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 ...
2
votes
1answer
35 views
What does “transform among themselves” mean?
I'm reading a script on atomic physics, and there's a chapter on irreducible tensors. I can't understand the meaning of "transform among themselves" in this context:
An arbitrary rotation of the ...
0
votes
0answers
18 views
transformation for Christoffel symbol of first kind
For Christoffel symbol of first kind, I have to show that
$$\bar \Gamma _{jkm} = \frac{\partial x^p}{\partial \bar x^j}\cdot \frac{\partial x^q}{\partial \bar x^k} \cdot \frac{\partial x^r}{\partial ...
0
votes
0answers
23 views
How to prove that symmetric traceless “transverse” tensor rank s in 4 dimensions has 2s + 1 independent components?
How to prove that symmetric traceless "transverse" tensor rank $s$ in 4 dimensions has $ 2s + 1$ independent components?
Let's have tensor
$$
F^{\mu_{1}\dots \mu_{s}}, \quad {F^{\quad ...
1
vote
2answers
45 views
Identity tensor as a tensor product of two vectors
Any second order tensor in a given basis can be expressed as a matrix. Also, as any second order tensor can be expressed a tensor product of two first order tensors (or vectors), I would like to find ...
3
votes
1answer
42 views
Help with notation in second order tensor.
I have seen recently this notation:
$$F=F_{ij}\,e_i\otimes e_j $$
Where $F_{ij}=\frac{\partial x_i}{\partial X_j}$ is the tensor matrix:
$$ F=\left[ {\begin{array}{cc}
...
3
votes
0answers
55 views
What is tensor, really?
How can one understands the definition of tensor from the purely formal point of view?
To what abstract structure this concept can be generalized?
0
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1answer
28 views
Synge & Schild Exercise 1.2
$x^1 = a \cos u^1 \\
x^2 = a \sin u^1 \cos u^2 \\
x^3 = a \sin u^1 \sin u^2 \cos u^3 \\
\vdots \\
x^{N-1} = a \sin u^1 \sin u^2 \sin u^3 \cdots \sin u^{N-2} \cos u^{N-1} \\
\displaystyle x^N = a ...
0
votes
1answer
59 views
Linear System where Coefficient Matrix is a Kronecker Product
I have a system of linear equations where the coefficient matrix and right hand side is given by a Kronecker product:
$(A_1 \otimes A_2) u = f_1 \otimes f_2$
My question: Is the solution simply ...
0
votes
0answers
55 views
explanation of riemann-christoffel curvature tensor [on hold]
Greetings Mathematics Stack Exchange:
Can anyone suggest a book or article that provides a simple, yet comprehensive explanation of the Riemann-Christofffel curvature tensor? I’ve studied the ...
1
vote
1answer
50 views
Decomposition of the Curvature operator and Matrix representation
I'm trying do this question from Peter Petersen's Book and I can't do some parts.
I know that
$$R=\frac{scal}{2n(n-1)}g\circ g+\left(Ric-\frac{scal}{n}g\right)\circ g+W$$
Where, $R$ is the ...
0
votes
2answers
47 views
Tensors - need materials to study
I want to study about tensors. Can you indicate me some materials, papers, books which I should begin. I tried last year to study but it seems to hard for me.
Thanks for help :)
1
vote
1answer
18 views
Summing over tensor indices
How can I prove that the product of two rank-2 tensors, one of which is symmetric and one is antisymmetric, must =0 when their indices are summed over?
3
votes
1answer
28 views
Non commutative tensor products / simple example?
In O'Neil's "Semi-Riemannian Geometry" it is stated that if $A$ is a $(r,0)$-tensor field and $B$ is $(0,s)$-tensor field then it is ""from the definition'' that $A \otimes B = B \otimes A$. I still ...
2
votes
1answer
24 views
Self-dual and anti-self-dual decomposition
Please take a look at the following:
Let $(M,g)$ be a four-dimensional oriented Riemannian manifold. The Hodge star operator $*$ obeys $**=Id$ acting on 2-forms. This allow us to decompose the ...
3
votes
1answer
41 views
O'Neil's problem 2.10 on Tensor Derivations. Literature on the subject.
I am having real trouble trying to understand a problem in O'Neil's "Semi-Riemannian Geometry" and I can't find much literature on the subject. I will expose the problem and I will be grateful to ...
0
votes
0answers
19 views
(1,3) curvature tensor in coordinates
I want to write the (1,3) riemann curvature tensor in coordinates, as a linear combination of basis vectors. Please, which are those basis vectors and why?
0
votes
1answer
93 views
Prove that the trace of a dyad uv is the dot product of u and v
$$
I'm\quad trying\quad to\quad demonstrate\quad that\quad the\quad trace\quad of\quad a\quad dyad\quad (tensor\quad product)\quad is\\ equal\quad to\quad the\quad dot\quad product\quad of\quad ...
