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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5
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1answer
59 views

Taylor Expansion of tensor moved by a flow.

I am reading Peter Petersen's notes on manifold theory and he introduces Lie Derivatives in the following way. "Let $X$ be a vector field and $F^t$ the corresponding locally defined flow on a smooth ...
2
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2answers
109 views

What is the modern approach to tensors?

I have recently started studying tensors a bit from the index notation point of view -- I understand contractions and the metric tensor and such well enough. However I've been told that this approach ...
4
votes
1answer
67 views

What are higher derivatives?

From Wikipedia: Higher derivatives can also be defined for functions of several variables, studied in multivariable calculus. In this case, instead of repeatedly applying the derivative, one ...
3
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1answer
34 views

Multivariable Calculus with Tensors

I'm looking for a book at the undergraduate level on multivariable calculus (for a 2nd course of multivariable calculus) that introduces and makes use of tensors to describe higher order derivatives ...
5
votes
1answer
116 views

Divergence theorem for a second order tensor

I want to integrate by part the following integral in cylindrical coordinates $$\int \vec{r} \times (\nabla \cdot \overline{T}) ~d^3\vec{r} $$ where $\overline{T}$ is a second order symmetric tensor ...
0
votes
1answer
37 views

Tensor contraction

Given that: $T_{i,j}=\lambda\theta\delta_{i,j} + 2\mu E_{i,j}$ Show that: $T_{i,i} = 3\theta \lambda + 2\mu E_{i,i}$ I didn't get the intuition behind tensor contraction, thus i can not solve this ...
4
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0answers
57 views

Matrix Calculus

My school offers Matrix Calculus class next semester. I have never heard about this subject before and got intrigued. After a short chat with professor I found myself unable to get rid of suspicion ...
0
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1answer
24 views

Lorentz transformation of electromagnetic field tensor

I need to calculate: $f^{\mu'\nu'}=L^{\mu}_{\kappa}L^\nu_\lambda f^{\kappa\lambda}$ Where $L^\nu_\lambda$ is the usual Lorentz transformation matrix I thought that I just needed to do some normal ...
1
vote
1answer
22 views

Relationship between euclidean metric in sphere of radius $r$ and the unit sphere.

I want to show $g_r=r^2g_1$ where $g_1$ is the (Riemannian) metric in the unit sphere induced by its inclusion in $\mathbb{R}^n$ and $g_r$ is the metric in the sphere of radius $r$ also induced by ...
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0answers
30 views

Understanding Extension of Scalars in a Vector Space

$\newcommand{\R}{\mathbf R}\newcommand{\C}{\mathbf C}$ Low-Tech Complexification: Let $V$ be a finite dimensional vector space over $\R$. We can forcefully make $W:=V\times V$ into a complex vector ...
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0answers
37 views

Levi-Civita tensor in curved space

In the book "Gravitation and cosmology" by Weinberg at the page 99-100. He defines the Levi-Civita tensor as $\epsilon^{0123}=+1$ from which he writes ...
1
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1answer
85 views

Multiplying two tensors of the Levi-Civita type

How to multiply two epsilons with one another? We know ...
0
votes
1answer
39 views

Covariant metric tensor of a subspace

Suppose $f_1,f_2$ and $f_3$ are vectors in a vector space $V$ with a dot product. Me assume that the vectors are linearly independent. What does it mean to find the covariant metric tensor of ...
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1answer
24 views

Deriving the Geodesic Equation

I found a derivation of the geodesic equation that includes this step as I write it: $$ \frac{d (g_{ab}\dot{x}^b)}{dt}=\frac{1}{2}\partial_ag_{bc}\dot{x}^b\dot{x}^c \Rightarrow \\ \\ ...
3
votes
1answer
110 views

Comparing metric tensors of the Poincare and the Klein disk models of hyperbolic geometry

I was trying to compare the metric tensor at the wikipedia pages of the Beltrami Klein model https://en.wikipedia.org/wiki/Klein_disk_model and the metric tensor of the Poincare disk model at ...
1
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0answers
34 views

How does a 4-tensor act as a linear trasformation of 2-tensors?

I'm trying to understand tensors by looking at this table and thinking about the various types of transformations the tensors represent. From the linked table, I tried looking up some of the less ...
1
vote
1answer
21 views

Locality of tensors part of definition?

I am wondering whether linearity with respect to scalar functions $f \in C^{\infty}(M, \mathbb{R})$ is part of the definition of a tensor? Let me explain it by referring to the Riemann curvature ...
3
votes
1answer
30 views

Why should the metrical groundform on a variety be a quadratic form?

