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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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 ...
2
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0answers
74 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
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
193 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
103 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
29 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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0answers
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 ...
1
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0answers
17 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 ...
1
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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 ...
1
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1answer
40 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 ...
3
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0answers
80 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 ...
0
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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 $f_1f_2...f_k(v_1,..,v_k)=\Sigma_{\delta\in S_k}f_{\delta 1}(v_1)...f_{\delta_k}(v_k)$ ...
0
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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 ...
0
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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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0answers
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 ...
4
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1answer
80 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. ...
0
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1answer
142 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
148 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
69 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 ...
0
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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 ...
3
votes
2answers
189 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 ...
3
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0answers
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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0answers
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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0answers
36 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 ...
2
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1answer
117 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. ...
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0answers
99 views

Superscripts and subscripts in tensors - interpretation as a matrix operation?

Consider the equation $$u_k = t_i U_i^k$$ I am told that subscripts represent covectors (row vectors) and superscripts represent column vectors. My intuitive interpretation of the above equation as a ...
9
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1answer
235 views

How can I determine the number of wedge products of $1$-forms needed to express a $k$-form as a sum of such?

This question was motivated by this related one: How "far" a differential form is from an exterior product . Let $\mathbb{V}$ be a vector space of dimension $n$ with underlying field ...
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2answers
88 views

Help with a paper about tensors

I came across something in a paper I am not able to understand jet. Unfortunately the author is kind of short with explanations. Maybe someone here can help me to understand this. $M^d \in ...
3
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1answer
65 views

Derivation of the Geodesic Equation

From page 40 of A. Schild and J. L. Synge's "Tensor Calculus", I'm having issues understanding the following mathematical steps ( I feel like it's simple algebra that I'm messing up. We have, $a_{rm}$ ...
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1answer
44 views

Polynomial and super-symmetric tensor

A quadratic function uniquely determines a symmetric matrix. Ok that’s easy. Now a homogeneous polynomial function $f(x)$ also uniquely determines a super-symmetric tensor. My question is how do I ...
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2answers
54 views

Einstein Summation - does the following equality hold: $a_{ij} x_i y_j = a_{ij} y_i x_j$

Does equality hold when $x_i = y_i$ and $x_j=y_j,$ and $ i, j = 1, ..., n $.
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2answers
40 views

Einstein Summation: How do I show $a_{ij} (x_i + y_j) \not= a_{ij}x_i + a_{ij}y_j $?

Einstein Summation: How do I show $a_{ij} (x_i + y_j) \not= a_{ij}x_i + a_{ij}y_j $?
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0answers
32 views

How to write an analogue to matrix-vector multiplation with an extra dimension in tensor notation

My background is severely lacking in tensor algebra, and after a few days of looking into tensors I am still not able to even formulate this question quite correctly; my apologies for that. I am aware ...
0
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1answer
32 views

Derivative of a contravariant tensor

Let $T$ be a contravariant tensor so it transforms under change of coordinates like $$ T^{i'} = T^i\ \frac{\partial x^{i'} }{\partial x^i} $$ In this it seems $T^{i'}$ is a function of the "primed" ...
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1answer
55 views

normal vectors in spaces where $n > 3$

I am reading Lovelock and Rund's book on Tensors and they make a statement that I wanted to validate about normal vectors in high-dimensional spaces. It should be remarked that the above ...
4
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2answers
267 views

Properties and notation of third-order (and higher) partial-derivatives

This question has been bothering me for quite a while and I still haven't found a satisfying answer anywhere on the internet or in any of my books (which may not be that advanced, mind you...). Since ...
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1answer
51 views

Simple question on symmetric tensors 2

This question is related to this one Simple question on symmetric tensors. To prove that a vector field $Z$ is Killing, we use the identity $$0=(L_Zg)(X,Y)=g(X,\nabla_YZ)+g(\nabla_XZ,Y)\ \ \ \forall ...
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1answer
44 views

Notation for proof with Tensors

I'm working on proving For a second order tensor $\mathbf{A}$,$\mathbf{u}\cdot\mathbf{A}\cdot\mathbf{u}=0$ for all vectors $\mathbf{u}$ if and only if $\mathbf{A}$ is skew symmetric. Now, I ...
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1answer
572 views

Difference between the Jacobian matrix and the metric tensor

I am just studying curvilinear coordinates and coordinate transformations. I have recently come across the metric tensor ($g_{ij}=\dfrac{\partial x}{\partial e_i}\dfrac{\partial x}{\partial ...
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1answer
20 views

Transpose of second order tensors inside of an expression

In a book I found the following expression: $\int_V (A_{ij} \delta B_{ij}) dV = \int_S (T_i \delta c_i) dS $ That apparently is equal to: $\int_V (\delta B_{ij}^T A_{ij} ) dV = \int_S (\delta ...
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1answer
43 views

Simple question on symmetric tensors

This question seems to be silly, but i am really confused. Suppose we have a symmetric $2$-tensor $\omega$, I want to prove $$\omega(X,Y)=0\ \ \ \forall \ \ \ X,Y \iff \omega(X,X)=0 \ \ \forall \ \ \ ...
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0answers
51 views

Tensor Rank for derivatives and integrals

How can I go about determining the rank of a tensor if I am taking a derivative or an integral? For example, I have a 2nd order Tensor, $A_{ij}$ and I take the derivative with respect to $x_i$ or ...
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0answers
34 views

How has the definition of a tensor changed since Tullio Levi-Civita's definition?

To get a good grounding in tensors, I'm reading the book *The Absolute Differential Calculus (Calculus of Tensors) (Dover Books on Mathematics) Paperback by Tullio Levi-Civita. I'll then move on to a ...
1
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1answer
46 views

Elegant Proof of the Product of Two Levi Cevita Tensors

Is their an elegant way to prove the product of two Levi Cevita tensors is equivalent to a determinant of a matrix of Kronecker deltas? I know that the anti-symmetry and cyclic nature should be easily ...
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0answers
39 views

Proof that $\displaystyle \sum_{i=1}^{i=3} \displaystyle \sum_{j=1}^{j=3} A_{ij} \delta_{ji} = A_{ij}$

Can someone provide a proof that $\displaystyle \sum_{i=1}^{i=3} \displaystyle \sum_{j=1}^{j=3} A_{ij} \delta_{ji} =A_{ij}$? I know that the Kronecker delta is identified with the identity matrix, but ...
0
votes
1answer
64 views

Metric Tensor Antisymmetry

The metric tensor on a Riemannian manifold is given as a symmetric $n \times n$ symmetric matrix (so $g_{ij} = g_{ji}$). Is there an intrinsic reason for this symmetry? Why can't it be antisymmetric ...