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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Proof that Kronecker Delta is Mixed Tensor

The book I am reading asks the reader to verify that the Kronecker Delta is a second-order mixed tensor with one contravariant and one covariant index as indicated: $$ \delta_j^i = \left\{\begin{...
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Infinitesimal Strain Tensor in a Cubic Crystal

I'm currently working through Vectors and Tensor in Engineering and Physics, and there's a problem regarding the strain tensor that I'm having a bit of trouble with. Given a cubic crystal with zero ...
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1answer
59 views

Geometric Significance of some features of the Exterior Algebra

I've been tinkering with differential forms for a while now, and I've had a few questions all rolled into one trying to understand them. The exterior derivative is quite natural to me - it looks just ...
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28 views

Series Relation

$\{A_n\}$ is a sequence of 4th order tensors. $lim_{n\rightarrow\infty}A_n = O_4$, where $O_4$ is the null 4th order tensor. The series $\sum_{n=1}^{\infty}A_n$ converge to a known tensor $B$. I ...
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Element-wise derivative of matrix logarithm

$E = ln(C) = -\sum_{a=1}^{\infty}\frac{1}{a}(I-C)^a$ I want to find a simple formula for $\frac{\partial E_{ij}}{\partial C_{pq}}$ $\frac{\partial C_{ij}}{\partial C_{pq}} = \delta_{ip}\delta_{...
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29 views

Linearize Matrix Equation

I want to find a linearized formula for G in terms of A. $G = B^TC^{-1}T(I+BA)$ $G$ is 4x2 $B$ is a constant matrix 2x4 $A$ is a variable matrix 4x2 $C = I + A^TB^T + BA + BAA^TB^T$, so $C$ is ...
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33 views

Symbol of differential operator and change of coordinates

Some time ago I posted the question about the change of coordinates in differential operator. Here is the relevant discussion Symbol of differential operator transforms like a cotangent vector The ...
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14 views

$Hom(E\times\stackrel{(r)}{\ldots}\times E,E)$ isomorphic to $\bigotimes_r^1 E$?

Let $E$ be a $n$-dimensional $\mathbb{R}$-vector space. Prove that: $$\begin{array}{ccll} \Phi:&Hom(E\times \stackrel{(r)}{\ldots} \times E,E)&\longrightarrow&\bigotimes_r^1 E\\ &\psi &...
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25 views

addition of tensors

The tensor algebra over a $n$-dimensional vector space $E$ is defined as the direct sum: $$\bigotimes E:=\bigoplus_{r,s>0}\Big(\bigotimes\nolimits_r^sE\Big)$$ An element of $\bigotimes E$ is ...
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2answers
352 views

What does shear mean?

As I understand it, the gradient of a vector field can be decomposed into parts that relate to the divergence, curl, and shear of the function. I understand what divergence and curl are (both ...
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0answers
29 views

surface gradient of the unit normal vector

I am reading a book that defines a curvature tensor as $\boldsymbol{K} = -\nabla_\pi \boldsymbol{\hat{n}}$ where $\boldsymbol{\hat{n}}$ is the unit normal vector of a surface and \begin{align*} \...
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Does this calculus argument involving rank-4 tensors make sense?

Edit: Completely rewritten to be shorter and easier to digest. Background and the Actual Question: I'm trying to derive a gradient formula (back propagation) for a machine learning application. The ...
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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

Let $V$ and $W$ be vector spaces over $\mathbb{C}$. Show that $0\otimes w = v\otimes 0 = 0 \in V \otimes W$.

Algebraically, the vector space $V \otimes W$ is spanned by elements of the form $v \otimes w$, and the following rules are satisfied, for any scalar $c$. The definition is the same no matter which ...
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1answer
28 views

How many arguments does the product $v\otimes f$ have?

Let $V$ be a vector space. I've learnt that the tensor $v\otimes f$, where $v\in V$ and $f\in V^*$, is given by $ v\otimes f: V^*\times V\to\mathbb{R}\\ \quad\quad\quad\quad\, (g,\omega)\mapsto v(g)f(...
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1answer
145 views

Why use geometric algebra and not differential forms?

This is somewhat similar to Are Clifford algebras and differential forms equivalent frameworks for differential geometry?, but I want to restrict discussion to $\mathbb{R}^n$, not arbitrary manifolds. ...
2
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1answer
81 views

$\hat{e_\mu } \cdot \hat{e^\nu } \neq \delta _{\mu} ^{\nu}$? Tensor algebra question.

Let $\hat{e_{\mu }}$ and $\hat{e^{\mu }}$ be the co- and contravariant basis vectors, respectively, for an arbitrary coordinate system Is it true that sometimes, $\hat{e_\mu } \cdot \hat{e^\nu } \neq \...
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16 views

Is the norm of tensor fields just Hilbert-Schmidt norm/ generalized $L^p$-norm?

