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

2
votes
1answer
62 views

Multilinear algebra some basics.

The wedge product of $p$ vectors in vector space $V$ is called a $p$-vector and the vector space generated by all $p$-vectors is denoted $\bigwedge^p V$ with the basis $e_I:=e_{i1}\wedge\dots\wedge ...
1
vote
1answer
85 views

Is it possible for a manifold to have a normal vector that is zero everywhere, if so, would this indicate that the manifold is non-orientable?

Basically I've been thinking about defining a non-orientable three-dimensional metric space via defining the normal vector and looking to see if there is two possible vectors for the same point. I'm ...
3
votes
2answers
553 views

How many independent components does a rank three totally symmetric tensor have in $n$ dimensions?

How many independent components does a rank three totally symmetric tensor have in $n$ dimensions? Needed for the irrep decompositon of $3\otimes 3\otimes 3$ in here. No idea where to start to ...
2
votes
1answer
54 views

Integrals in Index Notation and Orientation

I am wondering what is the correct way to write integrals in index notation. At first I thought $$\int_M f \varepsilon_{ij}$$ would be the index equivalent of $$ \int_M f dx \wedge dy $$ but I started ...
2
votes
0answers
122 views

Tensor Calculus Second Order Derivatives

I'm learning tensor calculus by myself through lectures and texts, and I'm presented with the problem of finding the first and second order derivatives of a scalar function of three variables that ...
4
votes
0answers
77 views

Is there a method of knowing which of the Christoffel symbols of second kind survive and vanish for a given metric?

I'm trying to solve a problem and the given $g_{ij}$ metric is $$\left[\begin{array}{cc}1&0&0\\0&(x^1)^2&0\\0&0&(x^1 \sin x^2)^2\end{array}\right]$$ The non-zero Christoffel ...
7
votes
2answers
95 views

Does a $p$-form eat $p$-vectors or $p$ number of vectors?

A bilinear form is another term for a $2$-form. So does it eat $2$ distinct vectors or a single $2$-vector?
6
votes
0answers
174 views

What is the difference between tensor calculus and exterior derivative type concepts?

I am trying to clarify terms in order to help me figure out what I'd like to study. I understand that $p$-forms and $p$-vectors are used with things like wedge products, exterior algebras, and a ...
3
votes
1answer
54 views

Geometry of the Covariant Derivative

Taking the standard covariant derivative from tensor calculus $ \nabla_{\mu}T_{s} =T_{s|\mu}=T_{s,\mu} $. Could this be geometrically interpreted as the directional derivate of a tensor that has extra ...
5
votes
0answers
81 views

Decomposition of order-$n$ tensors

If $V$ is a finite-dimensional vector space, then $V\otimes V\cong\mathbf{S}^2(V)\oplus\bigwedge^2(V)$. The first summand on the right is the symmetric part of $V\otimes V$ and the second summand is ...
8
votes
1answer
652 views

How can I derive the back propagation formula in a more elegant way?

When you compute the gradient of the cost function of a neural network with respect to its weights, as I currently understand it, you can only do it by computing the partial derivative of the cost ...
1
vote
0answers
63 views

Order of Riemann tensor indexes and the Ricci Identity

I have seen the Ricci identity written variously as $R_{ijk}{}^l x^k = (\nabla_i\nabla_j- \nabla_j\nabla_i) x^l$ $R_{ij}{}^l{}_k x^k = (\nabla_i\nabla_j- \nabla_j\nabla_i) x^l$ $R^l{}_{kij} x^k = ...
1
vote
1answer
38 views

Writing in Cartesian tensor form

Write the following in Cartesian tensor form $$(1) \nabla (\operatorname{div} G) \times \nabla\Omega$$ $$(2) (\operatorname{curl}(F)\times G)\cdot \nabla(Φ)$$ I have answers for these two questions, ...
0
votes
1answer
41 views

Definition of rank of an element in tensor algebra

Let $V$ be a vector space over a field $K$ and let $T(V)$ be its tensor algebra; that is, $$T(V)=\displaystyle\bigoplus_{k=0}^{\infty}V^{\otimes k}=K\oplus V\oplus V\oplus(V\otimes V)\oplus(V\otimes ...
1
vote
3answers
104 views

Intuitive transition from matrices to tensor-concept

I would like to know how to build intuition for the concept of a tensor using the following reasoning: If I conceive of a vector as an extension of the scalar concept, i.e. an $N \times 1$ "array of ...
0
votes
0answers
22 views

Matrix operation repeat matrix members

I am going to use C++ Armadillo library which handles matrices to generate matrix $B$ and $C$ from matrix $A$. $$ A=[M_0,M_1,\ldots,M_{n-1}]^T $$ $$ ...
5
votes
0answers
64 views

The metric and Kronecker's delta

I am reading some lecture notes for GR and it is currently showing how we are going to derive the field equations using a metric for a massive free particle with a metric ...
2
votes
2answers
72 views

Raising and Lowering Through Differentiation

I'm calculating the Christoffel symbols of the second kind which is of course defined as multiplying the symbol of the first kind multiplied by the contravariant metric. I was thinking of how to make ...
1
vote
0answers
30 views

