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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7
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0answers
382 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 ...
6
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0answers
520 views

Rotation invariant tensors

It is often claimed that the only tensors invariant under the orthogonal transformations (rotations) are the Kronecker delta $\delta_{ij}$, the Levi-Civita epsilon $\epsilon_{ijk}$ and various ...
5
votes
0answers
138 views

Mnemonic device for relationships between Hom and Tensor

Probably this is a stupid question, but nevertheless... Let $A$, $B$, $C$ and $D$ be rings, and $M$, $N$ and $K$ be appropriate bimodules between them. There are extremely well-known canonical ...
4
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0answers
152 views

Use of graph theory to determine tensor contraction ordering

I am considering using a computer program to execute tensor contractions like the following: $\displaystyle \sum_{ij}^{o} \sum_{ab}^{v} \sum_{KL}^{X} B_{ia}^{K} B_{ia}^{L} B_{jb}^{K} B_{jb}^{L} $ ...
4
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0answers
38 views

Is $H^{\alpha}_{ij,k}=H^{\alpha}_{ik,j}$ for submanifolds in $R^n$?

Consider $\Sigma^n\subset {\bf R}^{n+p}$ a submanifold and $\{e_i, e_\alpha\}$ an orthonormal frame where the Greek indexes stand for the normal vectors. Then define $H^{\alpha}_{ij,k}=e_k\langle ...
4
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0answers
157 views

What are spinor fields?

For example, a tensor field $T=T_{a,b}^{\ \ \ \ c}$ on a manifold $M$ should be thought as $$ T_{a,b}^{\ \ \ \ c}dx^{a}\otimes dx^{b}\otimes \frac{\partial}{\partial x^{c}} $$ with local coordinate ...
3
votes
0answers
62 views

Intuition about the Riemann and Ricci tensors

I have started an attempt to self-study Riemannian Geometry. I well understand all the algebraic properties of the Riemann tensor (With a symmetric connection), and how it gradually becomes less and ...
3
votes
0answers
62 views

dot product between vector and matrix

In my book on fluid mechanics there is an expression $$ \boldsymbol{\nabla}\cdot \boldsymbol{\tau}_{ij} $$ where $\boldsymbol{\tau}_{ij}$ is a rank-2 tensor (=matrix). Given that ...
3
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0answers
121 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?
3
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0answers
44 views

Equivalent definitions of tensors on finite dimensional spaces.

I have recently been studying various texts on differential geometry, and I am quite puzzled that various authors define the notion of a tensor quite differently. I have come across the following ...
3
votes
0answers
173 views

Gradient and Einstein summation

Suppose I have a vector in the covariant basis $\bar e_i$ and a metric $g$ and I want to obtain a unit vector $\bar u_i$ parallel to $\bar e_i$. I would write: $$\bar e_i = \hat u_i ||\bar e_i|| = ...
3
votes
0answers
197 views

Better Tensor Notation

I am in a General Relativity class, and I am finding the usual tensor notation very difficult to think about -- it seems like there are too many names to express something simple. E.g., I think of ...
2
votes
0answers
64 views

Covariant vectors and Dual spaces

There are contravariant vectors and their duals called covariant vectors. But the duals are defined only once the contravectors are set up because they are the maps from the space of contravariant ...
2
votes
0answers
69 views

Prove the Curvature Tensor is a Tensor

For an affine connection $\nabla$, prove the curvature R $R(X,Y,Z,\alpha)=\alpha(\nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z -\nabla_{[X,Y]}Z)$ with $X,Y,Z$ vector fields and $\alpha$ a co-vector, is ...
2
votes
0answers
72 views

Stress Tensor and the ubiquitous cube: a misrepresentation?

