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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3
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1answer
296 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
163 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
609 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
142 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
271 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
84 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
161 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
117 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
366 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
145 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
vote
0answers
84 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
284 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
89 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
229 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
448 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
87 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
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 ...
1
vote
1answer
275 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
205 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 ...
4
votes
2answers
273 views

$\det(A \otimes B - B \otimes A) = 0$ why? Why $rk(M) = n^2-n$ ? Why x and -x in Spec(M) ?

Let $A$, $B$ be $n\times n$ matrices. It seems $\det(A \otimes B - B \otimes A) = 0$. Moreover it seems that the kernel of $A \otimes B - B \otimes A$ contains $n$ vectors. Here is MatLab code to ...
1
vote
1answer
113 views

basic vector being hermitian

If the space has a mixed metric signature, not all the basis vectors are Hermitian. Nevertheless, they are defined to be self-adjoint under reversion. The vector transpose conjugate is, ...
4
votes
1answer
802 views

vector/tensor covariance and contravariance notation

As I looked over the Wikipedia article on covariance and contravariance of vectors and $\mathbf{v}=v^i\mathbf{e}_i$ is said as a contravariant vector while $\mathbf{v}=v_i\mathbf{e}^i$ is said as ...
1
vote
2answers
597 views

Tensor calculus - Christoffel symbol of the second kind

and I understand these parts up there, but I cannot understand how the second formula of the last equality leads to the third formula. Can anyone show me what relabeling indices rules are used ...
2
votes
0answers
161 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
vote
0answers
815 views

Conversion of motion equation from Cartesian to Polar coordinates: Is covariant differentiation necessary?

Say I have the following equation of motion in the Cartesian coordinate system for a typical mass spring damper system: $$M \; \ddot{x} + C \; \dot{x} + K \; x = ...
1
vote
1answer
195 views

What's are these index objects called? And $\mathrm{\LaTeX}$ \sum question

I want to refer to $$A_iB_jC_k$$ using $$\psi(ijk) = A_iB_jC_k$$ So that I can write out quite overwhelming-looking sums of ABC terms as sums of terms that look like 123, 231, 113, etc. If I am not ...
3
votes
0answers
206 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
1answer
138 views

Is this tensor question valid?

A tensor exercise in a text reads: If $T_i$ are the components of a covariant vector $T$, show that $S_{ij}:=T_iT_j-T_jT_i$ is an order 2 covariant tensor $S$. Am I missing something or is $S$ ...
1
vote
2answers
362 views

What is the difference between tensors and tensor products?

The tensor product $S\otimes_R T$ of $S$ and $N$ over $R$ is a module. A multilinear form $L:V^r \to R$ is called an $r$-tensor on $V$. On the one hand a tensor is a function sending elements of ...
5
votes
2answers
277 views

Unique symmetric covariant $k$-tensor satisfying $(\operatorname{Sym} T)(A,…,A)=T(A,…,A)$ for all $A \in V$

Let $T$ be a covariant $k$-tensor on a finite dimensional vector space $V$. I want to prove that the symmetrization of $T$ is the unique symmetric $k$-tensor satisfying the following condition: ...
0
votes
0answers
465 views

Inertia tensor transformation under coordinate change

Let $I(x)$ be an inertia tensor in matrix notation of a body in a coordinate system $x\in R^n$. Under a coordinate change $x=\phi(y)$, does the tensor transform as $Dx^TI(\phi(y))Dx$, where ...
5
votes
2answers
253 views

$e_1\otimes e_2 \otimes e_3$ cannot be written as a sum of an alternating tensor and a symmetric tensor

Let $(e_1,e_2,e_3)$ be the standard dual basis for $(\mathbb{R}^3)^\ast$. How can I show that $e_1\otimes e_2 \otimes e_3$ cannot be written as a sum of an alternating (or antisymmetric) tensor and a ...
2
votes
1answer
548 views

