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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69 views

What are the non-vanishing directions of $A_{a(b} B_{cd)} - B_{a(b} A_{cd)}$? ($A$ and $B$ symmetric tensors)

Question Consider two symmetric tensors $A_{ab}=A_{ba},\, B_{ab}=B_{ba}$ which are generally of full rank (non-degenerate, all eigenvalues non-zero but of general sign) and their tensor product under ...
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0answers
22 views

Metric tensor for just one index

I'm novice in the territory of tensor calculus. I know the utilization of the metric tensor to transform the covariant basis to contravariant one, vice versa: $Z^i = Z^{ij}Z_j$ (Eq. 1) I am going ...
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1answer
18 views

algebra operations

If $[x]=x^{i}\sigma^{i}$, Find an alternative form for the product $$ x^{i}n^{j}\sigma^{k}\epsilon^{ijk}=x^{i}(\vec{n} \times \vec{\sigma})^{i} $$ that has to be more compact than $$ [[x],\vec{\...
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0answers
39 views

What kind of operation is available here?

What kind of operation $\clubsuit$ is available here? $$ \underbrace{\left(v^{\mathrm T}A_{n\times n\times n}\right)}_{\in\mathbb R^{n\times n}}\underbrace{\left(v^{\mathrm T}B_{n\times n\times n}v\...
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0answers
54 views

Well definedness of Lie bracket (coordinates approach)

For practice I wanted to define Lie bracet in terms of coordinates and show that this definition is independet of the choice of coordinates. So I started with the definition $$[X,Y]:=X^i\partial_i(Y^j)...
1
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1answer
26 views

Covariant Derivatives and Swapping Indices

Okay,there's a covariant derivative of a rank 2 tensor. Swapping any indices gives a different tensor. Can we associate any physical significance to the swapping? For example, if I have a velocity ...
1
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1answer
44 views

Tensor Contraction Invariance

On page 86 of Bishop and Goldberg's Tensor Analysis on Manifolds you are asked to show that contractions are invariants. Rather than doing it awkwardly for an (r,s)-tensor (r is the contravariant and ...
0
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1answer
56 views

Checking that a two-form transforms correctly under Lorentz transformations

This is exercise $7.22$ in Supergravity by Freedman and Van Proeyen, but I did not understand it and would appreciate if you clear it out. Given the below, I still don't get how, if we define the ...
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0answers
36 views

Tensors in a vector bundle with infinite-dimensional fibers

If $\xi = (\pi,E,M)$ is a vector bundle such that each fiber has finite-dimension, and $V=\Gamma(M,E)$ is the space of smooth sections, then there is an isomorphism between $(\otimes^r V)\otimes(\...
1
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1answer
35 views

Multiplication of tensor and vector

How to do the multiplication of the multidimensional array $A_{n\times n\times n}$ and the vector $v_n$ (indices denote dimensions)? Can you kindly give suggestions or references? Thanks in advance.
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0answers
41 views

Connection with zero torsion and curvature

I remember encountering the following theorem many, many time ago: if a connection $\nabla$ has $R = 0$ and $T = 0$ then... The problem is that I cannot remember the conclusion. Was it that the ...
2
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1answer
30 views

Are Linear Transformations Always Second Order Tensors?

I've been reading a bit about tensors on Wikipedia (so correctness not guaranteed here) and I have a question. The order of a tensor $T$ is defined as $n+m$, where $n$ denotes the number of covariant ...
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0answers
23 views

Is every square matrix a tensor of 2nd order?

Is every square matrix a second order tensor? If not, what is an example of a square matrix, which is not a tensor? How can I prove that a matrix is in fact a tensor?
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0answers
72 views

Intuition Behind Dual Vectors

Recently I've been self-studying tensor analysis through the book titled "An Introduction to Tensors and Group Theory for Physicists," by Nadir Jeevanjee. I found it to be very intuitive, up until the ...
0
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1answer
37 views

What is considered to be the natural (injective) homomorphism $\frac{IL+J}{IL} \rightarrow I \otimes \frac{R}{L}$?

Let $R$ be a ring and $I,J,L \unlhd R$ such that $J \subseteq I$. What is considered to be the natural homomorphism $\frac{IL+J}{IL} \rightarrow I \otimes \frac{R}{L}$ ? Remark: It must be ...
0
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0answers
23 views

Covariant derivative of curved space basis vector is not 0. Why?

https://www.youtube.com/watch?v=jQTm-YyKWs0&index=24&list=PLlXfTHzgMRULkodlIEqfgTS-H1AY_bNtq In this video at 24:30 the teacher writes down the expression for the covariant dervative of the ...
0
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1answer
49 views

How many components of an antisymmetric rank five tensor on $ \mathbb{R}^5 $ are independent?

How many components of the a rank five tensor on $\mathbb{R}^5$ which is antisymetric under exchange of any pair of indices are independent? If we write the tensor $E_{i_1i_2i_3i_4i_5}$ then there ...
0
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1answer
47 views

Derive second fundamental from metric

If M is immersed in $\mathbb{R}^2$ and the metric is given by $g=e^{x^2+y^2}(dx^2+dy^2)$. How to calculate Gaussian curvature $K_M$? First, I use the Gauss equation $K_{\mathbb{R}^2}-K_M=\frac{1}{2}|...
1
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1answer
57 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 ...
1
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1answer
87 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
60 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 $u^1,...
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0answers
33 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, $$ \partial_\mu \partial_\nu \eta^{\mu \nu} \psi = 0 $$ ...
2
votes
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 ...
0
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1answer
42 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
33 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 ...
0
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0answers
22 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
55 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 ...
0
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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 $$\...
0
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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
59 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
25 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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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): $\frac{d}{d\lambda}=\frac{dx^{\mu}}{d\lambda}\...
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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 $$A_{abcd}=A_{...
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0answers
44 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 coordinates....
3
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0answers
63 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 $\dfrac{\...
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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 x^\mu}{\...
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1answer
73 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
39 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
48 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
58 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
31 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 \mathbb{Z}/...
0
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1answer
70 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
74 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}}} \, \underline{u})^T\...