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
24 views

Prove the exterior derivative of the following (n-1) form is zero

Let $\omega(x)=\frac{1}{{\parallel x \parallel}^n}\displaystyle\sum_{i=1}^{n}(-1)^{i-1}x_{i} dx_{1} \wedge \dots \wedge \widehat{dx_{i}} \wedge \dots \wedge dx_{n}$ be a differential $(n-1)$ form on ...
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. ...
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0answers
35 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
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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) ...
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 ...
0
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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
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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. ...
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0answers
20 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
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 ...
2
votes
1answer
44 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 = ...
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: ...
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
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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
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0answers
20 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 ...
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
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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 ...
1
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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
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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
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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 ...
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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
126 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
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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: $$ T : \underbrace{V^* \times \dots \times V^*}_{p} \times \underbrace{V \times \dots \times V}_{q} \to \mathbb{R} $$ ...
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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 $\textbf{r}_J$ ...
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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
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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
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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: $b^{11}=1$; ...
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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 ...
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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
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 ...
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 ...
0
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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 ...
1
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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: $$ ...
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
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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
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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 ...
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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 ...
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0answers
19 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 ...
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0answers
29 views

If a k-form vanishes in the neighborhood of p then it vanishes at p

Let w be a k-form defined in an open set A of $\mathbb R^n$. We say that w vanishes on x if w(x) is the zero tensor. Show that if w vanishes at each x in a neighborhood of $x_0$ then dw vanishes at ...
1
vote
2answers
28 views

Verify a Tensor Differential Identity

I would like to show that the ID below holds: $$(n \cdot \nabla n)=1/2 \nabla(n \cdot n)-(n \times(\nabla \times n))$$ Using the Einstein notation I've come up with: $$(n \times(\nabla \times ...
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0answers
13 views

Energy Conditions

Hi guys im trying to solve the energy conditions for a specific stress energy tensor but have come up at a stumbling block. How would i calculate the energy conditions for the following stress ...
0
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0answers
22 views

Writing a linear transformation in terms of elementary alternating k-tensors

Let T: $\mathbb R^m \rightarrow R^n $ be the linear transformation T(x) = Bx If $\alpha_I$ is an elementary alternating k-tensor on $\mathbb R^n$, the $T^*\alpha_I$ has the form $T^*\alpha_I = ...
0
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0answers
33 views

Given Hooke's law, prove the stiffness tensor is indeed a tensor

I have two symmetric, second-rank tensors $\sigma_{ij}$ and $e_{ij}$ related by $$\sigma_{ij}=c_{ijkl}e_{kl}.$$ Assuming $c_{ijkl}=c_{ijlk}$, why must $c$ be a tensor?
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0answers
12 views

Verify $T^*(f^\sigma)$ = $(T^*f)^\sigma$

Where $T^*$ is linear. $f^\sigma(v_1,...,v_k)$ = $f(v_{\sigma(1)},...,v_{\sigma(k)}) $T^*f(v_1,...,v_k)$ = $f(T(v_1),...,T(v_k))$ Attempt at the proof: I didn't use the fact that T is linear ...
1
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0answers
20 views

The determinant of $X_I$ = wedge product of elementary tensors

Let $(x_1,\dots,x_k)$ be vectors in $\mathbb{R}^n$; let $X$ be the matrix $X = [x_1 \cdots x_k]$. If $I = (i_1, \dots, i_k)$ is an arbitrary $k$-tuple from the set $\{1,\dots, n\}$, show that ...
4
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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 ...