Questions tagged [tensors]
Use this tag for questions that involve tensors. Tensors are fundamental tools for linear computations, generalizing vectors and matrices to higher ranks. Mathematica 9 introduces powerful methods to algebraically manipulate tensors with any rank and symmetry.
292
questions
0
votes
0answers
36 views
“PermutationReplace[1,#1] cannot be used as a part specification” [on hold]
I am looking for all permutations of the standard coordinates on $\mathbb{R}^7$ that leave the 3-form $\varphi=dx_1 \wedge dx_2 \wedge dx_3 + \dots$ (this is a form with stabiliser $G_2$) unchanged.
...
4
votes
1answer
30 views
Computing Higher Order Tensor of Variable Rank
The following code takes a vector x of variable length, computes the outer product of the vector with itself to form the matrix $\rho$ of dimension $2^n \times 2^n$....
4
votes
1answer
56 views
How to Creat a Symbolic Rank 4 Symmetric tensor
I would like to create a rank 4 symbolic tensor with this symmetries
(1) C_ijkl = C_jikl
(2) C_ijkl = C_ijlk
(3) C_ijkl = C_klij
is there any way to apply symmetry (3)?
symmetry (1) and (2) can ...
1
vote
1answer
40 views
Higher Order Tensor of Variable Rank
I would like to write a function, that matricizes a higher order tensor according to the following rule:
Let $\mathcal{A} \in \mathbb{C}^{I_{1} \times I_{2} \times \ldots \times I_{N}}$ be a tensor ...
1
vote
1answer
35 views
Independent permutation symmetry of a tensor [Using TensorSymmetry command]
Suppose I have this tensor $A_{ijkl} = \epsilon_{ik} \epsilon_{jl}+\epsilon_{il} \epsilon_{jk}$. Now I want to find all the independent permutation symmetries of the indices of this tensor. The answer ...
2
votes
1answer
66 views
How do I force this simplification?
I am trying to calculate:$$\epsilon_{ijk}n^iM1^j\epsilon^{lmk}n_lM2_m$$
Defining the vectors as:
M1={M1x,M1y,M1z}
M2={M2x,M2y,M2z}
n={nx,ny,nz}
and using
...
1
vote
1answer
37 views
How to define commutative covariant derivatives in xTensor
I'm new with working xAct package and got stuck defining covariant derivatives that commute with each other.
I want to simulate flat space-time. And here is my code
...
3
votes
1answer
51 views
How do I prove this formula with LeviCivitaTensor?
How do I prove that
$$ \epsilon_{ijk}\epsilon^{ijm}=2\delta_k^m ? $$
If I use
...
8
votes
4answers
362 views
How can I automate this tensor computation?
I am doing this work by hand, but it takes a lot of time and I make several calculation errors, so I was thinking to make Mathematica to calculate this for me, but I am stuck at the very beginning.
I ...
4
votes
1answer
97 views
Symbolic multiplication of tensors
I am trying to do symbolic calculations on Pauli Matrix Algebra. I want to find some way of making CircleTimes[a,b] ** CircleTimes[c,d] map to CircleTimes[a.c,b.d] for every a,b,c,d. I would also like ...
6
votes
1answer
126 views
Speeding up tensor contractions and multiplication
Consider a tensor $T\in\mathbb{R}^{N\times N\times N\times M}$ and two vectors $x,y\in\mathbb{R}^N$. I want to compute the $N\times M$ vector defined by $X_{ij}=\operatorname{tr}(x^\top T_{:,:,i,j}y)=\...
0
votes
0answers
33 views
Doubt about tensor producto of two columns vectors
I want to get the tensor product of two columns vectors for example:
a={1,2,3}
b={2,3,1}
psi0 = ArrayFlatten[TensorProduct[a, b]];
The size of psi0 is 3x3 but it ...
0
votes
0answers
35 views
How to compute gauge variation of expression?
Suppose I have a symmetric tensor field $h_{\mu\nu}$
I want to implement somehow the following gauge variation of this tensor field as follows
$\delta h_{\mu\nu} = \nabla_{\mu}\epsilon_{\nu} + \...
1
vote
0answers
23 views
Inverse fuction of TensorExpand
Is there a way in mathematica to factorize/simplify a dot product? I.e. I have something like a.b + a.c (obviously more complicated expressions) and I would like to factorize terms like
...
10
votes
2answers
544 views
Make symbols atomic, without losing their type
So, I'd like to define a matrix M, that does not decompose into it's constituents when I do things like Tr[M], but I also want it's type to be retained.
(By type, ...
