 views

Meaning of / intuition for contraction of tensors (in the Riemannian setting)

I'm currently taking a course in differential geometry, and we are, I'm guessing, finally going to start working with the Riemannian curvature tensor after having covered a lot of smooth manifold ...
3
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0answers
31 views

Derivative of product tensor (or matrix product )

if I define a "product tensor" of two second-order tensors as the second-order tensor: $\boldsymbol{C}=\boldsymbol{AB}$ such that $C_{ij}=A_{ik}B_{kj}$ does the product rule applies to ...
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0answers
13 views

Changing the coordinates of a specific metric tensor

As I have understood, the change of coordinates of a pseudo-Riemannian metric $g(x) \to \bar g(\bar x)$ is given by: $$\tag{1} \bar g_{\alpha\beta}(\bar x) = \sum_{\mu,\nu} \dfrac{\partial ...
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1answer
72 views

Curvature tensor for a particular Hilbert manifold

My question involves an infinite dimensional Hilbert manifold with a Riemannian metric. My question is: What is the form of the curvature tensor for a infinite dimensional Hilbert manifold with ...
3
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0answers
38 views

Proving a vector identity using Einstein summation

I need to prove $$\nabla \times [(u\cdot \nabla)u]=(u\cdot \nabla)(\nabla \times u)-[(\nabla \times u)\cdot \nabla]u+(\nabla\cdot u)(\nabla \times u)$$ So the ith component of each side using ...
0
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1answer
47 views

What is the definition of the tensor product $U⊗ V$ of two vector spaces $U$ and $V$?

All I want to know is the exact definition of the vector space that is the tensor product of two vector spaces, say $U$ and $V$, i.e. I want to know: - how its vectors are defined, - how does the $+$ ...
1
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1answer
54 views

Gradient of vector field notation

Working in 3D. I know that the gradient is a vector operator defined as $\nabla = [\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}]$. The gradient of a scalar ...
1
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1answer
30 views

cardinality of orbits

Let $G=\mbox{GL}(\mathbb{Z}/p\times\mathbb{Z}/p )\times \mbox{GL}(\mathbb{Z}/p\times\mathbb{Z}/p ) \times \mbox{GL}(\mathbb{Z}/p\times\mathbb{Z}/p )$, $p$ prime, act on $(\mathbb{Z}/p\times ...
0
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1answer
69 views

group ring Z(G) is isomorphic to tensor algebra T(G)

Well, I've been learning Tensor algebra and related topics. So, I wonder tensor algebra is related to somehow familiar with me, so called group ring. $\mathbb{Z}$: a ring of integers. Let G be a ...
4
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1answer
72 views

Divergence operator of higher order and intrinsic point of view

Let $\underline{u}$ be a $1$ - order tensor (say a column vector) I want to prove that : $\underline{\operatorname{div}} \left( (\underline{\underline{\operatorname{grad}}} \, ...
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0answers
41 views

Nice way of proving that the pullback of a tensor field is smooth?

I'm going through some routine exercises in studying smooth manifolds. This one is 12.27 (d) from Lee, Introduction to Smooth Manifolds If $F:M\to N$ is smooth, and $B$ is a covariant $k$-tensor ...
4
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1answer
56 views

Map Laplacian in terms of covariant derivatives

I stumbled upon the following definition: Let $\mathcal{M}$ be a manifold, $g_{ij}$, $h_{ij}$ be two Riemannian metrics on $\mathcal{M}$, $\psi : \mathcal{M} \to \mathcal{M}$ be a ...
3
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0answers
26 views

Rotation of eigenvectors of a time-dependent tensor

I have a symmetric, real tensor in ${\mathbb R}^3 $ where the components are continuous functions of time: $${\mathbf D} = \left( \begin{matrix} d_{1,1}(t) & d_{1,2}(t) & d_{1,3}(t) \\ ...
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0answers
32 views

Kronecker delta representation of a matrix (Quantum raising / lowering operators)

The Kronecker Delta is commonly used to represent a diagonal matrix: $$ a_i \delta_{ij}=\left( \begin{array}{ccc} a_1 & 0 & 0\\ 0 & a_2 & 0\\ 0 & 0 & a_3 \end{array}\right) $$ ...
1
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1answer
27 views

divergence theorem applied to a tensor dotted with a vector

Is my expression for the divergence theorem correct? $\int_{V}\underline{v}.div(\underline{\underline{\tau}})dV=\int_{S}\underline{v}.\left(\underline{\underline{\tau}}.\underline{n}\right)dS$ and ...
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2answers
63 views

Does the Divergence Theorem hold for arbitrary tensor fields?