1
vote
0answers
87 views
Rank-2n tensor algebra eigenvalue equation
Im interested in resources and work done on the eigenvalue equation for rank-2n tensors:
$$
M_{ij}A_{j} = \lambda A_{i} \\
$$
$$
M_{ijkl}A_{kl} = \lambda A_{ij} \\
$$
$$
M_{ijklmn}A_{lmn} = \lambda ...
1
vote
1answer
39 views
Problems with tensor notation
I've got a question for the mathematically more educated for I am a humble engineer having a hard time:
$\kappa = \left( \delta_{ij}-n_in_j\right)\displaystyle\frac{\partial u_i}{\partial xj} - ...
1
vote
2answers
45 views
How to reduce an order 3 tensor to an order 2 tensor?
Are there any techniques to reduce an order 3 tensor to an order 2 tensor?
For example, I have an $m \times m \times p$ tensor and I want to reduce it to a $m \times m \times 1$ tensor.
Thanks
4
votes
1answer
51 views
Basic property of a tensor
In Jost's Riemannian Geometry and Geometric Analysis (6th ed.) on page 142 there is the following remark concerning the torsion tensor.
Remark. It is not difficult to verify that [the torsion ...
1
vote
1answer
61 views
Tensors: intrinsic versus index notation
I consider the following equality:
$$ \bar{\bar{T}}=T_{ij}\mathbf{e}_i\otimes\mathbf{e}_j \tag{1}$$
The double bar notation is used to say the quantity is a second rank tensor. Is there more ...
1
vote
1answer
85 views
Divergence of stress tensor in momentum transfer equation
Let suppose that we work in a 2D cartesian coordinates. what will be x and y components of
$\nabla.\left[-p I+\mu \left(\nabla \text{u}+(\nabla \text{u})^T\right)-\frac{2}{3} \mu (\nabla.\text{u})
I ...
2
votes
0answers
50 views
Multilinear or Tensor Regression?
Given input data $x_t\in \mathbb{R}^n$ and output data $y_t\in\mathbb{R}^m$, the closed form solution to $\min_A \sum_t \|y_t - Ax_t\|^2_2$ is given by $A = (XX^T)^{-1}XY^T$ where $x_t$ form the ...
3
votes
1answer
102 views
The divergence of the Weyl tensor
First, the Weyl tensor is given by
$$W_{ijkl}=R_{ijkl}-\frac{1}{n-2}(g_{ik}A_{jl}-g_{il}A_{jk}-g_{jk}A_{il}+g_{jl}A_{ik})$$
where, $A_{ij}$ is the Schouten tensor, given by
...
1
vote
1answer
75 views
Matrix tensor factorisation
Say we have a matrix $A$ expressed as the tensor $$A=\sum_{m=1}^Mx^{(m)}A^{(m)}$$ where $A$ and $A^{(i)}$ are $N\times P$ matrices and $x$ is a $M\times 1$ vector. I would like to decompose $A$ (or ...
1
vote
0answers
52 views
What are these physicists talking about? Dyadic green function?
I am interested in a mathematical explanation(in the sense that you say for example: is it a mapping from A to B) what a dyadic green function and the unit dyad actually is? I am reading this as ...
2
votes
1answer
94 views
What is the practical difference between abstract index notation and “ordinary” index notation
I understand that in "normal" index notation the indexes can be thought of as coordinates of scalar values inside a tabular data structure, while in the abstract index notation the can not. However, I ...
2
votes
0answers
60 views
Tensors: summing over indices
Would anybody mind teaching me how to work these indices?
Definitions:
Throughout the following, repeated indices are to be summed over.
Hodge dual of a p-form $X$:
$$(*X)_{a_1...a_{n-p}}\equiv ...
3
votes
1answer
47 views
Lie Derivative for Wedge Product of Vector Fields
I am having trouble here. The context is: Let $X$, $Y$ and $S$ be vector fields ina a manifold (we can assume it's $\mathbb{C}^2$ though I'm pretty sure this should work in any manifold), and we can ...
1
vote
1answer
33 views
Help with substituting definitions into tensor
I have 4 definitions for the following (Einstein summation) tensor
$A^{ijk}A^{*}_{ijk}=A^{111}A^{*}_{111}+3(A^{112}A^{*}_{112})+3(A^{122}A^{*}_{122})+A^{222}A^{*}_{222}$
If I have these 4 ...
0
votes
1answer
40 views
How does $A^{123}A^{*}_{123}$ look when expanded?
tensors are a new subject for me. I am trying to expand $A^{123}A^{*}_{123}$
Does it look something like the following?
...
0
votes
1answer
36 views
Eigenvalues of a second derivative
I have a function f(r) that describes a Gaussian random field. A second derivative can be formed $\nabla_i \nabla_j f(r)$. I am looking at a paper that claims that in finding the extremum, the ...
