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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37 views

Levi-Civita symbol (permutation tensor)

I was going over a past exam and the following two questions came up: Show that the Levi-Civita symbol $\varepsilon_{ijk}$ is a tensor. Evauluate the following: $\varepsilon_{ijk}\varepsilon_{ipq}\...
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1answer
120 views

Coordinate-free notation for tensor contraction?

I am not sure if I can prevent this question from being too vague or with too large an overlap with other similar math.SE questions, but I will do my best... A standard linear operation in tensor ...
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1answer
28 views

Tensor derivative

What is the result of $$ \frac{\partial^2 \left(A^{ij}y^ix^j+B^{ij}x^iy^j\right)}{\partial \bf x\partial \bf y} $$ where $i,j$ obey Einstein summation convention, $A,B$ are constant, ${\bf x}=[x^1,x^2,...
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0answers
29 views

Show a beam is in equilibrium given the stress tensor

Having some trouble with this: https://gyazo.com/0835bdaa8e01cb976765aac94555f6ef I know how to show that at x_2 = -h the surface traction is zero, but I'm not sure how to show it's in equilibrium? ...
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1answer
42 views

An example of tensor product

Let $$ \otimes:R\times R\rightarrow W $$ $$ f:R\times R\rightarrow R~~,~f(X,Y)=XY $$ $\otimes$ is tensor product, $W$ is a vector space, and $f$ is a bilinear may. As I know , we need to find a ...
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46 views

Symmetry of Rank Two Tensor with Mixed Components

I understand that a rank two tensor (t) is classified as symmetric if $t^{ij} = t^{ji}$ or $t_{ij} = t_{ji}$. Later in my reading, I came across the following quote: It is not useful to speak of ...
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1answer
17 views

Defining covectors when the basis is oblique

Given a $2$-dimensional vector space with an oblique unit length basis, say, $(f_1, f_2)$, what is the dual vector or covector corresponding to $f_1$, call it $\hat f_1$? There appear to me to be ...
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0answers
28 views

Tensor Eigenstates

I have the following equation: $$f_i^´(t+1)=\sum_{jk}R_{i|jk}\tilde{f_j}(t)\tilde{f_k}(t)$$ It is about evolution of a population. I use this equation in my python program in the following way: <...
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0answers
47 views

Why is lowering and raising index not affecting the value of a tensor?

I have two questions to ask about: For example I have a tensor $T(r,s)$ by which it means that $T$ operates on $r$ vectors and $s$ dual vectors. Take for example $T(3,0)$, so in this notation it ...
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5answers
92 views

In mathematics, what is an $N \times N \times N$ matrix?

In mathematics, what is an $N \times N \times N$ matrix? I think this is a tensor but definitions of tensors that I have read are so overly complicated and verbose that I have trouble understanding ...
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0answers
23 views

Raising index on covariant derivative

So suppose $X$ is some vector field and $t$ is a tangent vector to some curve on some smooth manifold. Then $t^a\nabla_a X$ gives the directional derivative of the vector field in the direction of $t$....
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1answer
16 views

What is the intuitive meaning of the partial derivate in coordinate transforms?

We learned that when changing coordinate system from $u^i$ to $u'^i$, a contravariant vector transforms like this (using the Einstein-convencion): $v'^i = \frac{\partial x'^i}{\partial x^j}v^j$, And ...
0
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1answer
33 views

Tensor product, coordinates

Find the coordinates of bivector u⊗v with the respect to cannonical basis and basis M = ((1,2),(1,3)), u = (1,1) v=(1,-2). Please help, does it even have the solution? After the tensor multiplication ...
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2answers
33 views

Representing a linear transformation as a tensor

I understand that a linear transformation from a vector space $V$ to a vector space $W$ is a rank-$2$ tensor. What I would like some help with is how exactly to represent specific linear ...
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1answer
42 views

Components of the metric tensor and its inverse

Let $g$ denote a metric tensor. Then Wald writes (in his book on general relativity): "The inverse of $g_{ab}$...is a tensor of type $(2,0)$ and could be denoted as $(g^{-1})^{ab}$. It is convenient, ...
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1answer
79 views

Confusion with abstract tensor notation

I am currently going through the abstract tensor notation in Wald's "General Relativity". I understand the purpose of it, but I need help understanding some of the conventions and definitions. So, ...
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0answers
23 views

Definition of the derivative of a 2nd-order tensor with respect to a scalar

The derivative of the (positive definite, symmetric, 2nd-order) tensor $\mathbf{C}(t)$ with respect to the scalar $t$ is defined as: $$ \frac{\partial \mathbf{C} }{\partial t} = \lim_{\Delta t\...
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1answer
44 views

Tensor product of dual vectors and vectors

I am reading "General Relativity" by Wald. At first he defines a tensor of type $(k,l)$ to be a multilinear map $T: V^* \times \cdots \times V^* \times V \times \cdots \times V \rightarrow \mathbb{R}$,...
2
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1answer
29 views

Tensor product with vectors

I just started reading Wald's "General Relativity" and I am on his section regarding tensors. He defines the outer product as an operation on tensors of type of $(k,l)$ and $(k', l')$ which gives a ...
1
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1answer
26 views

Foiling two tensor products

I have this problem in exterior algebra where I have a function B and B is defined in the following ways. $$ B\left( \left( \begin{array}{c} u \\ v \\ w \end{array} \right), \left( \begin{array}{c} x\\...
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1answer
39 views

Basic question about the metric tensor

So I am reading Barrett O'Neil's Elementary Differential Geometry, which defines the metric tensor $g_p$ on a surface $M$ to be a bilinear symmetric positive-definite function on the set of ordered ...
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0answers
28 views

Question about properites of tensor product

For $A$, $B$ and $I$ being 2 by 2 matrices and $I$ being the identity, Is $((A\otimes I\otimes I) + (I\otimes A\otimes I) + (I\otimes I\otimes A))\otimes B$ = $(A\otimes I\otimes I)\otimes B + (I\...
2
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1answer
41 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 $\...
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1answer
26 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
36 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 "...
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1answer
36 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) $\delta_{i\,j}...
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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 ...
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1answer
15 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
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1answer
25 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 ...
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1answer
50 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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1answer
46 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) $R^a_{\,b[cd;e]}=0$. ...
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1answer
30 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
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1answer
55 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 = \left<\frac{\...
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2answers
79 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: $\nabla_c\nabla_dZ^a-\nabla_d\nabla_cZ^...
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2answers
104 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 ...
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0answers
15 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.: $$t_{...
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0answers
42 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
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1answer
48 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 $0$:...
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1answer
91 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 ...
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2answers
63 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 ...
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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 ...
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0answers
23 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 ...
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1answer
31 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 ...
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0answers
18 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
20 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 ...
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1answer
35 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 $$f\left(...
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2answers
136 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 ...
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1answer
70 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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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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33 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 ${\alpha_i}'s$ ...