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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22 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
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0answers
17 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
18 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
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0answers
19 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
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0answers
17 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
24 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
7 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 ...
1
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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 ...
3
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1answer
28 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
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1answer
44 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
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2answers
28 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
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1answer
37 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 ...
0
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1answer
12 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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0answers
23 views

Proof by using the definition of gradient [closed]

How do you prove the following equation by using the definition of the gradient? Thanks.
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1answer
68 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
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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}) ...
0
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1answer
38 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 ...
0
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0answers
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 ...
1
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2answers
45 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 ...
0
votes
1answer
46 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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0answers
46 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 ...
0
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1answer
51 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 ...
0
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1answer
29 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 $T^{*}_p M$. ...
1
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1answer
25 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 ...
0
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2answers
24 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 ...
0
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1answer
46 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: $$a \dot\ (b\times c)=a_i(\epsilon_{ijk}b_jc_k)=\epsilon_{ijk}a_ib_jc_k.$$ ...
1
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1answer
71 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 ...
3
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1answer
52 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 ...
0
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0answers
15 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 ...
0
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0answers
13 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 ...
1
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1answer
56 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 ...
0
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1answer
69 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, ...
1
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1answer
45 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 ...
2
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1answer
22 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
13 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 ...
1
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1answer
32 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}= ...
0
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1answer
21 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 ...
1
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1answer
32 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 ...
1
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2answers
22 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 ...
0
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1answer
21 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
46 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 ...
0
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1answer
53 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 ...
2
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0answers
48 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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0answers
16 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 $N*N$ ...
1
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1answer
70 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 ...
3
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1answer
73 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 ...
1
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1answer
21 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 ...
0
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0answers
27 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 ...
0
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0answers
26 views

How to show a coordinate system is orthogonal by using metric tensor

For example, the paraboloidal coordinate \begin{align} x &= uv\cos\theta\\ y &= uv\sin\theta\\ z &= \frac{1}{2}(u^2-v^2) \end{align} the covariant metric tensor I calculated is $$ ...
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0answers
15 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 ...