I'm learning General Relativity and I can't understand why the distance function on space time is a quadratic form $$\textrm{d}s^2=g_{\mu\nu}\textrm{d}x^{\mu}\textrm{d}x^{\nu}$$ I explain it through ...
4
votes
1answer
86 views

Show that $\mathbb{C} \otimes_\mathbb{Z} \mathbb{C} \cong \mathbb{C} \otimes_\mathbb{Q} \mathbb{C}$

Show that $\mathbb{C} \otimes_\mathbb{Z} \mathbb{C} \cong \mathbb{C} \otimes_\mathbb{Q} \mathbb{C}$ This is not homework, it is part of an answer of Show that $\mathbb{A}_\mathbb{C}^2 \ncong ...
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vote
0answers
17 views

The space $V^{0}_{p}$ of p times covariant tensors and canonical isomorphisms

I have been studying tensor calculus by myself, but I have found the following claim in my book: The space $V^{0}_{p}=V^{*} \otimes \cdots \otimes V^{*}$ of $p$ times covariant tensors is ...
1
vote
1answer
20 views

Showing that two tensors are coaxial

Given two second-order tensors $\mathbf{C} = C_{ij}\mathbf{e}_i\otimes\mathbf{e}_j$ and $\mathbf{U} = U_{ij}\mathbf{e}_i\otimes\mathbf{e}_j$ with the following relation between $\mathbf{C}$ and ...
4
votes
2answers
66 views

A question about irreducible representation of symmetric group (permutation group) in tensor space and tensor contraction

In chapter 13 of the textbook of Group Theory in Physics by Wu-Ki Tung, Lemma 2 discusses the equivalence of two irreducible representations of GL(m) on ${T^i}_j$. In its proof, it simply mentioned ...
2
votes
2answers
43 views

Is there a way to factor out the middle tensor product?

Given a state $(|1\rangle \otimes |0\rangle \otimes |1\rangle) + (|0\rangle \otimes |0\rangle \otimes |1\rangle)$, is it possible to factorise out the $|0\rangle$ in the middle of both of them? ...
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0answers
21 views

Is this proof of the product of determinants in tensor notation correct?

I'll start with the matrix C which is the product of the matrices A and B. $$c^i_k = a^i_jb^j_k$$ The determinant of C is $$\frac{1}{3!}\delta_{ijk}^{rst} c^i_rc^j_sc^k_t $$ by the definition of ...
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2answers
79 views

Looking For a Coordinate Free Way to Prove This Linear Algebra 'Fact'

$$\newcommand{\mc}{\mathcal}$$ Let $V$ be an $n$-dimensional vector space over a field $F$. (We use $\mc L(V)$ to denote $End(V)$). For each $v\in V$, define $\Theta_v:\mc L(V)\to V$ as ...
1
vote
1answer
29 views

How to simplify this expression using tensor notaion?

$\nabla^2 (\phi A)-A \nabla^2 \phi -2(\nabla \phi \cdot\nabla)A$ Where $A,\phi$ are any sufficiently smooth vector and scalar fields respectively.
5
votes
2answers
69 views

Curvature tensors and bivectors

At the beginning of the paper "The curvature of 4-dimensional Einstein spaces", by Singer and Thorpe, the authors define the space $\mathcal{R}$ of curvature tensors of the vector space $V$ as the set ...
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0answers
50 views

Matrix transponse in tensor notation

In this paper http://www.ita.uni-heidelberg.de/~dullemond/lectures/tensor/tensor.pdf at the end of chapeter 2 the author says that in index notation a matrix is written as $A^\mu_{\;\;\nu}$ and its ...
4
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3answers
140 views

On Learning Tensor Calculus

I am highly intrigued in knowing what tensors are, but I don't really know where to start with respect to initiative and looking for an appropriate textbook. I have taken differential equations, ...
2
votes
1answer
39 views

When is shear useful?

I'd never heard of the shear of a vector field until reading this article. Shear is the symmetric, tracefree part of the gradient of a vector field. If you were to decompose the gradient of a vector ...
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0answers
25 views

Finding the basis one forms (covectors) corresponding to a particular formulation of basis vectors

This formulation of the basis may be wrong, or I may be missing something, but I can't see a way to formulate the covectors this particular basis: \begin{align} \vec{e}_0 &= \vec{x} + \vec{y} ...
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2answers
67 views

Dual tensor for partial derivative, if it has any meaning

I'm trying to find out some details about tensors, so my question maybe isn't quite correct. What if $\omega$ is volume form in $(x,y,z)$ coordinates, then how to understand that ...
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2answers
42 views

Contraction as Adjoint of Wedging

Let $V$ be an $n$-dimensional vector space. Given $\phi^1\wedge \cdots \wedge \phi^k\in \bigwedge ^k(V^*)$ and $v_1\wedge\cdots\wedge v_k\in \bigwedge^k(V)$, we write $$ \langle \phi^1\wedge ...
0
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0answers
17 views

Find all components of the tensors $T_{ij}=u_iv_j+v_iw_j$ and $S_{ijk}=u_iv_jw_k-v_iu_jw_k+v_iw_ju_k-w_iv_ju_k+w_iu_jv_k-u_iw_jv_k$.

