# Tagged Questions

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 ...

1answer
209 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.
0answers
63 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, ...
0answers
68 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 ...
1answer
127 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 ...
2answers
144 views

### Tensor operation on a vector spaces

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 ...
0answers
83 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 ...
0answers
64 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 ...
2answers
247 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 ...
1answer
71 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, ...
1answer
480 views

### vector/tensor covariance and contravariance notation

As I looked over the Wikipedia article: http://en.wikipedia.org/wiki/Covariance_and_contravariance_of_vectors is said as a contravariant vector and is said as covariant vector (or covector). ...
2answers
339 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 ...
0answers
108 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", ...
0answers
462 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 = ...
1answer
144 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 ...
0answers
132 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 ...
1answer
132 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 $S$ ...
2answers
207 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 ...
2answers
217 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: ...
0answers
325 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 ...
2answers
175 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 ...
1answer
240 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 ...
2answers
128 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 ...
1answer
105 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 $R^{\perp}:\mathfrak{X}(M)\times\mathfrak{X}(M)\times\mathfrak{X}(M)^\perp\rightarrow\mathfrak{X}(M)^\perp$ ...
1answer
196 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 ...
1answer
63 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 ...
1answer
1k 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: ...
2answers
95 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 ...
2answers
199 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 ...
3answers
720 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 ...
2answers
351 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 $m \times m$ ($y$ ...
1answer
373 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 ...
2answers
200 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 ...
1answer
157 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 ...
0answers
525 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 ...
2answers
675 views

### Mathematically Precise Definition of Covariant and Contravariant Transformation

I am trying to understand the meanings of "covariant transformation" and "contravariant transformation" and how they are related. I have read the related Wikipedia article and still feel I cannot ...
2answers
448 views

### What is the definition of tensor contraction?

According to Wikipedia's page on tensor contraction: In general, a tensor of type $(m,n)$ (with $m \geq 1$ and $n \geq 1$) is an element of the vector space \$V \otimes \ldots \otimes V \otimes V^* ...
1answer
180 views

### How did the author of the following paper compute the curvature matrix?

I would like to be shown how the curvature matrix K on page 7 in the paper "Regularisation Theory Applied to Neurofuzzy Modelling" (Bossley) is computed. ...
0answers
36 views

### Is there a particular name for a'long-small-small' tensor/array?

I'm thinking of a 3D array, with dimensions small,small,large. I've taken to saying 'sausage' as shorthand (and I'm sure there are worse NSFW descriptions) but is there a 'legitimate' description for ...
2answers
282 views

### Is correct to say that every tensor is a spinor but not every spinor is a tensor?

Can spinors be seen as a generalization of tensors,but with complex numbers?
1answer
84 views

### Matrix representing $\Lambda^k$(A)

Let V , W be finite dimensional vector spaces over R. Let A : V->W be a linear map. Choose bases of V and W and the corresponding bases of $\Lambda^k$(V ) and of $\Lambda^k$(W). How to show that the ...
2answers
615 views

### Prove the determinant of a tensor is invariant

Given is a second-order tensor $T$, and three arbitrary vectors, $u$, $v$ and $w$, defined in Euclidean point space $\mathcal{E}$. Prove that the determinant of the tensor $T$ \$\det T=\frac{Tu.(Tv ...
1answer
252 views

### Relation between metric tensor and second fundamental form

I'm confused with these definitions. The metric of certain space and the second fundamental form seem to be the same object. I don't know what else to say, this is a pretty straight forward question. ...
1answer
148 views

### Trouble deriving the Harris Corner Detection

I just started studying a small paper about the Harris Corner Detection. The problem is I don't understand how step 7 is derived from step 6. In step 7 the expression is expanded in a way that we get ...
1answer
172 views

### gradient of row vector multiplied by scalar

I'm trying to re-write $v (u x)$ where $v$ and $u$ are row vectors and $x$ is a column vector as some expression $M x$ (or $\bar{v}x$, etc.). The motivation is because I'm trying to compute the ...
1answer
205 views

### Taylor expansion in time of the time component of a stress energy tensor

Perform a taylor expansion in 3 dimensions in time on the time compontent of of $T^{\alpha \beta}(t - r + n^{i} y_{i})$ given that $r$ is a contstant and $n^{i} y_{i}$ is the scalar product of a ...
2answers
281 views

### In tensor notation in Spivak's Calculus on Manifolds, what is that character that looks like a 3?

For example, saying that $T$ is a k-tensor one might see $T\in 3^k(V)$, of course it's not actually a 3. It looks somewhat like Fraktur font Z: $\frak{Z}$. I couldn't detexify it, and it doesn't ...
2answers
1k views

### Tensors as matrices vs. Tensors as multi-linear maps

So I read the answers in this question, and don't feel that much closer to an answer about how tensors as multi-linear maps and tensors as "multi-dimensional" matrices are truly related. For ...
5answers
3k views

### An Introduction to Tensors

As a physics student, I've come across mathematical objects called tensors in several different contexts. Perhaps confusingly, I've also been given both the mathematician's and physicist's definition, ...
2answers
2k views

### Intuitive way to understand covariance and contravariance in Tensor Algebra

I'm trying to understand basic tensor analysis. I understand the basic concept that the valency of the tensor determines how it is transformed, but I am having trouble visualizing the difference ...
1answer
354 views

### Einstein notation - difference between vectors and scalars

From Wikipedia: First, we can use Einstein notation in linear algebra to distinguish easily between vectors and covectors: upper indices are used to label components (coordinates) of ...
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