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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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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2answers
71 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
29 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
48 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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1answer
77 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
52 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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2answers
153 views

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

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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1answer
46 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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0answers
44 views

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
39 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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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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0answers
25 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
35 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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0answers
48 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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41 views

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
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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1answer
74 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
12 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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1answer
22 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 $S\in \mathbb{R}^{d_U \times d_M \times d_C}$, ...
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2answers
51 views

“All scalars are invariant”: meaning in the context of changing basis

I'm interested in understanding a comment by Chapter 0 of McCullaugh (1987). To keep the question self-contained, I'll provide some context below. Thank you. Suppose I have vector space $V$ with $\{...
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25 views

Tensors as mathematical objects

Continuing my journey to understand Tensors, Maxwell's equations. Here is my current understanding. Is it correct? Tensors are mathematical objects, i.e., an entity in mathematical reality or a ...
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1answer
31 views

Induced metric on a one-sheet hyperboloid

I am trying to find the induced metric on a one-sheet hyperboloid. Suppose we use cylindrical coordinates $(r, \theta, z)$ for the ambient space in which the hyperboloid is embedded. The hyperboloid ...
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0answers
13 views

Compute directly in coordinates that $\sum a_i^v v^i_v = \sum a^U_jv^j_U$

I am working my way through Frankel's The Geometry of Physics: An Introduction and am a bit confused by the wording of problem 2.1.1 in the revised edition. The problem is stated as below if v is a ...
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1answer
34 views

(Geometric) Intuition behind Different Types of Rank 2 Tensor (Specifically Quadratic Forms)

This is essentially a follow-up to this question: Differences between a matrix and a tensor I think I have a good intuition/idea for the change of basis for a rank-(1,1) tensor ($A\vec{v} = \vec{w}$) ...
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14 views

Text introducing $T^{i,j}$-tensor algebra

I'm reading a lecture note here : http://www.cis.upenn.edu/~cis610/diffgeom7.pdf It introduces $T^{•,•}(M)$ the tensor algebra and says that this is a necessary tool in differential geometry. Well, ...
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0answers
38 views

Self-commuting of the covariant derivative: Menzel's Mathematical Physics

Menzel defines covariant differentiation as equivalent to partial differentiation with respect to the general coordinates. “To indicate the covariant nature of the differential operator, set $$\frac{\...
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35 views

Parallel transport for infinitesimal displacement

I have a question about the following calculation about the parallel transport of an infinitesimal vector. I read the following text but I do not understand where the expression for the components of ...
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36 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
117 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
27 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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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
39 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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44 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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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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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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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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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 ...
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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
30 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
78 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
42 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 $T: V^* \times \cdots \times V^* \times V \times \cdots \times V \rightarrow \mathbb{R}$,...
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1answer
27 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 ...
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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 ...