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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2
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1answer
36 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
votes
2answers
49 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
vote
1answer
30 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
vote
1answer
14 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 ...
0
votes
0answers
22 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) ...
0
votes
0answers
19 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
23 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? ...
1
vote
0answers
14 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
74 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
15 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
20 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 ...
-3
votes
1answer
25 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)?
0
votes
1answer
16 views

Little mistake with Levi-Civita symbol property

I have this equation $ \varepsilon_{ijk}B_k = \partial_iA_j - \partial_jA_i $ and I was asked to prove $\mathbf{B}=\nabla\times\mathbf{A}$, where $\mathbf{B}=B^i\mathbf{e}_i$ and ...
0
votes
1answer
22 views

How to convert $V \otimes W^*$ to a matrix space?

Namely let's say we have chosen basises $e_1, e_2, ... e_k$ for $V$ and $j_1, j_2, ... j_n$ for $W$. Now, since we can always just convert them separately, and then add the matrixes, how we represent ...
1
vote
2answers
43 views

maximizing a coordinate of $x^T A^T A x \leq r^2$

Given a vector $\mathbf{x} \in \mathbb{R}^n$, a scalar $r\gt 0$ and an invertible matrix $\mathbf{A} \in \mathbb{R}^{n\times n}$, I'd like to maximize one of the components $x_\alpha$ constrained by ...
0
votes
0answers
10 views

Why the gradient of the r vector is the identity map, geometrically speaking?

When doing some simple quantum mechanics problem involving commutators, I forgot the result of this expression $$\left[\vec{r} ,\hat{p}\right]$$ Thus I then brute force it using the definition of ...
4
votes
2answers
78 views

How do I understand constraints on high order derivatives of the Gauss Map?

I'm trying to understand the constraints resulting from differentiating an unit normal field $N$ on a surface $S$ in $\mathbb{R}^3$. If I write the unit-length constraint at a point $p \in S$, I ...
0
votes
0answers
32 views

How do you get the curvature tensor of the Schwarzschild Solution?

So, on the Wikipedia page on the derivation of the Schwärzschild solution , I get everything up to the part about the Ricci tensor. What were the components of the tensor that were used? Could ...
0
votes
1answer
33 views

What is the prerequisiste to study Tensors for application in signal processing?

I want to study Blind Source separation in signal processing for this I need to study Tensors and have a basic idea about rank, border rank and other concepts. Right now I am studying from ...
6
votes
2answers
109 views

Intuition about $v\otimes w$

In Physics and Differential Geometry usually tensors of type $(k,l)$ on a vector space $V$ over $\mathbb{F}$ are defined as multilinear functions $$f : \underbrace{V\times\cdots\times V}_{k \ ...
0
votes
0answers
16 views

Tensor as multi-dimensional array

I have a finite (but unknown) dimensional discrete object that I would like to represent as tensor (as a multi-dimensional matrix) with some basis. Can someone guide me on matrix-transformations that ...
2
votes
1answer
25 views

Showing $T$ equivalent to linear map

Let $T$ be a $(1,1)$ tensor over a vector space $V$. Let $\left\{e_a\right\}$ be a basis for $V$ and $\left\{f^a\right \}$ be its dual basis. Show that $T$ is equivalent to a linear map $V^* ...
2
votes
1answer
39 views

Relationsip between two definitions of the christoffel symbol?

When I first started learing about tensor calculus, the professor defined the Christoffel symbol as $$\Gamma ^a _{bc} = Z^a \cdot \partial_b Z_c $$ Where $Z^a$ is a contravariant basis vector and ...
0
votes
1answer
27 views

What is mean by “trace on any pair of indices”?

