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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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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55 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
63 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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85 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
91 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
42 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
36 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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29 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
92 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
89 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
86 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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1answer
55 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
344 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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2answers
99 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
48 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
38 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} ...
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1answer
21 views

Gradient of a function with base vectors

\begin{align} \nabla\left(x_1x_3\hat{e}_1+x_2x_3\hat{e}_2\right)&=x_3\left(\hat{e}_1\otimes\hat{e}_1\right)+x_3\left(\hat{e}_2\otimes\hat{e}_2\right)\\&+\mathbf{x_1\left(\hat ...
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1answer
44 views

Show $A_{ab}$ are the components of a tensor.

The question asks: "If $v_a$ are the components of a vector, show that in an arbitrary coordinate system that $A_{ab}$ are components of a rank-2 tensor, where:" $$A_{ab}= ...
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1answer
30 views

Progresive diagonalisation of a symmetric matrix

Given a symmetric real matrix $A$ there is an orthogonal transformation that brings $A$ to a block diagonal form, where $a \in \mathbb{R}$: $$ RAR^{T} = \left[ {\begin{array}{cc} a & 0 \\ 0 ...
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1answer
41 views

using a symmetric matrix to simplify expression involving antisymmetric matrices

I am trying to simplify the following expression --$$(\vec{A} \times \vec{B})\cdot\underline{\underline{S}}\cdot(\vec{C} \times \vec{D}) = (A_jB_k\epsilon_{jki})S_{im}(\epsilon_{mnp}C_nD_p)$$ $S$ is a ...
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2answers
28 views

Is this equation consistent with index notation?

Is the equation $$a_i+b_j=c_k$$ consistent with index notation? I think that the answer is yes, since the free index within each term is arbitrary, so it doesn't matter which one we pick. Is my ...
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1answer
29 views

Proving that $(\vec a \times \vec b) \times (\vec a \times \vec c)=\vec a(\vec a \cdot \vec b \times \vec c)$ using index notation.

I'm trying to prove that $$(\vec a \times \vec b) \times (\vec a \times \vec c)=\vec a(\vec a \cdot \vec b \times \vec c)$$ using index notation (i.e. Einstein sumnmation notation). Here's what I've ...
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2answers
57 views

How can Ishow that $(\vec a \times \vec b) \cdot (\vec a \times \vec b)=|\vec a|^2|\vec b|^2-(\vec a \cdot \vec b)^2$ using index notation?

I'm trying to use index notation (i.e. Einstein summation notation) in order to show that $(\vec a \times \vec b) \cdot (\vec a \times \vec b)=|\vec a|^2|\vec b|^2-(\vec a \cdot \vec b)^2$. Here's ...
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1answer
73 views

How to identifiy $V \wedge V$ with the space of all alternating bilinear forms

Let $\{ e_i \}$ be a basis for $V$, then the space of tensors $V \otimes V$ could be identified with the space of all formal sums $\sum_{ij} \alpha_{ij} (e_i, e_j)$ (I know a base independent approach ...
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0answers
70 views

Unnecessary Elements in the Tensor Product?

For vector spaces $U, V$ there exits a unique (up to isomorphism) vector space, denoted by $U \otimes V$, and a bilinear map $\eta : U \times V \to U \otimes V$ such that for every bilinear map $\xi : ...
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17 views

Questions about a special tensor transformation

Suppose tensor $U_{i\alpha\beta}$ with dimension $M*N*N$ satisfy following condition: $$U_{i\beta\alpha}=W^1_{\alpha\alpha'}W^2_{\beta\beta'}U_{i\alpha'\beta'}$$ where $W^1$ and $W^2$ are $N*N$ ...
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1answer
166 views

Partial derivative with respect to metric tensor $\frac{\partial}{\partial{g_{kn}}}(g_{pj}g_{ql})$

$$-\frac{1}{4\mu_0}F^{pq}F^{jl} \frac{\partial}{\partial{g_{kn}}}(g_{pj}g_{ql})=+\frac{1}{4\mu_0} F^{pq} F^{lj} 2 \delta^k_p \delta^n_j g_{ql}$$ I need to know how to derive ...
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1answer
101 views

Rank of tensors in terms of ranks of associated linear maps

Let $V$ be a vector space over a field $k$, let $w \in \otimes^l V$ be a tensors. We call $w$ a simple tensor if it can be written as $$ w=w_1 \otimes w_2 \otimes \ldots \otimes w_l, $$ where $w_i \in ...
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1answer
27 views

Nested square brackets in tensor indices

I know that using square brackets on tensor indicies denote the anti-symmetric part $$ T_{[ab]} = \frac{1}{2} \left( T_{ab} - T_{ba} \right)$$ I now have to prove that $$ T_{a [[bc]d]} = T_{a ...
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32 views

How does one express $\theta_{i_1\dots i_k}$ as a linear sum of $\theta_{i_{1'}\dots i_{k'}}$?

Let $\{dx^{i_1}\wedge\cdots\wedge dx^{i_k}\mid 1\leq i_1<\cdots<i_k\leq n\}$ and $\{dx^{i_1'}\wedge\cdots\wedge dx^{i_k'}\mid 1\leq i_{1'k}<\cdots<i_{k'}\leq n\}$ be two basis for the ...
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16 views

k-space tensor integral in statistical mechanics [duplicate]

k is the modulus of the vector k. Please help me to integrate the above tensor expression in the infinite domain of the vector k. I have tried to let u in the direction of kz and then transform the ...
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1answer
20 views

If $\mu(e_1,…e_n)=1$, then how to show that $\mu=f^1\wedge f^2…\wedge f^n$?