Extrinsic curvature tensors

I risk of sounding too vague, but I am interested if there are other tensors reflecting the extrinsic geometry of a submanifold other than the second fundamental form? The first fundamental form ...
3
votes
1answer
123 views

Riemannian curvature tensor of product manifolds

Let $(M_{1},g_{1})$ and $(M_{2},g_{2})$ be two Riemannian manifolds. Let $% R_{1}$ and $R_{2}$ be the (1,3)-type Riemannian curvature tensors of $M_{1}$ and $M_{2}$, respectively. Finally, let $R$ be ...
1
vote
1answer
68 views

Surface area of a 2-sphere in Abstract Index Notation

I believe the following completely specify a 2-sphere of radius 1 in AIN: $$ R_{ijkl}=\epsilon_{ij}\epsilon_{kl} \\ R_{ij}=g_{ij}\\ R_{ii}=g_{ii}=2 $$ It is easy enough to determine the area by ...
2
votes
0answers
31 views

Objects corresponding to Higher forms

If $Q$ is a quadratic form, then we know there exists matrix $A$ such that $Q=xAx'$ and $Q$ can be expressed as weighted sum of eigenvalues of $A$. If $H$ is a higher order form, then is there an ...
1
vote
1answer
178 views

Covariant derivative for a covector field

In the lecture we had the immersion $f: U \subset \mathbb{R}^2 \rightarrow \mathbb{R}^3$. Now we said that a map $\lambda : U \rightarrow T^*f$ is a covector field. Okay, that's fine. Also, we ...
1
vote
1answer
29 views

Notation in Srednicki's QFT

In the book Quantum Field Theory by Srednicki, equation 21 for the commutators of the generators of the Lorentz group is ...
2
votes
3answers
45 views

MATLAB: matrix multiplication with exponentiation

Normal matrix multiplication computes $C=A*B$, such that $C_{ij}=\sum_k{A_{ik}*B_{kj}}$ I want to compute D, such that $D_{ij}=\sum_k{e^{A_{ik}*B_{kj}}}$ Basically I want to exponentiate each ...
2
votes
1answer
124 views

Showing a property of a curvature tensor in $S^2$

Consider $S^2 \subset \mathbb{R}^3$. I need to show that if $$R_{ijkl} = -g(R(\partial_i,\partial_j)\partial_k,\partial_l)$$ is a curvature tensor in $S^2$ and $g$ is a metric also in $S^2$, then ...
0
votes
0answers
22 views

Extensions to higher dimensions by tensorization. Unitary DFT in 2D?

I have problem understanding the underlying concept of tensoration (if there is such term). Fist of all the unitary DFT is NxN. Is it 1D ? How does it look when we increase the dimension let say to ...
5
votes
0answers
93 views

Elastic wave equation in curvilinear coordinates: how do you perform a coordinate change?

The essence of this question is that I don't know how to convert an equation from Cartesian coordinates into curvilinear coordinates, and would like to know how, preferably using the language of ...
2
votes
1answer
45 views

Components of Torsion Tensor

So I am trying to calculate the components $T^i\,_{j k}$ relative to a coordinate basis, of the torsion tensor defined as: $$ \bf{T(u,v)= \nabla_{u}v-\nabla_v u-[u,v]} $$ Or: $$ T= T^i\,_{j k} ...
1
vote
3answers
88 views

Prove: Gravitation operator is invertible.

Let $(M,g)$ be a Riemannian manifold and $\Gamma(S^2M)$ the space of symmetric 2-covariant tensors. Define the gravitation operator as the map \begin{align*} ...
0
votes
1answer
89 views

Tensors: Maxwell stress on an isotropic elastic material

The Maxwell stress is $\sigma_{ij} = -\varepsilon_r[E_iE_j - 0.5E_kE_k\delta_{ij}]$. Assume that in a dielectric solid under study, the electric field vector is aligned with the 1 axis. That is, ...
1
vote
1answer
107 views

Tensors: traction free planes

Given the following stress tensor matrix, determine the value of $\sigma_{22}$ such that there is a traction free plane and determine the unit normal to this plane. $$ \sigma_{ij} = ...
1
vote
0answers
62 views

Apparently meaningless computation with the Hodge star operator

In the last lecture we started speaking about hodge star operator. Let $E$ be a $n$ dimensional vector space with a non degenerate bilinear form $g$. $\mathcal{O}$ the orientation line of $E$, i.e. ...
1
vote
1answer
57 views

Tensors, basic notation and components of

I'm trying to understand he basic notation(s) used to write out tensors, namely \begin{equation} T = T^{\nu_1,...,\nu_m}_{\mu_1,...,\mu_n} \frac{\partial}{\partial x^{\nu_1}} \otimes...\otimes ...
1
vote
1answer
44 views

Definition of Linear independence of algebraic $1$-forms

"Suppose that $a,b,c,d$ are linearly independent algebraic $1$-forms on $\mathbb{R}^n$". What does it mean for algebraic $1$-forms to be linearly independent? I have looked through my notes and ...
1
vote
0answers
37 views

