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

Tutorial about tensor

Description: I have not learned the course about tensor. However, when I learning Mathematica software to deal with my problem that came from my specity, there are ...
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1answer
52 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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31 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
44 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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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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22 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
21 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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Basic understanding of stress tensors in a fluid [migrated]

So, after having spent the last 9 hours attempting to understand the basic tenets of stress tensors in fluids, I can honestly say that I think I know less now than when I began. My questions are ...
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36 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 ...
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2answers
39 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
33 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
27 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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32 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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0answers
13 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
30 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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21 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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21 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
27 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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29 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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23 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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31 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
86 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
38 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 ...
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1answer
35 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
52 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
27 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
61 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
42 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
49 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
18 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
34 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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21 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
101 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} ...
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1answer
51 views

Decomposition of Tensor into a Product of Tensors

I was working on a text that made use of the assumption that, for some rotation matrix $a_{ij}$, and some tensor $U$, the tensor under rotation is represented by $U'_{\alpha}=a_{\alpha i}U_i$. It made ...
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1answer
108 views

Double dot product in Cylindrical Polar coordinates - Strain energy

I'm working with a problem in linear elasticity, and I have to calculate the strain energy function as follows: $$ 2W = σ_{ij}ε_{ij} $$ Where σ and ε are symmetric rank 2 tensors. For cartesian ...
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1answer
47 views

Differentiation of scalar fields using tensor notation

I'm learning tensor calculus to understand differential geometry. Please verify if I've understood how to employ Einstein's sum convention and index notation correctly. Suppose that $\varphi := ...
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1answer
57 views

Interpretation of $(r,s)$ tensor

A tensor of type $(r,s)$ on a vector space $V$ is a $C$-valued function $T$ on $V×V×...×V×W×W×...×W$ (there are $r$ $V$'s and $s$ $W$'s in which $W$ is the dual space of $V$) which is linear in each ...
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33 views

Why abstract index notation should not be confused with the Ricci calculus?

Considering this answer, it is mentioned that the range of indices $a, b, c,\dots$ are seen as abstract and coordinate-free and linear operations can be represented with them; and the range of indices ...
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64 views

How to prove that the Kronecker delta is the unique isotropic tensor of order 2?

Is there a way to prove that the Kronecker delta $\delta_{ij}$ is indeed the only isotropic second order tensor (i.e. invariant under rotation), i.e. so we can write $T_{ij} = \lambda \delta_{ij}$ ...
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1answer
31 views

Is it necessary for a linear map to be an automorphism to allow polar decomposition?

Bowen and Wang's Introduction to Vectors and Tensors I (pg. 168) states a general form of the polar decomposition theorem as Every automorphism A has two unique multiplicative decompositions $$ ...
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2answers
74 views

$\nabla \cdot (\mathbf{B}\mathbf{B} - \frac{1}{2}B^2 \tilde{1})=(\nabla \cdot \mathbf{B})\mathbf{B} - \mathbf{B} \times (\nabla \times \mathbf{B})$

Does someone know how to show this identity? $\nabla \cdot (\mathbf{B}\mathbf{B} - \frac{1}{2}B^2 \tilde{1})=(\nabla \cdot \mathbf{B})\mathbf{B} - \mathbf{B} \times (\nabla \times \mathbf{B})$
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1answer
64 views

Using metric to raise and lower indices

Everything I read on tensors makes it clear that using the metric matrix $g_{ab}$ and its inverse $g^{ab}$ to respectively lower and raise indices of a tensor is very important. As far as I know (and ...
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3answers
30 views

Tensor tranformation between basis?

If I am the basis vector $e_i$ into another basis to get $e'_j$ I use: $$e'_j=S_{ij}e_i$$ My text book says that $S_{ij}$ is the ith component of the vector $e'_j$ with respect to the unprimed basis. ...
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2answers
47 views

Kronecker delta symmetry?

This is quite a simple question for the Kronecker delta does $$\delta_{ij}=\delta_{ji}$$ My textbook implies it does but does not actually state it and hence am looking for clarification.
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1answer
13 views

Show that getting parallel transported does not change angle between them

I must tell you that I have never seen this kind of question in Tensor Analysis. Our professor had set up this question in our exam, but I don't know whether it belongs to Tensors or not. The question ...
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1answer
67 views

Which Subspaces do Antisymmetric Tensors Represent?

So antisymmetric tensors represent volumetric subspaces (I've asked this here instead of on phys.stackexchange because it seems like more of a math question)? How exactly would one know WHICH ...
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45 views

Relation between componenet and algebraic definition of covariant vectors

I studied contravariance and covariance concepts in following way: For any vector if we get its components by parallelogram way we achieve contravariant components, and if we want to get its ...
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3answers
73 views

Can you illustrate the use of coordinate-free notations that serve as an alternative to Einstein summation notation with an example?

"Abstract index" and "coordinate free notations" are often submitted as alternatives to Einstein Summation notation. Could you illustrate their use using an example? Here's a sum written in ...
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70 views

Gradient is covariant or contravariant?

I read somewhere people write gradient in covariant form because of their proposes. I think gradient expanded in covariant basis i , j , k so by invariance nature of vectors, component of gradient ...
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1answer
73 views

Space of Alternating $k$-Tensors Notation

I will be taking a Differential Geometry class in the Fall, so I decided to get somewhat of a head start by going through Spivak's "Calculus on Manifolds." Before reading, though, I saw the Addenda at ...