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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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 ...
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1answer
34 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 ...
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1answer
39 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 ...
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0answers
27 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 ...
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0answers
38 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 ...
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1answer
15 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 ...
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3answers
36 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 ...
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73 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 ...
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0answers
10 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 ...
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67 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
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1answer
36 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} ...
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3answers
80 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*} ...
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1answer
58 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, ...
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1answer
40 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} = ...
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40 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. ...
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1answer
42 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 ...
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1answer
39 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 ...
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32 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) ...
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30 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 ...
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0answers
46 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} + ...
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25 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 ...
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20 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 ...
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0answers
26 views

Derivative with respect to a tensor in Mathematica

I am trying to differentiate a tensor with respect to another one in Mathematica but I cannot do it. Could anyone please help? The following is the code: ...
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0answers
8 views

Assign a symbolic tensor with determinant greater than zero in Mathematica

I can create a symbolic tensor in Mathematica: $Assumptions = F \[Element] Arrays[{3, 3}, Reals] Is there anyway that I can tell Mathematica, that the ...
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1answer
41 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 ...
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1answer
30 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 ...
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1answer
46 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 ...
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2answers
35 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 & ...
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1answer
43 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 ...
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1answer
13 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 ...
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1answer
70 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 \\ ...
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1answer
30 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}) ...
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1answer
40 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 ...
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29 views

The basis of this particular tensor product

I'm studying Fulton's Algebraic Curves book and he defines the module of differentials $\Omega$ in the following manner (R is a ring containing an algebraically closed field $k$): Let's define the ...
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2answers
46 views

$(n-1)$-alternative tensor on E are decomposable

$E$ is a real vector space with dimension $n$ and $E^*$ is dual space of $E$. Assume $\alpha \in Λ^{n-1}(E)$ Show that there exists $\alpha_1,\alpha_2,...,\alpha_{n-1} \in E^*$ such that ...
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1answer
49 views

Metric and Convariant Tensor

$g_{ij}$ is the metric tensor. Show that $g^{ij}$ which satsifies $g_{ij}g^{jk}=\delta_i^k$ is a covariant tensor of rank $2$. I am not sure how to show this? Does it instead mean to show that ...
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54 views

double Hodge star operator

$*$ is the Hodge star operator acting on differential $k$-forms on $\mathbb{R}^{n}$ as below: $$*:Λ^k(\mathbb{R}^{n}) \to Λ^{n-k}(\mathbb{R}^{n})$$ $$\alpha \wedge (\star \beta) = \langle ...
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1answer
55 views

For which k,n the k-covector is decomposable (14-2 from Lee)

This is homework so no answers please The problem is: Find for which k, n, a k-alternating map $\omega$ can be written as $\omega=\omega_{1}\wedge...\wedge \omega_{k}$ were $\omega_{i}$ are ...
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1answer
36 views

Basis of a tensor

I'm new to tensors, and I need to understand what a certain basis actually is, how to visualise it. Say we have the $r$-dimensional vector space $T_p M$ and $n$-dimensional dual space $T^{*}_p M$. ...
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1answer
26 views

Differentiating a rank-2 tensor to some power in index notation.

If I have some rank-2 tensor $g_{ab}$ with components dependent on some coordinate system $x^a$, how do I do the following differentiation in index notation (assuming the $\dot x^d$ are independent of ...
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2answers
25 views

Inner product on the k-tensor space

This is homework so no answers please. The problem is "Given inner product vector space V, define an inner product on $T^{k}(V)$ by declaring the standard basis $\{e^{*}_{i_{1}}\otimes...\otimes ...
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1answer
55 views

Tensor notation of a triple scalar product

I want to write the tensor notation for $$[a\dot\ (b\times c)]a=(a\times b)\times (a\times c).$$ What I got so far is: $$a \dot\ (b\times c)=a_i(\epsilon_{ijk}b_jc_k)=\epsilon_{ijk}a_ib_jc_k.$$ ...
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1answer
78 views

Tensors and rotation matrix

$a_{ij}$ is a rotation matrix that satisfies $\hat{e}'_i=a_{ij}\hat{e}_j$. Show that $\epsilon_{lmn}a_{mi}a_{nj}=\epsilon_{ijk}a_{lk}.$ Using the result from above, how can I show that ...
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1answer
54 views

Integrating tensors on manifolds

When/how can you integrate tensors on manifolds and what does it mean? I imagine that line integrals of tensors make sense when you have a connection, since you can uniquely parallel transport all ...
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16 views

Tensor decomposition: How to determine the number of components (rank) of a CP decomposition?

I need to compute a best $rank-R$ CP decomposition of a tensor built from real world 3-dimensional data, of sizes approximately $100×200×300$. The best rank-R is determined by a given metric, which ...
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0answers
17 views

What is the relation between Kruskal tensor and CP decomposition?

In Matlab Tensor Toolbox there is a tensor type called "Kruskal tensors", I found its form is similar to the CP decomposition. Wikipedia mentioned "As such, many of the methods have been ...
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1answer
73 views

Cauchy Momentum Equation - Stress Tensor

I've been trying to understand the derivation for the Cauchy Momentum Equation for so long now, and there is one part that every derivation glides over very quickly with practically no explanation ...
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1answer
70 views

How does the representation of co-vectors change if we change the basis of a vector space $V$?

I'm trying to understand how vectors, differential forms and multi-linear maps in general transform under change of coordinates. So I start with the simplest case of vectors. Here's my own attempt, ...
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1answer
46 views

Manipulation of Tensors

I have an expression: $\eta^{\mu \nu} F_{\alpha \beta, \nu} F^{\alpha \beta}$ Where $\eta^{\mu \nu}$ is the Minkowski metric, F is an antisymmetric tensor, and the comma on the middle tensor denotes ...
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1answer
23 views

Evaluating the Lie derivative of the metric

From the Wikipedia definition of the Lie derivative of a tensor along a vector field, we have, $$\mathcal{L}_X g_{\mu\nu} = X^\lambda \nabla_\lambda g_{\mu\nu} + (\nabla_\mu X^\lambda)g_{\lambda \nu} ...