So, a heads up, this is my first post and I'm a fairly new user. Additionally, my math knowledge tops out at vector calculus and ODEs, but don't shy away from answering beyond my understanding should ...
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1answer
28 views

Show $\delta_{KL}$ is a Cartesian tensor [closed]

By using the definition of kronecker delta $\delta_{KL}$, show that $\delta_{KL}$ is a Cartesian tensor, that is $$\delta ' _{MN}=L_{MK}L_{NL} \delta_{KL}$$ under the rotation $X_K=L_{MK}X' _M$. I ...
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0answers
41 views

Derivative of dual basis vectors in terms of Christoffel symbols

How can I demonstrate from $$ \frac{\partial \mathbf{e_j}}{\partial x^i} \equiv \Gamma_{ij}^k \mathbf{e_k} $$ what the value of $$ \frac{\partial \mathbf{e^j}}{\partial x^i} $$ (with the index now ...
0
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1answer
41 views

Show that $\epsilon_{ijk}\epsilon_{ljk}=2\delta_{il}$

Question: Show that $\epsilon_{ijk}\epsilon_{ljk}=2\delta_{il}$ where $\epsilon_{ijk}$ is the Levi-civita symbol and $\delta_{ij}$ is the Kronecker delta symbol. My attempt: $\epsilon_{ijk}$ assumes ...
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0answers
35 views

Easy Tensor Notation: What is $\partial_\mu\partial^\nu x_\nu$

Part of a far larger piece of work... I know it should simplify $\partial^\nu x_\nu\partial_\mu$ but I don't know how. Similarly, does it change for the reverse? In other words I want to simplify: ...
2
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5answers
177 views

How to Prove $(A \times B) \otimes C + (B \times C) \otimes A + (C \times A) \otimes B = [(A \times B) \cdot C] \bf{I}$?

Question Assume that $A$, $B$, $C$, and $D$ are four vectors in $\mathbb{R}^3$. Then I want to show that $$ {\bf{M}} \equiv (A \times B) \otimes C + (B \times C) \otimes A + (C \times A) \otimes ...
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0answers
37 views

Programming nested sums in Matlab for graph-based statistic

I have an undirected graph $G=(E,N)$, where $E$ is the set of edges and $N$ is the set of nodes, of which $|N|=n$. It's convenient to represent the edges via a (symmetric) adjacency matrix $B$. I ...
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1answer
46 views

Tensor algebra and symmetric algebra

Reference Book : Dummit - abstract algebra 3rd edition (page 455). But, I think $ $ $=$ $ $ notation is not correct. It may be corrected to $\cong$ (an isomorphic notation). Because $S^2 (V)$, ...
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2answers
62 views

When is a vector a tensor? Is invariance under affine transformation required?

Several resources, such as this (pages 11 & 15) and this, state that position vectors are not rank one tensors while displacement vectors are, since only displacement vectors are invariant under a ...
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0answers
19 views

SU(3) tensor methods in representations [duplicate]

I am trying to understand the Georgi chapter of tensor methods in $SU(3)$ representations, and I don't know how to resolve the tensor product of 2 matrices in a 2 heavy quark + 2 light antiquark ...
2
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0answers
32 views

Tensor formula in SU(3) representations

I am trying to understand the Georgi chapter of tensor methods in $SU(3)$ representations, and I don't know how to resolve the tensor product of 2 matrices in a 2 heavy quark + 2 light antiquark ...
1
vote
1answer
30 views

Confusion about tensor derivation in wiki article

I can't follow the derivation of the gradient of a vector field in cylindrical coordinates from this wiki page My problem is with the $e_r \otimes e_\theta$ term. Expanding the definition from a ...
4
votes
2answers
100 views

Matrix notation in tensor transformations

In special relativity one looks at coordinate transformations that consist of combinations of Lorentz boosts, rotations and reflections - members of the Lorentz group. Under an arbitrary ...