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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75 views
Tensor product of algebra
Can we find define a norm on tensor product $C(X) \otimes C(Y)$ such that the norm completion of $C(X)\otimes C(Y)=C(X\times Y)$
And can we define a norm on tensor product $L^1(X)\otimes L^1(Y)$ such ...
5
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0answers
240 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 ...
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132 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
35 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
88 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
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0answers
32 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
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0answers
82 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:
$$\bar e_i = \hat u_i ||\bar e_i|| = ...
3
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0answers
85 views
What are spinor fields?
For example, a tensor field $T=T_{a,b}^{\ \ \ \ c}$ on a manifold $M$ should be thought as
$$
T_{a,b}^{\ \ \ \ c}dx^{a}\otimes dx^{b}\otimes \frac{\partial}{\partial x^{c}}
$$
with local coordinate ...
3
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0answers
132 views
Better Tensor Notation
I am in a General Relativity class, and I am finding the usual tensor notation very difficult to think about -- it seems like there are too many names to express something simple. E.g., I think of ...
2
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0answers
109 views
Extending Tensor Fields defined on Manifolds to Ambient Space
I am currently reading about tensor fields on manifolds, and I came across two comments that sound contradictory to me.
The first comment is made in the book by James Munkres "Analysis on Manifolds", ...
2
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0answers
527 views
Invariant proof of the Contracted Bianchi Identity
In "Riemannian Manifolds: An Introduction to Curvature," John Lee states the following lemma:
Lemma 7.7 (Contracted Bianchi Identity): The covariant derivatives of the Ricci and scalar curvatures ...
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0answers
23 views
Solving tensor Identities
For my homework I am required to prove the following identity. Here $\bf{u}$ is a vector.
$\nabla \bf{uu} = u(\nabla.u) + (u.\nabla)u$
Only thing I understand in this equality is lest hand side is ...
1
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0answers
15 views
Metric spaces and curvature
Suppose that it is given that the Riemann curvature tensor in a metric space of dimension $d\geq2$ can be written as $$R_{abcd}=k(x^a)(g_{ac}g_{bd}-g_{ad}g_{bc})$$ where $x^a$ is a vector in the ...
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0answers
42 views
What is the needed background to study tensors?
What is the needed background to study tensors?
I do not plan to study it but, I want to know the background!
It's a matter of curiosity .
Thanks.
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0answers
34 views
To show that something is a four-vector
I hope this question is not too inane... it would be really helpful for me to have this cleared up. I want to know what I need to show to demonstrate that something is a four-vector. I have checked ...
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0answers
32 views
Tensor compact/matrix form.
I have got this tensor $S_{ij} = \frac{1}{2}(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i})$
Anyway I solve it for my problem and get
$$ S_{ij} =
\left( \begin{array}{ccc}
0 ...
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0answers
63 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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33 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 ...
1
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0answers
151 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 ...
1
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0answers
42 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
49 views
What is the last index of a third-order tensor called?
In a third-order tensor I guess the first and second index would be called row and column respectively but is there a name for the third index?
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0answers
43 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 ...
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0answers
128 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
84 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 ...
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0answers
90 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 ...
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0answers
57 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
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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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 ...
1
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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 ...
1
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0answers
470 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 = ...
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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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0answers
8 views
Computational Complexity of Tensor Decomposition
I am studying tensor decomposition techniques such as the CP model (a.k.a., PARAFAC), and the Tucker model.
My reference paper is "Tensor Decompositions and Applications".
I need a survey about the ...
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0answers
53 views
What is mathematics (in physics) of this tensor equation?
I want to know what is $C^{a}C_{a}$ if $C^{a}=A^{a}+B^{a}$ and $C_{a}=A_{a}+B_{a}$?
a. this one $A^{a}A_{a}+B^{a}B_{a}$ or
b. this one $(A^{a}+B^{a})(A_{a}B_{a})$
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0answers
18 views
Norm of tensor object
Suppose I have a $3\times2 \times 2$ tensor object $M$. What is then $|M|$ ?
Thank you for your support!
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0answers
17 views
what is the status of the theory of multilinear systems of equations?
