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 ...

learn more… | top users | synonyms

21
votes
4answers
10k views

Intuitive way to understand covariance and contravariance in Tensor Algebra

I'm trying to understand basic tensor analysis. I understand the basic concept that the valency of the tensor determines how it is transformed, but I am having trouble visualizing the difference ...
1
vote
2answers
66 views

Sum of low rank tensors

How high can the sum of $k$ low rank $m\times m\times\dots \times m$ tensors of rank $t$ be? Is there a good upper bound?
1
vote
1answer
20 views

When do exterior and tensor algebras commute with dual spaces?

Suppose $V$ is a vector space, and $V^*$ is its dual space. Furthermore, let $\Lambda(V)$ be the exterior algebra of $V$, and let $T(V)$ be the tensor algebra. When do the following two statements ...
1
vote
1answer
24 views

Finding the Gradient of a Vector Function by its Components

In Multivariable Calculus, we can easily find the gradient of a scalar function (producing a scalar field) $f : \mathbb{R^n} \to \mathbb{R}$, and the gradient function would produce a vector field. ...
1
vote
0answers
33 views

symmetric and alternating tensors in differential geometry

The following is an excerpt from Chern's Lectures on Differential Geometry: I don't see how the proof shows the other direction of the set inclusion. Would anybody explain the logic in the ...
0
votes
1answer
24 views

Levi civita and kronecker delta properties?

I'm trying to grasp Levi-civita and Kronecker del notation to use when evaluating geophysical tensors, but I came across a few problems in the book I'm reading that have me stumped. 1) ...
3
votes
2answers
96 views

Geometrical Interpretation of Tensors (intuition)

How would one describe to a non-mathematician (an undergraduate physicist actually- so please do use mathematics-i am not even sure if they can be described without mathematics!) what do tensors ...
0
votes
1answer
11 views

How to rewrite $\frac{\partial \rho u_i u_j}{\partial x_j}$ in vector notation

I want to rewrite this index notation expression to a vector notation /symbolic notation. $$\frac{\partial \rho u_i u_j}{\partial x_j}=\frac{\partial \rho }{\partial x_j}u_i u_j+\rho\frac{\partial ...
1
vote
1answer
22 views

Lie derivative of two differnt size related tensors

Let $\bar{M}=I\times M$ be a pseudo-Riemannian manifold equipped with metric $\bar g=-dt^2\oplus f^2g$ where $(M,g)$ is a Riemannian manifold, $I$ is an open connected interval and $f$ is a positive ...
2
votes
1answer
41 views

Affine manifolds which are *not* locally symmetric

Let $M$ be a manifold equipped with an affine connection $\nabla$. By standard theorems of differential geometry we have convex normal coordinate balls around every point in which geodesics are axis. ...
0
votes
0answers
19 views

Intuition about antisymmetrizing tensor equations

I was looking at the symmetries of the Riemann tensor, and tried to prove a couple of properties, namely If $\nabla$ is torsion-free, then: (i) $R^a_{\,[bcd]}=0$, and (ii) $R^a_{\,b[cd;e]}=0$. ...
0
votes
1answer
26 views

What does this colon in tensor notation mean?

I was reading a paper earlier an found the following: "The tensors satisfy orthogonality $$ <S_{:,j,:,:}|S_{:,j',:,:}> =0 $$ if $j \neq j' $. Here $<S_{:,j,:,:}|S_{:,j',:,:}>$ is the ...
0
votes
1answer
22 views

Is this tensor contraction correct?

If I start off with the $\left(p,q\right)$-tensor given by $$T_{i_1,\dots,i_p}^{j_1,\dots,j_q}e_{i_1}\otimes\dots\otimes e_{i_p}\otimes\varepsilon^{j_1}\otimes\dots\otimes\varepsilon^{j_q}$$ and I ...
4
votes
2answers
39 views

Proving the Ricci identity

I'm trying to prove the Ricci identity Let $Z^a$ be a vector field, $R^a_{\,bcd}$ the Riemann curvature tensor and $\nabla$ a torsion-free connection. Then: ...
2
votes
1answer
43 views

Finding the Gradient of a Tensor Field

Finding the Gradient of a Scalar Field I understand that you can find the gradient of a scalar field, in an arbitrary number of dimensions like so : $$grad(f) = \vec{\nabla}f = ...
0
votes
2answers
49 views

Equivalent definition of metric compatibility for a connection: what does $\nabla g$ mean?

So the definition I know for metric compatibility is: $$Xg(Y,Z)=g(\nabla_XY,Z)+g(Y,\nabla_XZ),$$ which make sense, as $g(Y,Z)$ is a smooth function from the manifold to reals and we think of $X$ as ...
2
votes
1answer
74 views

What is the most general/abstract way to think about Tensors

In their most general and abstract definitions as Mathematical Objects : A Scalar is an element of a field used to define Vector Spaces A Vector is an element of a Vector Space. Since a Scalar ...
0
votes
0answers
14 views

How to show that a tensorial field,$t$, $k$-times covariant over a differential variety $M$ is differentiable by its components

How can i show that a tensorial field,$t$, $k$-times covariant over a differential variety $M$ is differentiable if and only if the following functions (The components of t) are differentiable.: ...
0
votes
0answers
19 views

Is it possible to construct a pseudo-inverse of a 4-order tensor?

