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 ...

learn more… | top users | synonyms

2
votes
1answer
70 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
90 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
101 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 ...
4
votes
1answer
107 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
2answers
219 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
105 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
108 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
87 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
105 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
126 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 ...
2
votes
1answer
2k 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 ...
3
votes
0answers
137 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 ...
4
votes
1answer
330 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
104 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
496 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 ...
4
votes
1answer
636 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 they can not. However, ...
2
votes
0answers
80 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
2answers
217 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
35 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
44 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
136 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
44 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
290 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
81 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 ...
11
votes
5answers
658 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
351 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 ...
11
votes
1answer
741 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
66 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
59 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
78 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? ...
11
votes
2answers
1k 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 ...
4
votes
2answers
162 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
82 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
156 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
1k 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
40 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 ...
1
vote
1answer
55 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 ...
2
votes
1answer
118 views

How do you express the covariant cross product?

If the covariant cross product is given by $\mathbf{AxB}= \varepsilon^{ijk}A_{j}B_{k}$, then the Levi-Civita tensor must transform contravariantly for the indices to contract. But according to this ...
1
vote
1answer
130 views

Maxwells Equations in a curvilinear coordinate system

I was wondering if anyone would be able to explain to me how the author goes from equation 3 to equations 4(a,b,c) in this paper. I am confused how to treat the contravariant metric tensor in these ...
1
vote
1answer
335 views

Contravariant Metric Tensor

Okay, so I have exactly ZERO experience with tensors and this project I am working on involves tensors. I have looked through a bunch of online resources, and attempted to look for textbooks (not ...
16
votes
5answers
10k views

Differences between a matrix and a tensor

What is the difference between a matrix and a tensor? Or, what makes a tensor, a tensor? I know that a matrix is a table of values, right? But, a tensor?
1
vote
0answers
51 views

can I normalized the tensor rank in this way?

Is there such a thing as a "normalized tensor rank" for non-square tensors (i.e. a tensor with different sizes along each mode)? For example: If a 3rd order tensor (dimensions = 60 x 120 x 30) has ...
6
votes
2answers
494 views

Special case of the Hodge decomposition theorem

I am trying to prove the following special case of the Hodge decomposition theorem in differential geometry for an $n$ component vector field $V_i$ in $\mathbb{R}^n$. I have very little knowledge of ...
5
votes
3answers
1k views

Multiplying 3D matrix

I was wondering if it is possible to multiply a 3D matrix (say a cube $n\times n\times n$) to a matrix of dimension $n\times 1$? If yes, then how. Maybe you can suggest some resources which I can read ...
4
votes
0answers
358 views

Tensors as mutlilinear maps

I am aware that many books on differential geometry define tensors as multilinear maps. Namely $$ V\otimes W := L_2(V^*\times W^*,\Bbb F) $$ I am also aware that this space is isomorphic to the tensor ...
1
vote
1answer
78 views

Represent in a matrix form: $\sum_{ijl}(F_{il}-F_{jl})^2W_{ij}$

How can I represent this in a matrix form: $\sum_{ijl}(F_{il}-F_{jl})^2W_{ij}$ where all the entries are real and $W$ is a known(constant) matrix and $F$ is a rectangular matrix. When I say matrix ...
1
vote
1answer
155 views

Is the third order term of a Taylor approximation a covariant tensor of rank 3 restricted to (h,h,h)?

Recently reading Peter Lax's linear algebra text, he had a very concise way of writing the Taylor approximation up to second order: $f(x+h) = f(x) + l(h) + \frac{1}{2}q(h) + \|h\|^2 \epsilon(\|h\|)$, ...
3
votes
1answer
529 views

Basis of vector fields on manifold

For a typical real manifold of $n$ dimensions, the basis of a tangent vector space $T$ at point $p$ is $$\frac{\partial}{\partial x_i},\ldots,\frac{\partial}{\partial x_j}$$ So is the general basis of ...
5
votes
1answer
955 views

What is the metric tensor on the n-sphere (hypersphere)?

I am considering the unit sphere (but an extension to one of radius $r$ would be appreciated) centered at the origin. Any coordinate system will do, though the standard angular one (with 1 radial and ...
1
vote
1answer
122 views

Tensors Inner Product

If I had two tensors of rank $3$ and wanted their inner product, what would it be? Also, how could I represent the process with indices and please explain that? Could someone demonstrate this with ...