1
vote
0answers
41 views
Print 4x4x4x4 tensor as 4x4 matrix of 4x4 matrices [closed]
I'd like to print the LeviCivitaTensor[4] tensor as 4x4 block matrix of 4x4 matrices
The following code works to display the LeviCivitaTensor[3] tensor as a 3x1 vector of 3x3 matrices.
...
0
votes
0answers
34 views
CompiledFunction::ctfa - Argument should be a rank 4 tensor of machine-size real number
Dear Mathematica community,
I got the following error line:
CompiledFunction::ctfa "Argument {<<1>>} at position 1 should be a rank 4 tensor of machine-size real number"
This is my function:...
1
vote
1answer
68 views
xAct, xTensor: How to avoid clash of indices?
Please refer to the picture below. In the first line, I define the angular momentum vector $\vec{L} = \vec{R} \times \vec{P}$ using the Levi-Civita tensor $\epsilon^{i}_{jk}$. The definition relies on ...
3
votes
0answers
50 views
Lie-algebra valued non-abelian differential forms in Mathematica
I am trying to implement wedge product of Lie-algebra valued differential forms in the non-abelian case. Concretely, I am intereste in 1- and 2-forms, that is, $A$ and $F$. Is there a way of doing ...
2
votes
1answer
63 views
Outer (dyadic) product between vectors of the same index in two lists
I have two lists of vectors and I want to take the outer product between elements in the lists of the same index. Using either Outer or TensorProduct works for e.g. the first two indices:
...
0
votes
1answer
28 views
TensorReduce causes “inhomogeneous dimensions” error
I want to use TensorReduce to realize the following property of wedge:
...
1
vote
1answer
38 views
How to keep the form of tensor wedge, instead of using tensor product?
Why is an expression with full form of
TensorProduct[TensorWedge[v1, v2], w1]
changed into
...
2
votes
1answer
66 views
Efficient computation of matrices involving large sums of KroneckerDelta's
I was wondering if one could benefit from Mathematica's rich linear algebra methods for diagonalizing 2nd rank tensors. Namely, in the context of systems (fluids) comprised of capsule-like particles, ...
0
votes
1answer
94 views
How to construct — based on physics-type notation — a magical simplex $\mathcal{W}$ of bipartite qutrits?
I have a short Mathematica program:
...
2
votes
1answer
32 views
Maximally expand tensor series?
I am trying to obtain a power series expansion in some real parameter, but all the terms are arbitrary products of tensors. I.e. I want to expand an expression containing sums/products/powers of this ...
2
votes
1answer
49 views
Contraction of square tensors
Let there be tensors A and B
A = Outer[Times, {1, 0}, {2, 0}]
B = Grad[{f[x, y], g[x, y]}, {x, y}]
with output
...
0
votes
0answers
92 views
How can I define another metric (disformal transformation) in xAct?
I'm using the xAct package. I want to define two metrics with the disformal transformation relation
$\qquad \bar{g}_{\mu\nu}=A(\phi) g_{\mu\nu}+B(\phi)\nabla_\mu \phi\nabla_\mu \phi ,$
where $g_{\...
2
votes
2answers
85 views
How to define an antisymmetric symbol?
I want to work with linear expressions involving the formal symbol $w[a_1,...,a_n]$, and I would like Mathematica to know that expressions such as
...
0
votes
1answer
42 views
Performing a simple operation with operators [closed]
I have a map defined as
$\qquad \Phi(X) = a^2\, Tr[X] |0\rangle \langle 0| + b^2\, Tr[X] |1\rangle \langle 1| + a\,b\, Tr[\sigma_z X] |0\rangle \langle 1| + a\,b\, Tr[\sigma_z X] |1\rangle \langle ...
0
votes
0answers
41 views
How to save a multidimensional matrix in separated files for each dimension
I need to export to a file a multidimensional matrix, each dimension to a file.
For example:
...
0
votes
1answer
87 views
Exporting a tensor
I have a complicated and large tensor what can be shown in the simplest form as follows
How can I export it in this form?
3
votes
0answers
69 views
Lowering the index of the Riemann curvature tensor in Mathematica
I am looking to do some calculations in GR.
For a given metric, I can calculate the affine connection and the Riemann tensor as,
...
1
vote
1answer
82 views
I want to perform a simple tensorial contraction
I want to perform a simple tensorial contraction like, if KroneckerDelta[i, j] is contracted with some arbitrary tensor A_{lkj} (not-necessarily symmetric) it should give the answer as A_{lki}. Is ...
2
votes
2answers
147 views
How does one plot a 3 dimensional table of numbers?
I've just spent three hours searching the documentation and this website for an answer.
I have a rank 3 tensor:
t = {{{1, 2}, {3, 4}}, {{5, 6}, {7, 8}}}
How ...
