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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2
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2answers
37 views

$4$-vectors and indices

I am reading through some material about $4$-vectors. And came across the following for which an explanation woud be greatly appreciated. The index for $\partial_\alpha$ can be raised giving ...
2
votes
2answers
66 views

Splitting a tensor

Is it possible to write $$\int d^3x \,\,\, x_i\,\,x_j\,\,\,f(\vec x)$$ where $f(\vec x)$ is some function of the position and the indices indicate which component, as a sum of a traceless tensor and ...
4
votes
2answers
194 views

Confusion when dealing with tensors.

I don't understand how tensors work, can someone please explain? In particular, in the context of Electromagnetism, the dual of the field tensor $F$ is $$(*F)^{\mu\nu}:={1\over ...
4
votes
2answers
150 views

Tensor Components

I would like to ask something On the Barrett Oneill's Semi-Riemann Geometry there is a definition of tensor component: Let $\xi=(x^1,\dots ,x^n)$ be a coordinate system on $\upsilon\subset M$. If $A ...
3
votes
1answer
174 views

Difference between tensor and tensor field?

I couldn't get the difference between tensor and tensor field. I'm learning from Barret O'neill's Semi-Riemann Geometry and here are the definitions: if $A:(V^*)^r \times V^s\to K$ transformation is ...
5
votes
1answer
220 views

A tensor calculus problem

If the relation $a_{ij}u^iu^j=0$ holds for all vectors $u^i$ such that $u^i\lambda_i=0$ where $\lambda_i$ is a given covariant vector, show that $$a_{ij}+a_{ji}=\lambda_iv_j+\lambda_j v_i$$ where ...
8
votes
2answers
648 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 ...
4
votes
1answer
97 views

Connection between dual space V* and negation P^c

Notice the following similarity between the vector space dual and negation in propositional logic: $$ V^* \equiv V \rightarrow F $$ $$ P^c \equiv P \rightarrow \bot $$ Is there some general notion ...
0
votes
1answer
114 views

How does Kronecker $\delta_{ijkl}$ determined?

Let's have symbol $\delta_{ijkl}$. Is it equal to 1 for $i = j = k = l$ and $0$ in the other cases?
3
votes
1answer
516 views

What does cubic symmetry mean?

Is it an invariance under rotations around x-, y-, z-axis? Does this invariance separately include rotations around an arbitrary $(x$, $y$, $z)$ axis?
1
vote
1answer
74 views

Meaning of tensor expression

If $\epsilon$ is an alternating unit tensor and $\mu$ is any arbitrary tensor, then what does the expression $\epsilon : \mu = 0$ mean ? I came across this is some textbook I was reading. Is is ...
5
votes
3answers
577 views

Definition of a tensor for a manifold

While reading Nakahara's geometry, topology and physics. I came across the following definition of a tensor. A tensor $T$ of type $(p, q)$ is a multilinear map that maps $p$ dual vectors and $q$ ...
1
vote
1answer
223 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 ...
2
votes
1answer
129 views

Question about stiffness tensor

Let's have a stiffness tensor $$ a^{ijkl}: a^{ijkl} = a^{jikl} = a^{klij} = a^{ijlk}. $$ It has a 21 independent components for anisotropic body. How does body symmetry (cubic, hexagonal etc.) ...
3
votes
1answer
2k views

Christoffel symbol transformation law

It is known that the transformation rule when you change coordinate frames of the Christoffel symbol is: $$ \tilde \Gamma^{\mu}_{\nu\kappa} = {\partial \tilde x^\mu \over \partial x^\alpha} \left [ ...
1
vote
1answer
101 views

A basis for k-tensors

Let $V$ be a vector space of dimension $n$ with basis $e_1, \dots, e_n$. Let $a_1, \dots, a_n$ be the dual basis for $V^\ast$. Show that a basis for the space $L_k(V)$ of $k$-linear functions on $V$ ...
15
votes
3answers
3k views

What is a covector and what is it used for?

From what I understand, a covector is an object that takes a vector and returns a number. So given a vector $v \in V$ and a covector $\phi \in V^*$, you can act on $v$ with $\phi$ to get a real number ...
4
votes
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 ...
3
votes
2answers
385 views

Tensor invariants under isometries

UPD Equivallent formulation --- how do you find all the independent isometric invariants of a tensor? In what follows $V$ is a real inner product space. I want to understand how does one find all ...
6
votes
0answers
143 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 ...
3
votes
0answers
191 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|| = ...
1
vote
1answer
54 views

Are constant numbers (rank-0 tensors that are fixed) considered as symmetric tensors?

So there are some interesting symmetric rank 0 tensors, for example the Kronecker delta ..... But my question is more simple than that..... Is a number like 1 , 5, e , etc considered symmetric? I ...
2
votes
3answers
346 views

Vector Calculus - Curl of Vector

I'm asked to prove the following identity, using index notation: $(\nabla\times A)\times A=A \cdot\nabla A - \nabla(A \cdot A)$ However, when I work it out, I find that the actual solution should ...
5
votes
1answer
498 views

Tensor product and Kronecker Product

Is there any difference between tensor product and Kronecker Product?
1
vote
2answers
319 views

How do I calculate numerically a tensor in polar coordinates?

