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
30 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
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 ...
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1answer
49 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
58 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
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: $$ ...
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1answer
40 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
32 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
17 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
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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1answer
47 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 ...
3
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1answer
85 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 := ...
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1answer
56 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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0answers
24 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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0answers
10 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 $$ ...
3
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2answers
65 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
56 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
29 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
65 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
64 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
68 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
338 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 ...
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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 ...
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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} + ...
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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 ...
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1answer
19 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 ...
0
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1answer
32 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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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
34 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 ...
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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 $a_i^j b_j^k$ ...
2
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1answer
30 views

Building intuition for tensors in machine learning

I'm trying to understand tensors in the context of machine learning, but all the resources that mention tensors that I've found so far were building the intuitions through physics applications. As ...
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1answer
47 views

A question about vector fields and divergence

I am reading the paper http://www.goshen.edu/physix/mathphys/gco/TensorGuideAJP.pdf in order to gain a basic understanding about tensors. I had some difficulties about understanding some definitions. ...
2
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1answer
74 views

Introductory questions about tensors

I am trying to understand the concept of tensors. I seem to understand that they are generalization of vectors: They are subject to similar basis transformations with vectors but I am somewhat ...
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0answers
35 views

Time Evolution of Deformation Gradient Tensor in Lagrangian Frame

I found the following proof in a paper: $\frac{D\mathbf{F}}{Dt} = \frac{D\frac{\delta\mathbf{x}}{\delta\mathbf{X}}}{Dt} = \frac{\delta\frac{D\mathbf{x}}{Dt}}{\delta\mathbf{X}}=\frac{\delta ...
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0answers
37 views

Dimension of the space of tensors obtained by making partial symmetrizations and skew-symmetrizations.

Let $A=(a_{i_1\dots i_k})_{i_1,\dots,i_k=1}^n$ be a higher order cubic tensor or hypermatrix. The following two facts are well-known and are easy to prove: ${(\bf 1) }$ The dimension of the ...
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0answers
61 views

Planetary motion integral

I was reading Planetary Motion (page 117) in Barry Spain's Tensor calculus, and stupidly enough, I didn't understand this. The equations are: $$\frac{d^2\psi}{d\sigma^2} + ...
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1answer
20 views

Boundedness of Riemannian curvature gradient

I'm reading a paper by Wan-Xiong Shi "Deforming the metric on complete Riemannian manifolds". And there is a statement without proof. It can be summarized as follows: Let $B(x_{0},\gamma)$ be a ...
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1answer
65 views

Question Symmetric 2-Tensors on Vector Fields

I have a question on a direct computation. How would one compute the following $$ (dx \odot dy ) \left(\frac{1+uv+xy}{1+xy} \frac{\partial}{\partial v}, \frac{\partial}{\partial y} \right) $$ and$$ ...
2
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1answer
69 views

Construct tensors from differential forms?

Let $(M,g)$ be a Riemannian manifold, differential forms are defined using tensors, could we define a tensor using a differential form? For example, if $\omega$ is a two-form on $M$ which is expressed ...
0
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1answer
16 views

sum representation by/ determinant of elementary tensors

Consider the bijective linear map: $\alpha : K^2 \otimes K^2 \to Mat(2 \times 2,K), \alpha(v \otimes w) = vw^t, v, w \in K^2$ , where $K$ is an arbritrary field. First I want to show that every ...
0
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1answer
29 views

Quick question about contravariant and covariant tensors

I have seen many different notations to denote contravariant/covariant and mixed tensors. For example, I think the notation $\omega^{v}_{\,\,\,\mu}$ stands for a mixed tensor, where one index ...
0
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1answer
28 views

Covariant vectors

As far as I'm aware, covariant vectors are defined by how they transform: But I've also heard that the covariant components of a vector are defined as the dot product of the vector and the various ...
3
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1answer
39 views

Tensor Notation

I'm just starting to learn about tensors, and have a question. I'm looking at the statement $\Lambda_{\mu}\,^{\alpha}= \eta_{\mu\nu}\eta^{\alpha\beta}\Lambda^{\nu}\,_{\beta}$ What is the difference ...