3
votes
1answer
27 views
$\nabla \varphi \overset{?}{=} \nabla \cdot \varphi \bar{\bar{I}}$ where $\varphi$ is scalar, $\bar{\bar{I}}$ is identity tensor
I am trying to determine if these two are equivalent. I have a function written with both terms, and this is the only discrepancy. The gradient increases the rank of the scalar to a vector, while the ...
4
votes
1answer
181 views
Derivation or Intuition of Formula for Levi-Civita Symbol
http://www.ees.nmt.edu/outside/courses/GEOP523/Docs/index-notation.pdf spouted off and threw out with no motivation $$\epsilon_{ijk} = \frac{1}{2}(i - j)(j - k)(k - i) \, \forall \, \, k \in \{1, 2, ...
1
vote
0answers
50 views
Independency of the frame of reference of the strain rate tensor
I've got a problem regarding tensors.
Premise: we are considering a fluid particle with a velocity $\mathbf{u}$ and a position vector $\mathbf{x}$; $S_{ij}$ is the strain rate tensor, defined in this ...
9
votes
5answers
219 views
Book on tensors
Can anyone recommend me a book on tensors with an intuitive approach? I have some course notes on that subject, but it's really abstract and theoretical. I want to understand why tensors were ...
0
votes
1answer
91 views
Double dot product of two tensors [duplicate]
I have a problem that makes me very confused...
I have two tensors that must be multiply.
A is second order tensor and B is fourth order tensor.
I know when multiplying two tensor with double dot ...
7
votes
0answers
150 views
A user's guide to Penrose graphical notation?
Penrose graphical notation seems to be a convenient way to do calculations involving tensors/ multilinear functions. However the wiki page does not actually tell us how to use the notation.
The ...
1
vote
1answer
44 views
The gradient of a function is an alternating one-tensor
I'm currently reading Spivak's Calculus on Manifolds and I seem to have hit a snag in Chapter Four: Integration on Chains. Spivak develops tensors, vector fields, alternating tensors and differential ...
0
votes
1answer
43 views
Suficient condition for tensor product of vector spaces..
Can anyone help me showing the following:
Let $E$, $F$, $G$ and $H$ vector spaces and $\varphi:E\times F\rightarrow G$ a bilinear map. If for every $\psi:E\times F\rightarrow H$ bilinear there is an ...
1
vote
1answer
52 views
(Complex) Projective Space
I followed a course in projective geometry and I'm not sure about 2 things:
If I have 6 lines in projective space (IP³) with commun secant, why are
the 6 corresponding tensors linearly dependent?
...
7
votes
2answers
186 views
Do I understand metric tensor correctly?
So I've been studying vectors and tensors, and I'm trying to understand metric tensors.
As I understand them, besides a vast array of explanations, they provide an invariant distance between vectors ...
3
votes
2answers
94 views
For covariant tensors, why is it $\bigwedge^k(V)$, not $\bigwedge^k(V^*)$?
In learning the very basics of differential geometry, I have seen the exterior product defined a couple of ways: First, I have seen it as the image of the covariant tensors (which I believe are ...
3
votes
1answer
58 views
Simplifing formulas using tensor notation
Im trying to symplify formulas like:
$$\operatorname{div}(\operatorname{rot}\vec{F}),\qquad \operatorname{rot}(\operatorname{rot}\vec{F}) $$
or something more strange like:
...
3
votes
1answer
77 views
Alternating tensors vs $p$-vectors
Is there a reason to differentiate between alternating tensors and $p$-vectors? More precisely, is the exterior algebra always isomorphic to the subalgebra of alternating tensors?
Thanks.
6
votes
1answer
278 views
Vorticity equation in index notation (curl of Navier-Stokes equation)
I am trying to derive the vorticity equation and I got stuck when trying to prove the following relation using index notation:
$$
{\rm curl}((\textbf{u}\cdot\nabla)\mathbf{u}) = ...
0
votes
1answer
26 views
Coordinate-free definition of pseudotensors
How to define pseudotensors (particularly, pseudovectors) in a coordinate-free form? Can it be defined on a manifold (like a tensor field)?
Or may be the objects that physicists model via ...
0
votes
0answers
26 views
Tensor transformation for tensor of a special form
The components of the tensor $A^{ij}$ are $A^{12} = A^{21} = A$, whereas all the other components are zero. I am asked to write $\bar{A}^{ij}$, following a transformation to a new coordinate system, ...
1
vote
1answer
32 views
Relation of Hodge dual to antisymmetric part of the
I have a question in reaction to an article by M. Born and L. Infeld (cf. [1]) concerning the relation between the hodge dual of the electromagnetic tensor and the antisymmetrization of its ...