You can formulate the question also like this: What is the easiest way of calculating directed derivative of a function if its values are evaluated in a cartesian grid? a) fit a (spline) surface, ...
2
votes
1answer
511 views

Correct name for multi-dimensional array/matrix/tensor

What is the correct name for an n-dimensional array in mathematics? I have seen the following: nD-Matrix nD-Array nD-Tensor Which is the right way?
2
votes
0answers
160 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 ...
3
votes
1answer
452 views

Taking derivatives in index notation

So I'm working out some calculus of variations problems however one of them involves a fair bit of index notation. I'm familiar with the algebra of these but not exactly sure how to perform ...
3
votes
1answer
309 views

Understanding tensor divergence notation in an integral

Given a smooth tensor valued function $\sigma:R^2\rightarrow R^{2\times2}$, I'm trying to show that $\int_\Omega \nabla\cdot\sigma=\int_{\partial\Omega}\sigma n$, where $\Omega$ is a connected ...
4
votes
0answers
164 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 ...
2
votes
1answer
56 views

Question on tensor calculation in Reimannian geometry

Given a Riemannian manfiold $M$ with metric $g=(g_{i,j})$. Let $T=T_{A,B,C,\dots}^{a,b,c,\dots}$ be a tensor on $M$. I would like to compute for example $T_{A,B,C,\dots}^{a,n,c,\dots}g_{n,m}$. We ...
6
votes
2answers
670 views

Are matrices rank 2 tensors?

I know that this is sometimes the case, but that some matrices are not tensors. So what is the intuitive and specific demands of a matrix to also be a tensor? Does it need to be quadratic, singular or ...
4
votes
1answer
144 views

How to transform one nonsquare matrix into another

I am modeling the effect of neural activity on synaptic strength. My question, though, is mathematical. I have the following differential equation: $ \tau_{W} ...
4
votes
2answers
280 views

Is there such a thing as discrete Riemannian geometry?

General relativity expresses gravity as a curvature in space time, created by the stress energy tensor. $$T_{\mu\nu} \approx R_{\mu\nu} - \frac{R}{2} g_{\mu\nu}$$ Given I put the fact that energy is ...
2
votes
1answer
87 views

Is the inverse to a monoidal equivalence also monoidal?

Let ${\cal C,D}$ be two categories, and let $$ F:{\cal C} \to {\cal D}, ~~~~~~~~~~~~~~~~~ G:{\cal D} \to {\cal C}, $$ be an equivalence of categories. Let us now further assume that ${\cal C}$ can ...
3
votes
2answers
165 views

Why do you need tensors of rank $>2$?

Question from someone just starting to study tensors (sorry if it's silly): So I understand (maybe?) that tensors are basically about coordinate transformations (and things that are invariant under ...
1
vote
1answer
120 views

Contravariant Tensors

Sorry for the initial mistake. $\tau\lambda^a\mu^b\lambda^c\mu^d=0$ should read $\tau_{abcd}\lambda^a\mu^b\lambda^c\mu^d=0$. However, my approach to this problem is to introduce vectors, $\alpha$ and ...
4
votes
1answer
388 views

Torque calculation, to achieve clean spin+tumble

Here's a pencil-like robotic spaceship carrying an experiment, it is a solid mass 100m long, 100 inches thick and weighs 1000kg. We're in deep solar space 100au above the sun. Assume we can apply ...
0
votes
1answer
151 views

Why would anti-symmetric (0,2) tensor be traceless?

As it is, why would anti-symmetric (0,2) tensor be traceless? Is it because trace should allow any variable for its indices?
1
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0answers
88 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 ...
1
vote
1answer
288 views

What does “$(n,m)$-tensor” mean?

I know the meaning of tensor, but I forgot the meaning of "$(n,m)$-tensor". What do $n$ and $m$ refer to? Thanks.
0
votes
0answers
90 views

generalized pythogorean theorem in tensor form

I looked up http://mathworld.wolfram.com/MetricTensor.html but it does not seem to provide me exact formula of generalized pythogorean theorem using metric tensor and tensor. So, can anyone tell me ...
4
votes
1answer
235 views

How do I do “calculations” with tensors?

I just started to read about tensor products and tensors and I understand that a tensor product $V \otimes W$ is a space used to replace bilinear maps $V \times W \to U$ with linear maps $V \otimes W ...
1
vote
1answer
479 views

How to generate the inverse of a order 3 tensor

Is it possible to generate an inverse of an order 3 tensor? If so, how? I have been searching for a couple days, and cannot seem to find anything online to help with this.
0
votes
0answers
89 views

Is there a rigorous exposition of 'tensor methods' for finding lie group representations?

I've seen tensor methods in physics for finding lie group representations, as in Wu-Ki Tungs Group Theory in Physics, which uses tensors physics style, ie with indices; and Cvitonovics Birdtracks, ...
1
vote
0answers
86 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 ...
1
vote
1answer
282 views

Index notation clarification

Previously, I have seen matrix notation of the form $T_{ij}$ and all the indices have been in the form of subscripts, such that $T_{ij}x_j$ implies contraction over $j$. However, recently I saw ...
4
votes
2answers
210 views

Tensor operation on a vector space

From the various definitions provided in the article https://en.wikipedia.org/wiki/Tensor, the tensor seems always to be defined, even in the more abstract forms, as a multilinear map, from a product ...
1
vote
0answers
112 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 ...
2
votes
0answers
84 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 ...