What is the current status of the theory of multilinear systems of equations?
I have a particular interest for multilinear homogeneous systems of the form
$A_1 \otimes \cdots \otimes A_r) (x_1 ...
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0answers
31 views
An exact sequence from tensors
Let $V$ be a vector spaces. Why is the following sequence exact?
$S_{[table]}V \rightarrow V\otimes \Lambda^{2}V \rightarrow \Lambda^{3}V$
where suffix table is a diagram of a table with first row [ ...
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0answers
37 views
Einstein notation non-repeating indices
I forget the rule for Einstein notation. If I have something like the gradient:
$$\vec\nabla f = \frac{\partial f}{\partial x_i} = \langle \frac{\partial f}{\partial x}, \frac{\partial f}{\partial ...
0
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0answers
47 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 ...
0
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0answers
41 views
Tensor Products, various defintions
I came across a definition for the tensor product which differs from the standard definition. This book defined the tensor product of vector spaces $V$ and $W$ as the space $L(V,W,\Bbb K)$ of bilinear ...
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0answers
37 views
Differentiation of a vector (in index notation) with respect to its square
I have a formula which contains the derivative $\partial_{z^2}$(with respect to the square of a vector $z^\mu$). However, as the functions inside the formula might not directly depend on $z^2$ but on ...
0
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0answers
123 views
p-norm of hessian matrix of NxNxN dimension
I am trying to analyze a problem using the norm of second derivative of a vector-valued function
F = [ f1(x1,....,xn) ; f2 (x1,....xn);...;fn(x1,.....,xn)]. We assume that all fi functions are twice ...
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0answers
30 views
Differential geometry question relating to four-velocities
Let $$V^a={dx^a\over d\mu}$$ be a 4-velocity.
Is there any reason for this to be true: $$V_aV^b\nabla_b V^a=0$$ where $\nabla$ is the covariant derivative, WITHOUT assuming that the path is a ...
0
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0answers
75 views
Tensor calculus solution-why?
The text I read says that $\displaystyle\frac{\partial^2
x^\alpha}{\partial x^\delta
\partial x^\gamma}\frac{\partial x^\delta}{\partial
x^\beta} = 0$ leads to the solution $x^\alpha =
...
0
votes
0answers
68 views
contraction with the metric tensor
What does mean "$\wedge_0^4V $ is the space of 4-vectors whose contraction with the metric tensor of the space $V$ vanishes" how can we formulate this set?
this means $i_gT=0$ for tensor $T$?
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0answers
39 views
Extension of Convolution theorem
Is it possible to extend the convolution theorem to convolve tensors, as we do with discrete matrices?
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0answers
124 views
A few problems on tensor calculus which i could not solve.
1.if $A_{ij}$ is a skew-symmetric tensor,then show that $(B^i_jB^k_p+B^i_pB^k_j)A_{ik}$=0.
2.if the relation $a_{ij}u^iu^j=0$ holds for all vectors $u^i$ such that $u^ip_i$ where $p_i$ is a given ...
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0answers
24 views
Difference between Multiway data and tensors?
This is a very basic question, what is the difference between multiway data and tensors, from my point of view they look very similar, and a tensor would look like the space to define the multiway ...
0
votes
0answers
80 views
Derivative of a vector with respect to a matrix
I am at an impasse. I don't know if homework is allowed on here or not, so if it isn't, someone delete this.
Given:
$H_{\gamma} = C_{\beta \beta} v_{\gamma} + C_{\beta \varepsilon} C_{\varepsilon ...
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0answers
39 views
Tensor problem with spatial dependence cancelation
Consider the following tensor expression written in indicial notation
\begin{equation}
c_{ij} = A_{ijkl}(x_i) b_{kl}(x_i)
\end{equation}
Here $A_{ijkl}$ and $b_{kl}$ are dependent on $x_i$ but ...
0
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0answers
137 views
Structure tensor of a function and the distribution of gradients
In computer vision, one often computes what's known as the structure tensor of an image. The structure tensor of a an image (i.e. a function) is a matrix that, I quote from Wikipedia.
"summarizes ...