I am new here, and a physicist, so excuse me in case I do not use the right jargon but I think this is a question for the math community: In 1-D, I have a signal, represented by a vector v of ...
2
votes
1answer
40 views

What is the skew-symmetric part of the covariant derivative of a one-form?

This is a followup question to here. Let $E \to M$ be a vector bundle with connection $D := \nabla$. Extend $D$ to $E^*$ and $\text{Hom}(E, E)$. Let $E = TM$ here, and suppose that the torsion is ...
1
vote
0answers
28 views

Quadratic dual help

I'm trying to figure out the whether the quadratic dual of the following algebra is anti-commutative. My algebra is $A=T(V)/J$ where $J=\langle I\rangle$ and $I\subseteq\bigwedge^2(V)$. Firstly I ...
0
votes
0answers
21 views

Commutation of Tensor Products as operators

Suppose I have unitary operators $$A: \mathbb{C}^{2^k} \rightarrow \mathbb{C}^{2^k}$$ $$B: \mathbb{C}^{2^j} \rightarrow \mathbb{C}^{2^j}$$ For some $k,j \in \mathbb{Z}, j,k \ge 0$. How do show that ...
0
votes
1answer
20 views

On a proof that the metric volume form is parallel wrt to the Levi-Civita connection

In the context of (semi-)Riemannian geometry, the following fact is well-known: if a (semi-)Riemannian manifold $(M,g)$ is oriented, then the unique volume form $\epsilon = \mathrm{vol}_g$, induced by ...
0
votes
0answers
17 views

Quadratic dual of an algebra

Hi I'm trying to figure out the whether the quadratic dual of the following algebra is anti-commutative. My algebra is $A = T (V)/J $ where $J=<I> $ and $I \subseteq \wedge ^2 (V) $. Firstly I ...
79
votes
5answers
11k views

An Introduction to Tensors

As a physics student, I've come across mathematical objects called tensors in several different contexts. Perhaps confusingly, I've also been given both the mathematician's and physicist's definition, ...
0
votes
0answers
13 views

How to write clebsch gordan decomposition using tensor notation

Let be $G$ a Lie Group and $\textbf{N}$ its complex representation. It is well know that any state $|\ ab\ \rangle\in \textbf{N}\otimes\textbf{N} = \oplus_I\textbf{r}_I$ can be decomposed through the ...
1
vote
1answer
23 views

Tensor field acting on vector fields and covector fields gives a function?

If I have a $(p,q)$ tensor field that acts on $p$ covector fields and $q$ vector fields then does $T\left(X_1,\dots,X_p,Y_1,\dots,Y_q\right)$ return a function $f$ defined on the manifold by ...
1
vote
2answers
119 views

A Pure Maths Approach to Thinking About Vectors

Introduction Generally most students are introduced to the concept of Vector as something that has both a "magnitude and direction" and Scalars as something that only has a "magnitude and no ...
3
votes
1answer
65 views

Vector as a tensor

If we define a $(p, q)$-tensor $T$ to the vector space $V$ as a multi-linear map: $$ T : \underbrace{V^* \times \dots \times V^*}_{p} \times \underbrace{V \times \dots \times V}_{q} \to \mathbb{R} $$ ...
1
vote
0answers
12 views

Ordering of basis elements of a Lie-group representations tensor product

Let's consider a Lie Group $G$ and its complex representation $\textbf{N}$. Let's consider the decomposition $$ \textbf{N}\otimes\bar{\textbf{N}} = \bigoplus_J \textbf{r}_J $$ where $\textbf{r}_J$ ...
4
votes
1answer
2k views

How many independent components does a rank three totally symmetric tensor have in $n$ dimensions?

How many independent components does a rank three totally symmetric tensor have in $n$ dimensions? Needed for the irrep decompositon of $3\otimes 3\otimes 3$ in here. No idea where to start to ...
1
vote
0answers
24 views

Why the trace norm of a tensor is a good approximation to its rank?

In many literature, they use the trace norm of a tensor to approximate its rank, which is defined as follows: $$\|\mathcal{X}\|_{\ast}:=\sum_{i=1}^{n}\alpha_i\|X_{(i)}\|_{\ast}$$ where ${\alpha_i}'s$ ...
0
votes
0answers
34 views

Levi-Civita property on non-Euclidean space

I saw there are similar questions but it seems that no one of them gives an answer to my problem. I want to prove the following equality: \begin{equation} \frac{1}{(n-p)!p!}\varepsilon_{i_1\ldots ...
0
votes
1answer
28 views

Does the contraction of the tensor with the Kronecker delta always yield the same tensor?