So I'm just learning about Stress Tensors and have had an on going confusion and I'm wondering if the cube diagrams may to be blame. Usually when a stress tensor is introduced they show a cube and ...
2
votes
0answers
52 views

Factoring tensors by linear transformation

Let $T_n$ be tensors $\left( V^n \rightarrow \mathbb{R}\right)$ of order $n$. Now let $\sim$ be equivalence on $T_n$ that $A\sim B$ iff there is regular matrix $M$ that $A\circ M = B$. By $A\circ M$ ...
2
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0answers
80 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 ...
2
votes
0answers
73 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 ...
2
votes
0answers
149 views

Constant tensors and covariant derivatives

I seem to have trouble with an elementary computation, and figure it may help others if faced with a similar situation. The basic question is as follows: if I have a tensor field $T$ on some ...
2
votes
0answers
153 views

Extending Tensor Fields defined on Manifolds to Ambient Space

I am currently reading about tensor fields on manifolds, and I came across two comments that sound contradictory to me. The first comment is made in the book by James Munkres "Analysis on Manifolds", ...
1
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0answers
4 views

Index notation interpretation

I'm having some confusion with index notation and how it works with contravariance/covariance. $(v_{new})^i=\frac{\partial (x_{new})^i}{\partial (x_{old})^j}(v_{old})^j$ $(v_{new})^i=J^i_{\ ...
1
vote
0answers
46 views

What exactly do the terms of $(f \circ g)'''$ mean?

Say, $g: X\to Y$ and $f: Y\to Z$ are smooth. One can find $(f \circ g)'''$ by using the Faà di Bruno's formula: $$(f \circ g)''' =(f'''\circ g)(g')^3 + 3(f''\circ g)g'g'' + (f'\circ g)g'''$$ But my ...
1
vote
0answers
18 views

Hessian matrix of $g\circ f$

Say, $f:\mathbb R^n\to\mathbb R^k$ and $g:\mathbb R^k\to\mathbb R$ are both $C^2$. I'd like to express the Hessian matrix of $g\circ f$ $$\left( \frac{\partial^2(g\circ f)}{\partial x_i \partial ...
1
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0answers
15 views

Symbolic cancellation in tensor notation of derivative

Start with this: $\frac{\partial f}{\partial x'^i} = \frac{\partial f}{\partial x^j} \frac{\partial x^j}{\partial x'^i}$ I think(?) the $\partial x^j$s cancel and this simplifies to ...
1
vote
0answers
23 views

How is the multiplication between a multidimensional tensor with a matrix defined?

I am thinking this calculation in the following way but I am wondering if it is correct. Can anybody explain to me please? For example, I have a 3-way tensor $T^{u×i×t}$. How do I multiply this ...
1
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0answers
23 views

Is there a name for this type of tensor rank?

Let $A\in\mathbb{R}^{n_1\times n_2\times n_3 \times n_4}$ be a tensor. Suppose that $k$ is the minimum integer there exist matrices $X_1,\ldots,X_j\in\mathbb{R}^{i_1\times i_2}$ and ...
1
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0answers
22 views

Isomorphism between $T^k_{l+1}(V)$ and $\mathcal{L}\left( (V^*)^k \times (V)^l\right)$.

V is a real $n$-dimensional vector space. I'm using the notation as in John Lee's Riemannian Manifolds: an Introduction to Curvature for the set of $\binom{k}{l}$ tensors and the set of multilinear ...
1
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0answers
18 views

Number of isotropic tensors of rank N

Could someone please help me out with this problem? I would like to know what the number of distinct isotropic tensors of rank N is. I am new to the field and got confused by apparently contradictory ...
1
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0answers
45 views

Curvature tensor on the sphere

While reading R. Hamilton paper "Three-manifolds with positive Ricci curvature" I came across the sentence: For the sphere we have $$R(u,v,u,v)=R_{ijkl}u^{i}v^{j}u^{k}v^{l}>0,$$ which is the ...
1
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0answers
25 views

Prove $\frac{1}{N!}\varepsilon_{i_1\dots i_N}\varepsilon_{j_1\dots j_N}A_{i_1 j_1}\dots A_{i_N j_N} = \det A$

Is there any simple way to prove the following: $$\frac{1}{N!}\varepsilon_{i_1\dots i_N}\varepsilon_{j_1\dots j_N}A_{i_1 j_1}\dots A_{i_N j_N} = \det A. \tag{$1$} $$
1
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0answers
39 views

Question about tensor product of homomorphisms

I've come to think about this problem when reading a proof in Commutative Algebra by N. Bourbaki. Say, let $R$ be a commutative ring, given 3 $R-$modules $A$, $B$, $C$, and the $R$-homomorphism $f:B ...
1
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0answers
22 views

Significance of bump function in the proof.