Let V be a n dimensional vector space, $\mu$ be an antisymmetric n tensor.(i.e, a real valued multilinear functional with n inputs) If there exists a basis for $V$, say, {$e_1,e_2,...,e_n$}, such that ...
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1answer
39 views

$k$-space tensor integral in statistical physics

$$Q=\int_{\text{all space}} \frac{\hbar \nu_g \mathbf{k}\mathbf{k}}{\exp[(\hbar \nu_g |\mathbf{k}|-\mathbf{k}\cdot\mathbf{u})/k_B T]-1}d\mathbf{k} $$ Please help me to integrate the above tensor ...
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78 views

Computing volume element in spherical coordinates

Suppose $y = (r, \theta^1, \theta^2)$ are spherical coordinates in $(\mathbb{R}^3,g)$. What is the $d\text{vol}$ in these coordinates? I solved it but I don't know if it's right. My solution: We ...
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1answer
56 views

How can we show that $f_1f_2…f_k=0$ iff $\exists j$ st $f_j=0$?

Assume $V$ is an n dimensional vector space. $f_1,...f_k\in V^*,v_1,...,v_k\in V$ Define the symmetric k tensor $f_1f_2...f_k(v_1,..,v_k)=\Sigma_{\delta\in S_k}f_{\delta 1}(v_1)...f_{\delta_k}(v_k)$ ...
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1answer
28 views

Then, how can we show that $\forall i,j\in \mathbb Z, s.t.:1\leq i<j\leq n$, $e_i\wedge e_j $ is a basis vector for $\wedge^2(V) $?

Let $V$ be a n dimensional vector space. Suppose $x,y\in V, f,g\in V^*$. Define $f\wedge g(x,y) = det \left( {\begin{array}{cc} fx & fy \\ gx & gy \\ \end{array} } \right)$ Then, how can we ...
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2answers
22 views

Then, if $k>n$, can it be shown that $\psi$ is the trivial one which sends every element to $0 $?

Let $\psi:V\times \cdots \times V\to \mathbb R$ be an antisymetric $k$ tensor on $V$, which is $n$ dimensional. Then, if $k>n$, can it be shown that $\psi$ is the trivial one which sends every ...
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1answer
32 views

Confused by indicial notation term $u_{j,ij}$

I am confused by the indicial term $u_{j,ij}$ and cannot find it treated in discussions of tensor/indicial/Einstein notation even though it is an important term in linear elasticity. Working off ...
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14 views

Finding Polar Components by Raising/Lowering Indices

This is (I think) a simple question—I'm just making sure everything's correct: I'm given a vector field, $v^a$, which has constant Cartesian components $v^x = 0$ and $v^y = 1$. I'd like to find its ...
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1answer
79 views

Is there an easy way to reason about expressions involving lots of indices?

I have been reading some Riemannian geometry recently. So far, I think I am understanding the concepts well enough. However, I am finding it difficult to translate some of the notation into meaning. ...
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1answer
122 views

Confusion when applying Tensor transformation law to $\partial_{[a,v_b]}$

What I'm trying to show is that, if $v_a$ is a covector field, $\partial_{[a, v_b]} = \frac{1}{2}(\partial_a v_b - \partial_b v_a)$ transforms like a type $(0,2)$ tensor. First of all, a type ...
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2answers
129 views

Question about cross product and tensor notation

I am a bit rusty on tensor algebra and calculus and may use some wrong terminology, but I know that the cross-product can be expressed in tensor notation with the aid of the ...
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0answers
64 views

Curl of Deviatoric Stress Tensor In Index Notation

I'm taking the curl of the deviatoric stress tensor in index notation, and I've ran across something that I can't seem to be able to simplify. The issue is shown in the following portion of the curl ...
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1answer
14 views

When to use transformation of variable and when transformation of differentials

I was reading the book: Mathematical Methods in the Physical Sciences by M. Boas and I came across this statement; I wasn't quite sure why this was the case. Is it because in the curvilinear ...
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2answers
186 views

What is the scalar product of tensors?

Given there a vector space $V$ with a scalar product $g(v_1,v_2)$ on it, what is the scalar product on, say, $V \otimes V^*$ ? According to Jeffrey Lee's "Manifolds and Differential Geometry" (see ...
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95 views

Is the identification between symmetric tensors and homogeneous polynomials useful?

The general question: Given an $n$-dimensional vector space $V$ over a field $k$, there exists an identification $$\mathrm{Sym}^d(V) \sim k[x_1, \dots, x_n]_d$$ between the space of symmetric order ...
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28 views

General Tensor Assistance

Sorry if this is a stupid question, but it might help me grok things if I can connect from something that's intuitive to me. Consider a transformation from Cartesian coordinates to spherical ones: ...
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1answer
27 views

Tensors furnish representations of the group

I'm bad at english, so what exactly does it mean in simple english that Tensors furnish representations of the group?
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35 views

The effect of the Levi-Civita symbol on matrix elements

Suppose the matrix $O$ is orthogonal i.e. satisfies $$\tag{1} O^TO = 1 $$ and is also special $$\tag{2} \det O =1. $$ One can write equation $(2)$ as $$\tag{2'}\varepsilon^{i_1i_2\dots ...
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1answer
110 views

What is contracting a tensor actually doing?

I'm learning about tensors, and have a vague idea regarding what contracting a tensor means—but I'm still not sure of exactly what it's doing. Maybe someone here can put it in more intuitive terms. ...