As I am rather comfortable with functional analysis language and was new to Riemannian geometry, I am curious when inspecting the norm of Ricci tensor, which is: $$|\mathrm{Ric}|^2=g^{ij}g^{kl}R_{ik}...
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6answers
1k views

Tensor Book Recommendation Request

Requirements Tensors Intuitive + Practical Reason for Tensor Introduction Current Knowledge Course Notes Abstract + Theoretical
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2answers
34 views

Vector as directional derivate? [duplicate]

The Poor Man’s Introduction to Tensors - by Justin C. Feng. https://web2.ph.utexas.edu/~jcfeng/notes/Tensors_Poor_Man.pdf On page 3, "Before I can tell you what a tensor is, I must tell you what a ...
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1answer
30 views

Understanding metric tensor notation

I am trying to understand if there is a conventional way to read super- and subscript notation of metric tensors. Is there a canonical way of doing this? For instance, what is the difference between ...
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3answers
834 views

Why is a linear transformation a $(1,1)$ tensor?

Wikipedia says that a linear transformation is a $(1,1)$ tensor. Is this restricting it to transformations from $V$ to $V$ or is a transformation from $V$ to $W$ also a $(1,1)$ tensor? (where $V$ and $...
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2answers
210 views

Which concepts in Differential Geometry can NOT be represented using Geometric Algebra?

1. It is not clear to me that linear duals, and not just Hodge duals, can be represented in geometric algebra at all. See, for example, here. Can linear duals (i.e. linear functionals) be ...
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1answer
58 views

Einstein notation difficulties

I'm just learning the Einstein index notation, and came across this derivation in a textbook. I couldn't follow the steps. Can someone please help me out? The first order differential equation: $$\...
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2answers
128 views

8th order isotropic tensor

Does anyone knows what is the general form of an $8$th order isotropic tensor? $2$th order is $\delta_{ij}$, $4$th order it is $\lambda \delta_{ij} \delta_{kl}+\mu(\delta_{ik} \delta_{jl} + \delta_{il}...
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2answers
126 views

What's an infinite dimensional or function version of a tensor?

A function $f$ is like an infinite dimensional vector with the norm $|f| = \int^b_a f(t)^2 \, \mathrm{d} t $ and dot product $f \cdot g = \int^b_a f(t) g(t) \, \mathrm{d} t $ where appropriate ...
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1answer
22 views

Trace reverse tensor/matrix operation “carrying through” an operator

For some second rank tensor $h_{\mu\nu}$ on a Riemannian manifold with metric $g_{\mu\nu}$, one can write the trace-reverse of it as: $\bar{h}_{\mu\nu}=h_{\mu\nu}-\frac{1}{2}g_{\mu\nu}h$, where $h=h_{\...
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1answer
48 views

How to represent matrix multiplication in tensor algebra?

How can we represent matrix multiplication in tensor algebra? Even if we assume all matrices represent contravariant tensors only, clearly matrix multiplication does not correspond to the ...
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2answers
179 views

What is the “taxonomy” or “hierarchy” (partial ordering) of algebraic objects used to attempt to capture geometric intuition? [closed]

What follows is a list of terms all of whose relationships to one another I have never fully succeeded in establishing, despite having spent much of 6-8 years trying to so. Feel no need to give ...
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2answers
282 views

what's the relationship of tensor and multivector

what's the relationship of multivector in geometric algebra and tensor? Is tensor a subset of multivector?
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2answers
265 views

using the kronecker product and vec operators to write the following least squares problem in standard matrix form

I have a least squares problem with the following form: $$ \min_\mathbf{X} ~ \sum_{i=1}^n | \mathbf{u}_i^\top \mathbf{X} \mathbf{v}_i - b_i |^2 $$ where $\{\mathbf{u}_i\}_{i=1}^n$ and $\{\mathbf{v}...
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2answers
74 views

Defining a Riemannian metric

Let $M$ be a smooth manifold. I have seen a Riemannian metric be defined in many ways: A smooth choice of an inner product $g_p:T_pM\times T_pM\to\mathbb{R}$ which is symmetric and positive-definite,...
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1answer
54 views

What exactly are operations involving tensors… In terms of their indices

So I have heard that tensor operations involve the faces of the rectangular prism. These are matrices right, and different properties of those matrices say things about the tensor? Could someone ...
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1answer
36 views

Why is the Christoffel symbol of the 2nd kind symmetric in lower indices?

