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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0answers
7 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 ...
16
votes
1answer
710 views
Proof of $\det(\textbf{ST})=\det(\textbf{S})\det(\textbf{T})$ in Penrose graphical notation
For two matrices $\textbf{S}$ and $\textbf{T}$, a proof of $\det(\textbf{ST})=\det(\textbf{S})\det(\textbf{T})$ is given below in the diagrammatic tensor notation.
Here $\det$ denotes the ...
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3answers
25 views
notation question (bilinear form)
So I have to proof the following:
for a given Isomorphism $\phi : V\rightarrow V^*$ where $V^*$ is the dual space of $V$ show that $s_{\phi}(v,w)=\phi(v)(w)$ defines a non degenerate bilinear form.
...
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votes
1answer
21 views
How do I compute the Laplacian of a function in terms of a given (general) coordinate transformation?
Consider a coordinate transformation $\boldsymbol{x} = \boldsymbol{x}(\boldsymbol{\xi})$ (with Jacobian $\partial \boldsymbol{x}/\partial \boldsymbol{\xi})$, the scalar function $f(\boldsymbol{x}) = ...
1
vote
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 ...
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1answer
23 views
Handling more than three indices/super indices, tensorial calculus
I need to code an equation such as the following one :
$$ \frac{\partial u^j}{\partial q^i} = \frac{\partial \mathrm A^j_{pl}}{\partial q^i}\dot q^p \dot q^l + \frac{\partial \mathrm B^{jl}}{\partial ...
1
vote
0answers
15 views
Metric spaces and curvature
Suppose that it is given that the Riemann curvature tensor in a metric space of dimension $d\geq2$ can be written as $$R_{abcd}=k(x^a)(g_{ac}g_{bd}-g_{ad}g_{bc})$$ where $x^a$ is a vector in the ...
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1answer
26 views
The operator in Tensor algebra.
Let $V$ be a vector space over a field $K$.
We define the $k^{th}$ tensor power of $V$:
$$T^kV = V \otimes V \otimes ... \otimes V$$
We contruct $T(V)$ as the direct sum of $T^kV$ for $k=0,1,2,...$
...
1
vote
1answer
56 views
How to solve a tensor differential equation?
Essentially, How does one solve the tensorial differential equation $$\frac{dx^a}{d\tau}=A^a{}_bx^b$$
where $x^a$ is a 4-vector and $A^a{}_b$ is a $(1,1)$ tensor.
The original Problem
How does ...
2
votes
1answer
28 views
Is a $1^{\mathrm{st}}$ rank tensor identical to a vector?
As far as I know, we define vectors as elements of a vector space, then there is an isomorphism (by choosing a basis) from the vector space to tuples of components in some field, $\mathbb{F}$ say. ...
1
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0answers
42 views
What is the needed background to study tensors?
What is the needed background to study tensors?
I do not plan to study it but, I want to know the background!
It's a matter of curiosity .
Thanks.
1
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0answers
34 views
To show that something is a four-vector
I hope this question is not too inane... it would be really helpful for me to have this cleared up. I want to know what I need to show to demonstrate that something is a four-vector. I have checked ...
3
votes
1answer
44 views
“Inverse” of tensor product
I am trying to figure out something. I have a 4-tensor $\phi_{i \, j \, k \, \ell}$ and I know that $\phi = A \otimes B$, being $A$ and $B$ two matrices. With indices, I know this:
$\phi_{i \, j \, k ...
1
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0answers
32 views
Tensor compact/matrix form.
I have got this tensor $S_{ij} = \frac{1}{2}(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i})$
Anyway I solve it for my problem and get
$$ S_{ij} =
\left( \begin{array}{ccc}
0 ...
0
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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 $(A^{a}+B^{a})(A_{a}B_{a})$
2
votes
1answer
37 views
Prove that a tensor field is of type (1,2)
Let $J\in\operatorname{end}(TM)=\Gamma(TM\otimes T^*M)$ with $J^2=-\operatorname{id}$ and for $X,Y\in TM$, let
$$N(X,Y):=[JX,JY]-J\big([JX,Y]-[X,JY]\big)-[X,Y].$$
Prove that $N$ is a tensor field of ...
2
votes
1answer
36 views
Contraction of the second Bianchi identity
The second Bianchi identity is
$${R^a}_{b[cd;e]}=0$$
And contracting it with respect to $a$ and $e$ we get
$${R^a}_{b[cd;a]}=0 \Leftrightarrow $$
$${R^a}_{bcd;a}+R_{bc;d}-R_{bd;c}=0$$
What I don't ...
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votes
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!
1
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1answer
71 views
Quotient theorem for tensors
Can somebody please explain to me how the following statement is true?
The Riemann curvature tensor $R^c_{dab}$ is given by the Ricci identity $$(\nabla_a\nabla_b-\nabla_b\nabla_a)V^c\equiv ...