Show that $f:V\to W$ is a $(1,1)$-tensor

I'm currently reading Nakahara's Geometry, topology and physics (about tensors), and came across with the following proposition (exercise 2.12, p.99): Show that a linear map $f:V \to W$ is a (1,1) ...
2
votes
0answers
52 views

Bivector into orthogonal components

Suppose I have a metric $g$ and a bivector $ F $ on a four-dimensional vector space. It seems I can always decompose $ F $ into four mutually orthogonal vectors $a,b,c,d$ $$ F = a\wedge b + c\wedge d ...
2
votes
0answers
94 views

Calculating Hydrodynamic Interaction Tensor

I'm a bit of a newbie when it comes to Tensor calculus. Please excuse me as I learn... Given the Oseen tensor, $\mathbf{T}(\mathbf{R}) = (8\pi \eta R)^{-1} \left[ \mathbf{I} + ...
2
votes
0answers
51 views

Objectivity or frame invariant

I am not sure it is maths or physics question. A stress tensor of a nematic liquid crystal is given as follows $$ T_{ij}=-P\delta_{ij}-n_{k,i}n_{k,j}+\widetilde{T_{ij}} $$ where ...
1
vote
0answers
28 views

Tensor operator

I have come across the following expression: H:E where, H = e(levi-cita symbol)*a constant which means a 3rd order tensor with 27 components E = 2nd order tensor, now, what does H:E mean? I know ...
1
vote
1answer
53 views

algebra of polynomials of non-commuting variables

In Hatcher's book algebraic topology page 496-497: the Steenrod algebra $\mathcal{A}_2$ is the algebra over $\mathbb{Z}_2$ formed by polynomials of non-commuting variables $Sq^1,Sq^2,\cdots$ modulo ...
3
votes
1answer
34 views

Levi-Civita help

$\epsilon_{ijk}$ is the Levi-Civita tensor which is totally anti-symmetric. Let $A^{ijk}$ be a totally symmetric matrix. Is it true that $$\epsilon_{ijk}A^{ijk}=0?$$ I know this is the case for ...
1
vote
1answer
72 views

Determinant of a 2nd rank tensor help and inverse!

I have the following 3x3 matrix $$U_{ij} = g_{ij} + \epsilon_{ijk}u_k$$ and I want to find its inverse using the fact that it can be written as the linear combination of its symmetric part and its ...
1
vote
2answers
108 views

Levi Civita Symbol: from 4 to 3 indices.

In Four-dimensional space, the Levi-Civita symbol is defined as: $$ \varepsilon_{ijkl } =$$ \begin{cases} +1 & \text{if }(i,j,k,l) \text{ is an even permutation of } (1,2,3,4) \\ -1 & ...
3
votes
1answer
65 views

Formal adjoint of divergence

We define the so-called conformal Killing operator $K$ mapping (1,0) vectors to (0,2) tensors by $$K(X)_{ab} = \frac{1}{2}\nabla_aX_b+ \nabla_bX_a -\frac{2}{3}(\text{div}X) g_{ab}.$$ Here $g$ is the ...
2
votes
0answers
109 views

Converse of Euler Homogeneous Thm. How to show that $\lambda \mathbf{x}\cdot \frac{d}{d\lambda}(\nabla{f(\mathbf{\lambda x})})=\mathbf{0}$?

So basically I read someone else's answer to a question regarding Euler Homogeneous function theorem source: http://quant.stackexchange.com/questions/8911/what-is-exactly-eulers-decomposition Also ...
0
votes
1answer
31 views

Symmetric/Antisymmetric Co and Contravariant Tensors

Show that if the contravariant tensor $A^{ab}$ is symmetric and the covariant tensor $B_{ab}$ is antisymmetric, then $A^{ab}B_{ab}$ $= 0$ I have tried plugging in and expanding definitions ...
6
votes
1answer
81 views

Is it possible to build a tensor with the following properties?

I am searching for a tensor in 4-dimensional space-time with two indices that satisfy: \begin{eqnarray} M_{;\mu }^{\mu \nu } &=&0, \\ M^{\mu \nu } + M^{\nu\mu}&=&0, \nonumber \\ ...
1
vote
1answer
34 views

evaluation of $\nabla \cdot ( \boldsymbol{B}(\boldsymbol{x}) \cdot \boldsymbol{B}^{T}(\boldsymbol{x}) )$

i have a symmetric positive matrix $\boldsymbol{D}(\boldsymbol{x})$ which can be decomposed as: $\boldsymbol{D}(\boldsymbol{x})$ = $\boldsymbol{B}(\boldsymbol{x}) ...
1
vote
1answer
59 views

Showing $\det(I_E+f)=\sum_{p=0}^{n}\mathrm{tr}(f^p)$ where $f^p:Λ^{p}(E) \to Λ^{p}(E)$ and $f:E\to E$

$E$ is a vector space with dimension $n$ and $f:E\to E$ is a linear map and for every $p=1,2,3,...$ we have $f^p:Λ^{p}(E) \to Λ^{p}(E)$ which is defined as below ...