0
votes
1answer
44 views
\[CircleTimes] (tensor symbol use) in infix conversion (solved)
Reformulated problem.
I am using the code:
...
20
votes
4answers
859 views
What is the definition of Curl in Mathematica?
I have a usual mathematical background in vector and tensor calculus. I was trying to use the differential operators of Mathematica, namely Grad, ...
1
vote
0answers
70 views
Evaluating covariant derivative for perturbed metric given background metric
I want to use Mathematica to evaluate an expression like;
$h^{\alpha\beta}_{\,\,\,\,\,|\mu}h_{\alpha\beta\,|\nu} +\mbox{similar}$
where $h_{\alpha\beta}$ is the perturbation to a specified metric (...
2
votes
1answer
112 views
How to apply Partial differentiation w.r.t. tensors?
Let's say I have an expression like $\,a^{I}=2b^{I}+3c^{I}$ where $I$ stands for an arbitrarily large set of indices. It's known that $\,\frac{\partial a^K}{\partial c^{L}}=3\,\delta^K_L$ (equals a ...
3
votes
1answer
38 views
Why doesn't this Kronecker Product work with columns, but with rows?
Using the formula given in this math.stackexchange answer by the user greg
$$\eqalign{
vec(M\otimes dK) &= \left(\pmatrix{I_T\otimes (M \cdot e_1)\cr I_T\otimes (M \cdot e_2)\cr \vdots \cr I_T \...
1
vote
0answers
42 views
Typography of Mixed Tensor Index Notation
How can one properly and quickly write a mixed "tensor" such as $\Gamma{}_{1}{}^{2}{}_{3}$?
Shortcuts such as esc Gamma esc, Ctrl+^, Ctrl+_ are prefered. I intendd to use this in an inline equation ...
2
votes
1answer
35 views
Obtain argument of tensor product
I want to obtain the argument of a TensorProduct as a list of elements by applying another function.
For example, I wish to write a function h which, when applied to a tensor product, produces the ...
1
vote
0answers
104 views
Solving the values of Christoffel symbols from a given metric tensor
Is there any easy mathematica package by which I can solve the Christoffel symbols from a given metric tensor?
0
votes
1answer
66 views
Derivative of a function in undefined dimension
I have a scalar function defined on a $ n $-dimensional manifold:
$ f(x_1, x_2, ..., x_n) $, where $ n $ is undefined, and $x_i$ are the coordinates.
How to define something like "$∂_af∂^af$"?
(I'm ...
2
votes
2answers
171 views
Kronecker product $ n $ matrices
I would like to write code to realize the Kronecker Product of $ n $ matrices, for instance when $ n=4 $ and the matrices are Pauli matrices
...
2
votes
1answer
102 views
Define tensor as a derivative
I have the tensor which is expressed in terms of coordinate vector. I want to define tensor which is the derivative of the former tensor with respect to the coordinate axis:
$$
X = (x_1, x_2, x_3) ...
1
vote
0answers
84 views
Tensor Contraction Speed Up (Matrix Product States)
I'm recently trying to implement some tensor contractions in Mathematica for use in Matrix Product State algorithms.
Here's the operation I want to perform
$$
M^{\sigma_{i}\sigma_{i}'[i]}_{(b_{i-1},...
2
votes
0answers
104 views
Exterior products of differential forms
In $ \mathbb{R}^4 $ I have the forms $ \omega_1=z\;\mathrm dx+t\;\mathrm dy+x\;\mathrm dz+y\;\mathrm dt $ and $ \omega_2=t\;\mathrm dx+z\;\mathrm dy+y\;\mathrm dz+x\;\mathrm dt$.
I want to compute ...
1
vote
0answers
52 views
Calculating Christoffel Symbols [duplicate]
I am totally new in the astrophysics field and I need to calculate Christoffel symbols for a model. The line element for the FLRW metric and the geodesic equation is given below: ...
3
votes
1answer
91 views
Symbolic decomposition of a tensor given an assumption
I am relatively new to Mathematica. Suppose I have a matrix
M = {{a^2, a b, 0}, {a b, b^2, 0}, {0, 0, 0}}
How can I get Mathematica to decompose this into an ...
3
votes
4answers
173 views
Fast way to create $[ I_4\otimes e_1,\ \dots ,\ I_4 \otimes e_T]$?
Is there a fast way to construct this matrix?
$\left[\begin{array}{c}
I_4\otimes e_1\\
\vdots \\
I_4 \otimes e_T \end{array}\right]$
$e_i$ is the $i$-th column of the matrix $I_T$, $\otimes$ is the ...