2
votes
1answer
58 views
Parallel Transport along a curve
We had this homework assignment for our geometry course, and we couldn't figure it out, any ideas on how to do this:
Consider the Poincare model of Lobachevsky plane,
$H^2=\left\lbrace{ ...
1
vote
0answers
63 views
Riemannian curvature and its application on covariant derivative of tensors
This identity can be generalized to get the commutators for two
covariant derivatives of arbitrary tensors as follows $ \begin{align} T^{\alpha_1 \cdots \alpha_r}{}_{\beta_1 \cdots \beta_s ; ...
2
votes
1answer
27 views
Regarding the definition of covariant derivative and its use on basis vector fields
we find that for general vector fields ${\mathbf v}= v^ie_i$ and ${\mathbf u}= u^je_j$ we get :$\nabla_{\mathbf v} {\mathbf u} = \nabla_{v^i {\mathbf e}_i} u^j {\mathbf e}_j = v^i \nabla_{{\mathbf ...
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0answers
17 views
what is the status of the theory of multilinear systems of equations?
What is the current status of the theory of multilinear systems of equations?
I have a particular interest for multilinear homogeneous systems of the form
$A_1 \otimes \cdots \otimes A_r) (x_1 ...
1
vote
1answer
58 views
Gradient with respect to a matrix variable
I want to find the gradient of the function $\mathcal{F}_1$ with respect to the matrix $\mathbf{X}$ (differentiate with respect to $\mathbf{X}$):
$$
\mathcal{F}_1 (\mathbf{X}; \mathbf{\lambda})= ...
0
votes
1answer
16 views
Regarding confusion of basis tensors and the usage of tensors.
Let us for example give a tensor example of following: $X = X^i \partial_i$. According to mny knowledge, in this case $\partial_i$, basis, is treated as tensor (otherwise, $X$ as tensor won't be ...
1
vote
1answer
27 views
Finding an “inverse” of a deviatoric tangent
I have have a material model, defining the deviatoric stress for a nonlinear fluid:
$\boldsymbol{\sigma}_{\mathrm{dev}} = f(\dot{\boldsymbol{\varepsilon}}_{\mathrm{dev}})$
Now I wish to find the ...
2
votes
2answers
41 views
Tensor notation and rules
I have a few questions about tensors:
I appreciate that $g^{\alpha\beta}=g^{\beta\alpha}$ but when contracting say $T^{\sigma}_{\mbox{ }\;\mu\nu\rho}$ to $T_{\;\;\mu\nu}$, first of all can it be ...
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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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votes
1answer
91 views
Proving the symmetry of the Ricci tensor?
Consider the Ricci tensor :
$R_{\mu\nu}=\partial_{\rho}\Gamma_{\nu\mu}^{\rho}
-\partial_{\nu}\Gamma_{\rho\mu}^{\rho}
+\Gamma_{\rho\lambda}^{\rho}\Gamma_{\nu\mu}^{\lambda}
...
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votes
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 ...
2
votes
1answer
29 views
If $T$ is a $k$-tensor and $S$ is an $l$-tensor, then $\text{Alt}(T \otimes S) = (-1)^{kl} \text{Alt}(S \otimes T)$
Could someone please help me with the following algebra question? I know it should be easy, but the textbook leaves the proof to the reader and I am having a hard time with it. Thank you in advance.
...
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1answer
34 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 ...
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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 ...
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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 ...
-1
votes
1answer
61 views
Dot product between two vectors or vector and 1-form?
When we take a dot product between two vectors of a vector space, we actually "act" by a 1-form (dual vector) on a vector. So why most books define the dot product between vectors? Of course with the ...
0
votes
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
votes
3answers
91 views
How to differentiate a differential form?
Please explain me the idea of differentiating differential forms (tensors). Example:
compute d(xdy + ydx)
The answer is known, we should have 0. What's the rule?
0
votes
0answers
121 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 ...
28
votes
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, ...
5
votes
0answers
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 ...
1
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0answers
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
vote
2answers
144 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, ...
0
votes
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 ...
1
vote
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 ...
0
votes
1answer
118 views
How can I write a tensor in MATLAB?
My project is about tensors and I must write the program in MATLAB.
How can I write a tensor in MATLAB?
Is there anybody to help me?
Can you explain me what is the code for import a tensor?
thanks for ...
0
votes
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 =
...
1
vote
0answers
149 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
vote
1answer
65 views
Tensor and conservation
I am reading a book "The Early Universe " by Kolb and Turner. On P.48, it says
For $T^\mu\,_\nu=\operatorname{diag} (\rho, -p,-p,-p)$,
the $\mu=0$ component of the conservation of stress energy ...
2
votes
1answer
82 views
Tensors in math and physics
I know how tensor product f two modules is defined in communtative algebra. But there is also a concept of tensors used in physics. Are these two concepts related? If yes, can someone explain me ...
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 $T$?
