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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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.
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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, ...
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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 ...
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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 ...
2
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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 ...
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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 ...
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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 ...
4
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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 ...
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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, ...
3
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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).
...
1
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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 ...
2
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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", ...
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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 = ...
1
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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 ...
3
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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 ...
2
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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 $S$ ...
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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 ...
5
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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:
...
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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 ...
5
votes
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 ...
1
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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 ...
5
votes
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 ...
2
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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 $R^{\perp}:\mathfrak{X}(M)\times\mathfrak{X}(M)\times\mathfrak{X}(M)^\perp\rightarrow\mathfrak{X}(M)^\perp$ ...
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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 ...
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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 ...
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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:
...
1
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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 ...
5
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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 ...
5
votes
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 ...
1
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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 $m \times m$ ($y$ ...
1
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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 ...
2
votes
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 ...
2
votes
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 ...
2
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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 ...
4
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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 ...
5
votes
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^* ...
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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.
...
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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 ...
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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?
1
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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 ...
1
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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 ...
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votes
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.
...
1
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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 ...
0
votes
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 ...
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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 ...
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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 ...
4
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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 ...
28
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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, ...
8
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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 ...
1
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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 ...