Let us consider the $a^{ij}$ tensor, so that $a^{11}=1$; $a^{12}=2$; $a^{21}=3$; $a^{22}=4$. Let us consider $b^{jl}=a^{ij}\delta_{i}^{~l}$. I think that coordinates of $b_{jl}$ will be: $b^{11}=1$; ...
1
vote
1answer
875 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 $T_{ij} = \lambda \delta_{ij}$ ...
1
vote
0answers
43 views

How to interpret $a_i^k g_{kj}g^{jl}$? (Tensor index notation)

I have the equation $$a_i^k g_{kj}=-b_{ij},$$ where $a_i^k$ are given by: $a_i^k \cdot\vec r_k=\vec n_i$. Here, $g_{ij}$ and $b_{ij}$ are the matrices that determine the first and second fundamental ...
0
votes
1answer
24 views

invariance of $ds^2$ and transformation properties of $dx^i$

Invariance of $ds^2$ and transformation properties of $dx^i$ $$ ds^{\prime2} = ds^{2}$$ $$g_{ij}^{\prime}dx^{\prime i}dx^{\prime j} = g_{ij}^{\prime}\frac{\partial x^{\prime i}}{\partial ...
0
votes
0answers
17 views

Combining two isotropic tensors

When you combine two isotropic tensors do you get an isotropic tensor? e.g is $\epsilon_{ijk} \delta{ij} $ an isotropic tensor (I know it is equal to $0$ but can you prove this by saying it is a ...
1
vote
1answer
31 views

Arclength formula under the metric tensor on polar coordinates

I was reading: https://en.wikipedia.org/wiki/Metric_tensor#Examples Is it correct that in the polar coordinate example, just after the euclidean metric example, that distance is measured as: $$ ...
1
vote
1answer
42 views

When (and why) tensor product? When dot product?

This looks a lot like physics, but it is actually a math question! I will be omitting unnecessary constants for simplicity so the units might be off. I want to reduce the equation $-i\omega ...
1
vote
1answer
26 views

Counting independent components of Riemann curvature tensor

I'm having some trouble understanding the counting procedure for the number of independent components of Riemann curvature tensor $R_{iklm}$ in 4D spacetime. (The answer is supposed to be 20, but I'm ...
2
votes
1answer
207 views

Tensor Calculus Second Order Derivatives

I'm learning tensor calculus by myself through lectures and texts, and I'm presented with the problem of finding the first and second order derivatives of a scalar function of three variables that ...
0
votes
0answers
35 views

I'm not certain this makes any sense: Matrix Multiplication of Metric Tensor for calculating arclength

I was reading: https://en.wikipedia.org/wiki/Metric_tensor#Arclength Where in it gives the euclidean measure of distance as $$ ds^2 = E du^2 + 2 F du dv + G dv^2 $$ Equivalently as $$ ds^2 ...
3
votes
1answer
49 views

Kerr spacetime not symmetric?

I always see a term $dt \, d \phi$ in the Kerr-spacetime. Now assuming this means $dt \otimes d \phi$ this means that the Kerr spacetime is NOT(!) symmetric which is somehow non-sense. So do ...
0
votes
1answer
49 views

What is a Killing tensor?

Wikipedia gives the definition of a Killing tensor. Unfortunately, I don't know how to interpret the parentheses (it is also not explicitly explained in the link) and was therefore wondering whether ...
0
votes
0answers
26 views

Differential Form and Pullback of Composition of Smooth Functions (Proof Verification)

Let $f: \mathbb{R^3} \to \mathbb{R^2}$ be $C^\infty$ and let $g: \mathbb{R^2} \to \mathbb{R^3}$ be $C^\infty$. Let $h=g \circ f$ and denote the pullback of $h$ by $h^\ast$.Let $\omega$ be a smooth ...
1
vote
2answers
42 views

Proof of the contracted Bianchi identity

In proving the contracted Bianchi identity, I have problems understanding the contractions. Starting with the second Bianchi identity: $$R_{ijkl;m}+R_{ijlm;k}+R_{ijmk;l}=0$$ The first step is to ...
0
votes
0answers
18 views

clarification on what order of differential form means

I'm confused about what the order of a differential form is, from my understanding it is the k-tuple of vectors that the function takes in, but this doesn't seem right. For instance, according to my ...
4
votes
0answers
88 views

Eigendecomposition and eigenvalue scaling for tensor networks

I've recently been interested in studying tensors and networks of tensors. Consider the following quantity: $$Tr(A^N)$$ Where $A$ is a square matrix and $N$ is large. This is extremely easy - all we ...
1
vote
1answer
49 views

Tensor operator

I have come across the following expression: H:E where, H = e(levi-cita symbol)*a constant which means a 3rd order tensor with 27 components E = 2nd order tensor, now, what does H:E mean? I know ...