In the book Semi-Riemannian Geometry with applications to Relativity by Barrett O'Neill, in Chapter 2 (Tensors), he stated that: However, I don't get why he had to use a bump function in the proof. ...
1
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0answers
46 views

Metric on an open subset of $\mathbb{R}^d$ and Christoffel symbol of the second kind under diffeomorphisms

This is a follow-up question to this question I proved the following identity. $\Omega \subset \mathbb{R}^d$ is open and $g$ is a metric field on $\Omega$. Further $\Gamma_{\,kl}^j$ denotes the ...
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0answers
17 views

Trace of a tensor in a differential equation

If $Z$ is a rank-2 tensor, does the following differential equation mean anything to anyone: $\nabla^2Z+\frac{1}{c^2}\frac{\partial^2}{\partial t^2}tr(Z)=0$ The presence of this trace really blurs ...
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0answers
101 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
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0answers
246 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 ...
1
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0answers
65 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 ...
1
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0answers
66 views

What is the needed background to study tensors?

What is the needed background to study tensors? I do not plan to study it but, I want to know the background! It's a matter of curiosity . Thanks.
1
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0answers
64 views

To show that something is a four-vector

I hope this question is not too inane... it would be really helpful for me to have this cleared up. I want to know what I need to show to demonstrate that something is a four-vector. I have checked ...
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0answers
63 views

Tensor compact/matrix form.

I have got this tensor $S_{ij} = \frac{1}{2}(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i})$ Anyway I solve it for my problem and get $$ S_{ij} = \left( \begin{array}{ccc} 0 ...
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0answers
170 views

Riemannian curvature and its application on covariant derivative of tensors

This identity can be generalized to get the commutators for two covariant derivatives of arbitrary tensors as follows $ \begin{align} T^{\alpha_1 \cdots \alpha_r}{}_{\beta_1 \cdots \beta_s ; ...
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0answers
64 views

What is this tensor called?

Is there a standard name for this "generalized identity" tensor $x^{i j k ...} = I(i = j = k = ...)$ where $I$ is the indicator function, ie the tensor is 1 when all indexes are equal & zero ...
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0answers
40 views

Tensors and 4-vectors

This may be a very trivial question, but help would be appreciated. It has to do with P. 55 in these notes I don't understand why the fact that $$U_a\dot U^a=0$$ applied to $$U_a\nabla_b T^{ab}=\rho ...
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0answers
300 views

How to derive covariant derivative and Lie derivative of tensors

1) As title says, how does one derive the following equation for covariant derivate of tensor: $A^{\alpha}{}_{;\beta} = A^{\alpha}{}_{,\beta} + \Gamma^{\alpha} {}_{\gamma\beta}A^\gamma$ where ...
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0answers
130 views

Multi-dimensional array decomposition

My question is about decomposing a muti-dimensional array into a product of matricies. To ask the question I will work towards the tensor, and then ask the question about the reverse process. Let ...
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0answers
199 views

Showing symmetry of the stress tensor by applying divergence theorem to $\int\int_{\delta V(t)} \vec{x}\times \vec{t} dS$

I'm currently working through the symmetry of the stress tensor, in relation to viscous flow. I am looking at this by examining the conservation of angular momentum equation for a material volume ...
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0answers
79 views

Contravariance and covariance indice of tensors confusion

According to http://en.wikipedia.org/wiki/Tensor#Tensor_fields, $\hat{T}^{i_1\dots i_n}_{i_{n+1}\dots i_m}(\bar{x}_1,\ldots,\bar{x}_k) = \frac{\partial \bar{x}^{i_1}}{\partial x^{j_1}} \cdots ...
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0answers
85 views

Confusion with vectors and notation

Could someone please explain to me why $$\nabla (\dot{r}\cdot A)$$ take the following form in index notation? $$\left({\partial A_i\over \partial r^k}-{\partial A_k\over \partial ...
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0answers
111 views

Changing along a tensor field, the Lie Derivative

I can find considerable information about how to use the Lie Derivative to measure the change of a tensor field along a vector field, but I can't seem to find anything for the converse. What if I ...
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0answers
77 views

Relationship between Tensors of Different Rank

Simple question. Can one write every second rank tensor $T^{ab}$ as some finite sum $\sum U^aV^b$ with $U^a$, $V^b$ tensors? Apologies if this is an incredibly standard result - I don't own a textbook ...