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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0answers
23 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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1answer
258 views

Christoffel Symbols and the change of transformation law.

I have seen it written that the change of co-ordinate form is given by the following: $$ \tilde \Gamma^{i}_{jk} = {\partial \tilde x^i \over \partial x^\alpha} \left [ \Gamma^\alpha_{\beta ...
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3answers
56 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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2answers
178 views

Prove that the trace of a dyad uv is the dot product of u and v

$$ I'm\quad trying\quad to\quad demonstrate\quad that\quad the\quad trace\quad of\quad a\quad dyad\quad (tensor\quad product)\quad is\\ equal\quad to\quad the\quad dot\quad product\quad of\quad ...
3
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1answer
29 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 ...
0
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1answer
36 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
50 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 ...
2
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1answer
22 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
60 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 ...
4
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1answer
41 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 $M$ ...
1
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2answers
40 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: $$ ...
1
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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 ...
1
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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 ...
0
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1answer
91 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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0answers
18 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 ...
0
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1answer
48 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 ...
0
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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 ...
3
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1answer
95 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 ...
2
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1answer
46 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 := ...
1
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0answers
26 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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7answers
4k views

Does a “cubic” matrix exist?

Well, I've heard that a "cubic" matrix would exist and I thought: would it be like a magic cube? And more: does it even have a determinant - and other properties? I'm a young student, so... please ...
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0answers
18 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 $T_{ij} = \lambda \delta_{ij}$ ...
0
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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
68 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? $\nabla \cdot (\mathbf{B}\mathbf{B} - \frac{1}{2}B^2 \tilde{1})=(\nabla \cdot \mathbf{B})\mathbf{B} - \mathbf{B} \times (\nabla \times \mathbf{B})$
2
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1answer
59 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 ...
4
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1answer
265 views

Tensors as mutlilinear maps

I am aware that many books on differential geometry define tensors as multilinear maps. Namely $$ V\otimes W := L_2(V^*\times W^*,\Bbb F) $$ I am also aware that this space is isomorphic to the tensor ...
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0answers
79 views

Invariant tensors in adjoint representation

Suppose we have a simple Lie group $G$ with algebra $\mathfrak{g}=\{X_a\}$, where the generators $X_a$ are in some matrix representation. Is it true that the only invariant rank $n$ tensor in the ...
1
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1answer
74 views

tensor notation surprise

I'm trying to study tensors from several textbooks. One early example completely confuses me: Islam, Tensors and their Applications, in the "Preliminaries" chapter, gives this example (page 3, using ...
0
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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
43 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 ...
2
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1answer
66 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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0answers
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
70 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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0answers
65 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 ...
5
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1answer
69 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 ...
5
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3answers
342 views

Are vectors and covectors the same thing?

In Euclidean space, we usually don't distinguish between vectors and covectors (or dual vectors or 1-forms or whatever you want to call them) -- because the spaces overlap. However, a physicist ...
1
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1answer
223 views

Showing symmetry of the stress tensor by applying divergence theorem to $\int\int_{\delta V(t)} \vec{x}\times \vec{t} dS$

I'm currently working through the symmetry of the stress tensor, in relation to viscous flow. I am looking at this by examining the conservation of angular momentum equation for a material volume ...
0
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1answer
38 views

8th order isotropic tensor

Does anyone knows what is the general form of an $8$th order isotropic tensor? $2$th order is $\delta_{ij}$, $4$th order it is $\lambda \delta_{ij} \delta_{kl}+\mu(\delta_{ik} \delta_{jl} + ...
1
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1answer
38 views

formula of square of the covariant derivative

I am stuck with the calculation of $(\nabla ^2 \beta)(X,Y,Z_1,\dots,Z_r)$. In the following, capital letters are arbitrary vector fields. Suppose $\beta$ is an $(r,0)$ tensor. Denote $(\nabla ...
0
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1answer
23 views

Change of Eigenvalues of Ellipsoid Tensor during Rotation

I have an ellipsoid defined by the semiaxes $a,b,c$ and the orthonormal vectors $v_x, v_y, v_z$ describing the directions in which the axes point. The matrix ...
2
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2answers
154 views

Using the Levi-Civita alternating tensor and suffix notation to concisely write the vector product rule.

I am reading through a section on vector calculus in an electromagnetism book and it has started to use suffix notation and the Levi-Civita alternating tensor in order to prove some identities. Some ...
0
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1answer
33 views

A seemingly counterintuitive result on active and passive transformations of vectors

Let $\mathbf{v}$ be an element of a vector space with Euclidean $R^3$ as the underlying set. Assume the standard Cartesian basis $\{\mathbf{e^{(1)}, e^{(2)}, e^{(3)}}\}$ on it. Let $\mathbf{v^* = R ...
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2answers
647 views

Rotation invariant tensors

It is often claimed that the only tensors invariant under the orthogonal transformations (rotations) are the Kronecker delta $\delta_{ij}$, the Levi-Civita epsilon $\epsilon_{ijk}$ and various ...
0
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1answer
59 views

Levi-Civita symbol

Is the Levi-Civita symbol a tensor? R. A. Sharipov afirm (In "Quick Introduction to Tensor Analysis", page 30) that "...the Levi-Civita symbol is NOT a tensor..." ...
3
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2answers
165 views

Why do you need tensors of rank $>2$?

Question from someone just starting to study tensors (sorry if it's silly): So I understand (maybe?) that tensors are basically about coordinate transformations (and things that are invariant under ...
4
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1answer
87 views

Classical tensor analysis and Tensors on Manifolds

I learned tensors the bad way (Cartesian first, then curvilinear coordinate systems assuming a Euclidean background) and realize that I am in very bad shape trying to (finally) learn tensors on ...
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0answers
35 views

Let $K$ a field, $\operatorname{char}(K)=0$. Let $V$ a vector space over $K$, $\dim(V) \geq 1$, and be $f$ a $n$-tensor. Prove that $f \wedge f =0$

Let $K$ a field, with $\operatorname{char}(K)=0$. Let $V$ a vector space over $K$, $\dim(V) \geq 1$, and be $f$ a $n$-tensor ($f \in {\mathcal T}_n(V):=\Lambda^{n}(V)$), i.e., $f$ is an multilinear ...
0
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0answers
13 views

Decompose matrix multiplication?

Sorry, I can break my imagination... I would like to decompose matrix multiplication into some tensor-like operations. As we know, matrix multiplication is done the following way $a_i^j b_j^k$ ...
4
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2answers
150 views

Tensor Components

I would like to ask something On the Barrett Oneill's Semi-Riemann Geometry there is a definition of tensor component: Let $\xi=(x^1,\dots ,x^n)$ be a coordinate system on $\upsilon\subset M$. If $A ...