I am reading the book Riemannian Manifolds, written by John M. Lee. On page $53$, the author defines a connection on the tensor bundle $\text{T}_l^k(M)$ and says it satisfies (d) $\nabla$ commutes ...
1
vote
1answer
49 views

How to prove that below quantity is a Third Rank Tensor

$F^{ik}$ is an antisymmetric tensor. I want to prove that below quantity is a Third Rank Tensor. $$\dfrac{\partial F_{ik}}{\partial x^{l}} + \dfrac{\partial F_{kl}}{\partial x^{i}} + \dfrac{\partial ...
0
votes
1answer
46 views

Notation in symmetric and alternating products of forms

In Lee's Riemannian Manifolds book, we see that a Riemannian metric $g$ can be expressed locally in coordinates $(x^1, \ldots, x^n)$ by $g = g_{ij} dx^i \otimes dx^j$. Introducing the symmetric ...
1
vote
1answer
31 views

How to prove the skew symmetry of the Riemannian tensor?

I parallel transported a vector around a parallelogram formed by another two vectors to get the Riemannian tensor: $$R^m{}_{lkj} = \left( \partial_k \Gamma^m{}_{jl} + \Gamma^m{}_{kn} \Gamma^n{}_{jl} ...
2
votes
1answer
30 views

Are there any 3rd order tensors satisfying $e_{ijk} e_{lmk} = \delta_{il} \delta_{jm}$ in dimensions higher than three?

My question is simply wrote on the title. (I'm using Einstein's contraction rule.) In the case of three dimensions, I can construct the Levi-Civita-like tensor as follows. \begin{align} e_{ijk} = ...
2
votes
1answer
59 views

Minimize Product of Sums of Squared Distances

The Question Given two sets of vectors $S_1$ and $S_2$,we want to find a unit vector $s$ such that $$\{\sum_{u\in S_1}(\|u\|^2-\langle u, s \rangle^2)\} \cdot \{\sum_{v\in S_2}(\|v\|^2 - \langle v, ...
7
votes
0answers
60 views

Determinant of a tensor

Is there such a thing as the determinant of a tensor of rank $\gt 2$? I'm tried to think how it might be defined -- potentially like, the determinant of the tensor $A=a_{ijk}$ is ...
2
votes
0answers
35 views

Difference of two connections is a tensor

I am currently reading through Jost's Riemannian Geometry and Geometric Analysis and am seeking clarification of the statement in the title. Jost abstractly defines a connection on a vector bundle ...
1
vote
0answers
31 views

Why the following is not a tensor?

Say we have an arbitrary coordinate system, in which a position vector is represented by: $$\vec V=Z^i\vec Z_i$$ Where $\vec Z_i$ is a covariant basis. Now $\partial {\vec Z_i}/\partial Z^j$ term has ...
4
votes
2answers
45 views

Proof of a tensor identity involved in the derivation of the Einstein field equations?

On the wikipedia page for the einstein-hilbert action, the section for the derivation of the einstein field equations cites this identity: $$ \sqrt{g} \nabla_\mu A^\mu = \partial_\mu (\sqrt{g} A^\mu) ...
1
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0answers
29 views

Showing two tensors are equal

Let $T_{ijkl}$ be a 3-d tensor satisfying $T_{ijkl}=-T_{jikl}=-T_{ijlk}$, and $T_{ijij}=0$. Let $S_{pq}=-T_{rprq}$. I'm trying to show that $$T_{ijkl}=\varepsilon_{ijp}\varepsilon_{klq}S_{pq}$$ Now, ...
0
votes
0answers
22 views

Full time derivative of the function (Frank-Oseen energy). Calculus applied to physics problem

This question is about physical theory, but my question is pure nathematical, so I post it here. I don't think you have to know physics in order to answer it. I am studying liquid crystal theory with ...
2
votes
0answers
24 views

Tensor notation in vectors

I have the following expression $\partial_{x_a}(\partial_{x_b} \rho \partial_{x_b}\rho) - \partial_{x_b}(\partial_{x_a}\rho\partial_{x_b}\rho)$ How do I write this in vector notation? At least the ...
1
vote
1answer
32 views

Symmetries of the space form of riemann curvature tensor

We have $R_{abcd}=(g_{ac}g_{bd}-g_{ad}g_{bc})$ I need to establish the symmetry: $-R_{bacd}=R_{abcd}$ What I thought was just interchange a and b in the expression to get: ...
1
vote
0answers
16 views

Why is the border rank and rank different for order 3 tensors and above?

