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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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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0answers
13 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
101 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
115 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
57 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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0answers
56 views

How to prove a symmetric tensor keeps symmetric under rotation? [closed]

For example, $T_{ij}=T_{ji}$, prove $R_{il}R_{jm}T_{lm}$ is also symmetric. I know I need to prove $R_{il}R_{jm}T_{lm}=R_{jl}R_{im}T_{lm}$, and the fact that $R$ is antisymmetric might be helpful, ...
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1answer
13 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
184 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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0answers
94 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
27 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
34 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
106 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
76 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 ...
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1answer
213 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
87 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
61 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
43 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
53 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 ...
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1answer
31 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
54 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 ...
3
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2answers
157 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
47 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
41 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
413 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
40 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
43 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
32 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 ...
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1answer
37 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
38 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 ...
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1answer
52 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 ...
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105 views

Is Hodge star operation can be understood as contraction after tensor product of a $p$-form with the volume element?

By defintion, the Hodge star of a $p$-form $\omega_{a_1\cdots a_p}$ on a $n$-dimensional manifold is given by $*\omega_{b_1\cdots b_{n-p}}=\frac{1}{p!}\omega^{a_1\cdots a_p}\epsilon_{a_1\cdots ...
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3answers
303 views

Index notation for inverse matrices

I have a question: There is an standard way to write the inverse of a matrix in index notation?. The reason is that I don't want to write $(A^{-1})_{ij}$ or $(A^{-1})_i^j$ or $(A^{-1})^{ij}$ using ...
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1answer
54 views

Question about index notation on partial derivatives.

I've been studying quantum field theory a little bit and I've encountered a notation like the following: $$\mathcal{D}_{x,x'}=\frac{\partial}{\partial x^\mu}\frac{\partial}{\partial ...
3
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1answer
69 views

How to interpret tensor form PDE in terms of matrix algebra

From this mathwork page "c for system", the usual second order PDE is written in tensor form: $$ -\nabla\cdot(\mathbf{c} \otimes \nabla \mathbf{u})+\mathbf{a}\mathbf{u}=\mathbf{f} $$ and ...
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1answer
77 views

I need some help understanding the tensor algebra done this problem.

I often see equations rearranged across an equal sign and I have no clue what tricks and reasoning they are using to arrive at these solutions. The only resources I can find on tensor algebra only ...
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1answer
47 views

Role of metric in the matrix representation of Hermitian adjoint

I'm working through Jeevanjee's "An Introduction to Tensors and Group Theory for Physicists", and while trying to prove that the matrix representation $M(A^\dagger)$ of a Hermitian adjoint $A^\dagger$ ...
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2answers
71 views

Multiplication and derivation of 3D matrix

I have $A(q)=\begin{bmatrix}q_1 &q_2 & q_3\\ 2q_1 &3q_2 & 4q_3\\ 2q_1 &3q_1 & 10\\ \end{bmatrix}\tag 1$ $ q= {\left(\begin{array}{c}q_1\\q_2\\q_3\\q_4\\q_5\\q_6 ...
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2answers
66 views

Tensor Calculus and Differentiation

I've been reading various texts on tensor algebra and calculus in preparation for applications of it, and I find myself continuously having issues with the index calculus. I've been seeing: $$ ...
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1answer
114 views

connection laplacian on general vector bundles

As the title says, my question is about how to define the connection laplacian on general vector bundles. I think I understand how to define the connection laplacian on the tensorbundles: Let $M$ ...
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0answers
23 views

How can I compute the Lebesgue measure

Let $\mathcal{X}$ be a tensor whose frontal slices are defined by $X_1=\begin{bmatrix}{1}&{0}\\{0}&{1}\end{bmatrix}$ and $X_2=\begin{bmatrix}{0}&{1}\\{-1}&{0}\end{bmatrix}$. This is a ...
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1answer
38 views

Covariant and contravariant bases on a diffeomorphism

If we allow two domains $\Omega, \bar{\Omega}\in \mathbb{R}^3$, allow $\mathbf{\Theta}: \Omega \to \mathbf{E}^3$ and $\mathbf{\bar \Theta}: \bar \Omega \to \mathbf{E}^3$ to be two ...
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0answers
36 views

Two forms of application of metric tensor to get differential length

I'm reading the monograph "Foundations of Tensor Analysis for Students of Physics and Engineering With an Introduction to the Theory of Relativity" by Joseph Kolecki, now retired, of NASA. I have a ...
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1answer
306 views

expanding tensor notation of Navier stokes equation

I'm trying to expand the variable density and viscosity Navier-stokes equation for incompressible flows but I've had no luck so far. The Navier-Stokes in tensor notation is: $$ \rho \dfrac{Du_i}{Dt} ...