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.

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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
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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
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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 ...
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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 ...
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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 ...
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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 ...
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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
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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 ...
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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
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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
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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)=\...
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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 ...
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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} + \...
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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 ...
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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, ...
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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. ...
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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:...
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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
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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
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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: ...
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1answer
28 views

TensorReduce causes “inhomogeneous dimensions” error

I want to use TensorReduce to realize the following property of wedge: ...
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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
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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, ...
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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
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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
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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 ...
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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
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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 ...
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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 ...
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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: ...
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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?
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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, ...
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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
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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 ...
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1answer
44 views

\[CircleTimes] (tensor symbol use) in infix conversion (solved)

Reformulated problem. I am using the code: ...
20
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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, ...
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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
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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
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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 \...
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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
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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 ...
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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?
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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
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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
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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) ...
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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
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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 ...
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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
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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
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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 ...