Given three vectors, $\vec u=(u_1,u_2,u_3)$, $\vec v=(v_1,v_2,v_3)$ and $\vec w=(w_1,w_2,w_3)$. Find all components of the tensors $T_{ij}=u_iv_j+v_iw_j$ and ...
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0answers
8 views

Tensors which are symmetric and antisymmetric in overlapping groups

Say I have the following tensor $T_{abc}$ such that $$ T_{(a[b)c]} $$ Ergo, it is symmetric in indices $a$ and $b$ and antisymmetric in $b$ and $c$. Keeping in mind the various properties that ...
2
votes
0answers
29 views

$T(M) \cong R<x_1,\ldots,x_n>$ isomorphism question

I have that $T(M) = \bigoplus_{i =1}^{\infty} T^i(M)$ where $T^k(M) = \bigotimes_{i =1}^{k} M$. In a paper i have that to prove such isomorphism, we define: $$\Phi:R<x_1, \ldots, x_n> ...
0
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1answer
47 views

Trace in Einstein notation

I know quite well what the trace of a matrix is; however, I am not quite sure I understand the meaning of the 'trace' concept when applied to tensors. I would be very grateful to you if: 1) You could ...
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votes
0answers
61 views

Forming a (1,1)-tensor field from a (0,2)- and (2,0)-tensor field

Let $A$ and $B$ be 2-tensor fields on a manifold, contravariant and covariant respectively. Prove that there exists a smooth (1,1)-tensor field $C$ with components defined by $$C^i_j = ...
3
votes
1answer
50 views

Is there an “internal” definition of the tensor product?

We have the following "internal" definition of the direct sum: A vector space $V$ with subspaces $S,T$ is said to be the direct sum of $S$ and $T$ if $S + T = V$ and $S \cap T = \{0\}$. (Of course ...
2
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2answers
53 views

Is $a_i\mathbf e^i$ always equal to $a^i\mathbf e_i$?

The way that the covariant basis was described to me was that we could represent any vector $\mathbf a$ as either $\mathbf a=a_i\mathbf e^i$ or $\mathbf a = a^i\mathbf e_i$ (with the Einstein ...
1
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1answer
50 views

Resources on exterior algebra, wedge product, geometric product and tensors

I would like to learn exterior algebra, wedge product and geometric product along with their applications in physics. Is there a good source you can recommend? Should I study differential geometry in ...
1
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1answer
43 views

Tensor Notation Upper and Lower Indices

I want to ask what the difference between the tensors $T_i^{\; j}$ , $T_j^{\; i}$ , $T_{\; i}^{ j}$ , and $T_{\;i}^{j}$ are. In particular I am asking about the matrix representations of these tensors ...
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0answers
26 views

derivative of a tensor function

If e is a second order tensor and it's symmetrical, assuming abs() is a tensor function such that returns all the components of e be positive, I am interested the derivatives of the function abs(e) ...
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0answers
30 views

Pullbacks of symmetric tensors commute with products

The problem: Show that $$F*(AB)=(F*A)(F*B)$$ where F is a smooth map from a smooth manifold M to another smooth manifold N, A and B are symmetric tensor fields on N, and $F*$ denotes the pullback ...
2
votes
1answer
80 views

Matrix inversion via Levi-Civita symbols

Using Cramer's formula for the inverse of a matrix $M_{ij}$, is it possible to express the entries $(M^{-1})_{ij}$ in terms of the entries $M_{ij}$ using the Levi-Civita symbol and Kronecker deltas? ...
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0answers
19 views

Normalize tensors ($3$-by-$3$ matrix) so that the largest eigenvalue is 1?

I am trying to "normalize" a tensor $T$ (a $3$-by-$3$ matrix). The paper says ... the normalization of a tensor scales all eigenvalues so that the largest one equals to $1$. I am confused. ...
3
votes
2answers
89 views

How are these definitions of the inertia tensor the same?

I'm looking for some help in understanding the inertia tensor (not the physics, just the math). I'm trying to figure out how to convert between the wedge product and tensor product definitions. ...
2
votes
0answers
19 views

An extension of change of variables in double (and $n$-?) integrals - second-order Jacobian?

I'm aware that there are many, many questions regarding changing variables in double and triple integrals. The equation that typically pops up in textbooks is \begin{align} ...
2
votes
1answer
46 views

Proving vector Identities (Using the Permutation Tensor and Kroenecker Delta)

Prove the following vector identities by using permutation tensor and kroenecker delta. $$(\vec{A} \times \vec{B}) \times (\vec{C} \times \vec{D}) = (\vec{A} \cdot (\vec{B} \times \vec{D})) \cdot ...
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votes
1answer
30 views

derivative of a linear mapping

What is the derivative of a linear mapping A: R^n -> R^n? I assume it must be a tensor. In particular, if I have a linear function of a vector x, A(x), what is DA(x)?