The Dimension of the Symmetric $k$-tensors

I want to compute the dimension of the symmetric $k$-tensors. I know that a covariant $k$-tensor $T$ is called symmetric if it is unchanged under permutation of arguments. Also, I know that the ...
6
votes
2answers
161 views

Reference for densities and pseudoforms and non-tensorial representations of $\operatorname{GL}(n)$ and associated vector bundles

I'm looking for a reference that will set me straight on a few things. It started out with densities. In John Lee's book, "Introduction to Smooth Manifolds", densities on vector spaces are functions ...
2
votes
1answer
160 views

Simple problem with the normal curvature tensor

If $M$ is a s-R(semi-Riemannian) submanifold of a s-R manifold $\overline{M}$ the function $R^{\perp}:\mathfrak{X}(M)\times\mathfrak{X}(M)\times\mathfrak{X}(M)^\perp\rightarrow\mathfrak{X}(M)^\perp$ ...
1
vote
1answer
318 views

Index/Einstein notation to derive Gibbs/Tensor relations

In a few continuum classes I have seen indicial notation used to derive relations in Gibbs notation. However, Gibbs notation is valid for all coordinates while indicial notation is valid only for ...
0
votes
1answer
79 views

Help needed with tensors [duplicate]

Possible Duplicate: An Introduction to Tensors Recently I came across the concept of tensors and heard it is very difficult to understand. Is there a ...
0
votes
1answer
2k views

What is the divergence of a matrix valued function?

According to Wikipedia: The divergence of a continuously differentiable tensor field $\underline{\underline{\epsilon}}$ is: ...
1
vote
2answers
145 views

Intuitive Examples of (r,0) Tensors

It's easy to find "intuitive" examples of $(0, r)$ tensors or even $(k, r)$ tensors $( k, r > 0)$. For the purposes of this question, I am considering a tensor in the "classical" sense as being ...
6
votes
2answers
346 views

Index notation for tensors: is the spacing important?

While reading physics textbooks I always come across notation like: $$J_{\alpha}^{\quad\beta},\ \Gamma_{\alpha \beta}^{\quad \gamma}, K^\alpha_{\quad \beta}.$$ Notice the spacing in indices. I can't ...
92
votes
6answers
4k views

Does a “cubic” matrix exist?

Well, I've heard that a "cubic" matrix would exist and I thought: would it be like a magic cube? And more: does it even have a determinant - and other properties? I'm a young student, so... please ...
6
votes
3answers
2k views

Tensors, what should I learn before?

Here I will be just posting a simple questions. I know about vectors but now I want to know about tensors. In a physics class I was told that scalars are tensors of rank o and vectors are tensors of ...
1
vote
2answers
460 views

is there a way to solve the following tensor equation?

I have the following tensor (takes a vector of length $m$ and returns a matrix $m \times m$): $C(y) = A \operatorname{diag}(A^T y ) A^{-1}$ for some invertible matrix $A$ of size $m \times m$ ($y$ ...
1
vote
1answer
620 views

Extracting angular velocity tensor from orthogonal matrices

Let us suppose we have two orthogonal rotation matrices representing a three-dimensional rotations $$\mathbf{R}(t)$$ and $$\mathbf{R}(t+\Delta t)$$ How is it possible to extract the angular velocity ...
2
votes
2answers
344 views

Tensors of order 3

I'm wondering what a tensor of order 3 looks like, and what it's purposes are. I've seen them written down before, but they look like matrices; I'm probably not understanding the concept well. How is ...
2
votes
1answer
236 views

general (asymmetric) real rank-2 tensor visualization in 3d

I have general rank-2 real tensor in 3d space represented as a 3x3 real matrix $M$ (it is gradient of a vector field). I am writing some code to visualize it in several isolated points, this is what I ...
3
votes
1answer
839 views

Invariant proof of the Contracted Bianchi Identity

In "Riemannian Manifolds: An Introduction to Curvature," John Lee states the following lemma: Lemma 7.7 (Contracted Bianchi Identity): The covariant derivatives of the Ricci and scalar curvatures ...