I have consulted multiple books on tensors for physicists, but they all take for granted this relation: $\Gamma_{ij}^k = \Gamma_{ji}^k$ However, no proof is provided and I cannot find a single one ...
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1answer
49 views

Dual basis vectors and Basis one-forms

I'm studying Tensor Calculus on some MIT's notes (page 16) and I'm stuck at the point where it defines dual basis vectors. I have already studied basis one forms and I can't understand why we need to ...
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0answers
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Should I add $\frac {4 \pi}{3}\delta^3(\vec x) \mathbf{I}$ or $4 \pi \delta^3(\vec x) \hat x \hat x$ to the gradient of $\frac{\vec x}{x^3}$?

I know that divergence of $\frac{\vec x}{x^3}$ is $4 \pi\delta^3(\vec x)$ and also that trace of gradient of a vector is its divergence. When I take the gradient of $\frac{\vec x}{x^3}$ I get: $$\...
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1answer
41 views

How to handle the tensor $T^i_{ljk}$, given that $T^i_{jkl}=3T^i_{ljk}$?

$T^i_{~~jkl}$ is a tensor such that $T^i_{~~jkl}=3T^i_{~~ljk}$ is some coordinate system. Prove that $T^i_{~~jkl}=3T^i_{~~ljk}$ in all coordinate systems. The given answer says: \begin{align} \bar T^...
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0answers
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Contracting a symmetric tensor product with a covector

What I actually want to ask is about a problem of specific form, which I could not put in the title as I'm not certain on any short name for such problems. For given a rank-2 tensor $K^{\mu\nu}$ and ...
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26 views

Translation by tensors

According to this question, quaternions would not be the right choice to handle both rotation and translation. In the case of tensors, one might assert that the rotation would be possible by tensors, ...
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1answer
38 views

Given the parallel and perpendicular component of a vector in terms of another vector, how do you determine the tensor that connects both?

Sorry for the awkwardly phrased title, I wasn't sure how to properly word it. I want to do the following: I have a vector $\vec J$ and a vector $\vec E$ with the following relation (with the ...
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54 views

How can we visualize a tensor?

I would greatly appreciate it if someone could explain to me how to visualize a tensor in the analogous way that we visualize a vector. Thanks!
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1answer
53 views

Calculus of rank three tensor

Let $A(\alpha)$ be a matrix that depends to vector parameter $\alpha$. I want to approximate $A(\alpha+\Delta\alpha)$ using Taylor expansion. My work: $$ A(\alpha+\Delta\alpha) \approx A(\alpha)+\...
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How do I get from the universal product of the tensor product to other definitions.

I was wondering how you can "derive" the common (or "classical") definition of the tensor product before the universal property was established (I think there is no need to repeat it here), i.e. a ...
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1answer
54 views

Solve for third rank linear tensor equation $C_{[ij]k}U^jU^k=A_i$

Is there a way to solve a general tensor equation of the form, written in an arbitrary frame \begin{equation} C_{[ij]k}U^jU^k=A_i, \end{equation} for a tensor field $C$ of type $(0,3)$ (the square ...
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1answer
75 views

Why is $T$ in $\mathcal T_2^1(M)$?

Let $M$ be smooth manifold and $\nabla$ an affine connection on $M$. Then the torsion tensor of $\nabla$ is the map $T:\mathcal T(M)\times\mathcal T(M)\to\mathcal T(M)$ and $$T(X,Y)=\nabla_XY-\...
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1answer
13 views

benefit of trifocal geometry vs bifocal geometry?

I am at the moment trying to understand what kind of benefit I would have by using three cameras for stereo vision rather than two cameras? I mean, i would only have more constraints related to the ...
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46 views

Why can't 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 can't we just simplify this like: $G_{ab}=R_{ab}-\frac{1}{2}g_{ab}R=R_{ab}-\frac{1}{2}g_{ab}g^{...
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1answer
25 views

Transformation of fourth rank tensor and its matrix form

I would like to calculate transformation of fourth rank tensor, $$ C_{ijkl}=\Sigma_{m=1}^{3}\Sigma_{n=1}^{3}\Sigma_{p=1}^{3}\Sigma_{q=1}^{3}a_{im}a_{jn}a_{kp}a_{lq}C_{mnpq} $$ where $a_{xy}$ is ...
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38 views

How do I compute the gradient of a tensor?

From this paper, we have three matrices $U\in \mathbb{R}^{n\times d_U}$, $M\in \mathbb{R}^{m\times d_m}$, $C\in \mathbb{R}^{c\times d_C}$ and a tensor $S\in \mathbb{R}^{d_U \times d_M \times d_C}$, ...
2
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1answer
51 views

$G$-structure defined by a tensor

Let $M$ be an $n$ dimensional manifold with its bundle of linear frames $\pi:L(M)\to M$. Suppose $T_0$ is a tensor on $\mathbb R^n$ and $u\in L(M)$. We may view $u$ as a linear map $u:\mathbb R^n\to ...