Recall the definition of border-rank of a tensor T: border-rank(T) = the minimum r such that $\forall \epsilon > 0$ there exists an approximate tensor $T' = \sum^r_{i=1} u_i \otimes v_i \otimes ...
1
vote
2answers
32 views

Action of universal R-matrix of U_q(sl_2)

My question is really simple but requires a few definitions. No special knowledge of quantum groups should be needed, it is more about tensor algebra. Let $q \in \mathbb{C}$ with $q \neq 0, \pm 1$. ...
0
votes
1answer
22 views

Multiplying metric tensors.

Suppose I have the metric $g_{ab}$ in a k-dimensional manifold. Firstly, do metric tensors like this always commute? Is it always necessarily true that $g^{ab}g_{bc}=\delta^a_c$? What happens when I ...
0
votes
0answers
33 views

Why cant we define the Einstein Tensor in an easier way?

The Einstein tensor is defined as: $G_{ab}=R_{ab}-\frac{1}{2}g_{ab}R$ where $R=g^{ab}R_{ab}$. So why cant we just simplify this like: ...
7
votes
3answers
317 views

Why is the Riemann curvature tensor the technical expression of curvature?

According to my textbook on general relativity (Sean Carrol's book) and differential geometry, the Reimann curvature tensor is the technical expression of curvature. What makes the tensor so special? ...
1
vote
1answer
19 views

Tensor equations. Can I change an equation from covariant to contravariant?

Say I have a tensor equation like $G_{ab}=R_{ab}-\frac{1}{2}g_{ab}R$. Does this also imply that $G^{ab}=R^{ab}-\frac{1}{2}g^{ab}R$?
0
votes
1answer
11 views

Need definition of symmetric and antisymmetric tensor representations of a Lie algebra

I couldn't find a definitive answer online. Suppose we have a representation of a Lie algebra $(\pi,V)$. Consider the symmetric and antisymmetric vector subspaces of the $k$-th tensor product of ...
1
vote
2answers
53 views

Levi-Civita tensor

Show that $\epsilon_{ijk} A_{il} A_{jm} A_{kn} = \det(A) \epsilon_{lmn}$ where $\epsilon$ epsilon is the standard Levi-Civita symbol and A is a three dimensional matrix. I found the above ...
1
vote
0answers
36 views

What is this notation $\odot$ for?

(Note that symmetric algebra and symmetric tensor do not coincide when the characteristic is not $0$.) I'm reading this aricle:http://en.m.wikipedia.org/wiki/Symmetric_tensor And here it defines ...
0
votes
1answer
41 views

Are all 2D tensors in a specified flat metric equal to that same metric conformally scaled?

I have a tensor $T_{mn}$ where its indices coorespond to a flat metric $g_{mn}$. I want $T_{mn}$ to be a new metric $\tilde{g}_{mn}$, such that $T_{mn}(g_{rs}) = \tilde{g}_{mn}$. A theorem says that ...
1
vote
1answer
11 views

Is it okay to define k-th symmetric power of $M$ in this way?

I want to define the tensor algebra and related algebras in a very formal way. I will illustrate how I tried to define algebras below. Let $R$ be a commutative ring and $M$ be an $R$-module. ...
0
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0answers
18 views

What's actually $S^k(M)$?

Let $R$ be a commutative ring and $M$ be an $R$-module. Let $T(M)$ be the tensor algebra of $M$. Then, what is $S(M)$ (symmetric algebra and $S^k(M)$? Some articles define $S(M)$ as a quotient of ...
0
votes
0answers
16 views

Under what condition, does this universal property of tensor algebra hold?

Let $R$ be a commutative ring and $M$ be an $R$-module. Note that the tensor algebra $T(M)$ is a unital associative $R$-algebra. Below is the universal property of the tensor algebra. Theorem: ...