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.
520
questions
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How to transform a tensor (for example, the second order identity tensor) or matrix from cartesian coordinates to cylindrical coordinates?
First, I believe the unit tensor is a diagonal tensor with diagonal elements of 1,
regardless of whether it is in Cartesian coordinates or cylindrical coordinates.
Then, I wanted to verify this using ...
0
votes
1
answer
64
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Gradient of tensor expressions
I would like to compute gradients of tensor expressions without having to instantiate particular tensors and for arbitrary sizes.
Example 1 :
For example, let's say I want to minimize $C=\sum_j(\sum_i ...
0
votes
0
answers
43
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Christoffel symbol of 2nd kind [duplicate]
How to calculate Christoffel symbol of 2nd kind for a given metric tensor in Mathematica ? I tried a few programming but I'm not getting desired output .
1
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1
answer
51
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NonCommutativeMultiply in Matrix Multiplication
Is there a way to preserve the order of elements in the multiplication of matrices when the elements themselves are matrices?
For example, if I have:
...
0
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1
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40
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A problem with indices in tensor
I want to create a tensor similar to the one shown in the first picture, but I'm encountering some issues with indices. Could anybody assist me?
Here's my code:
...
2
votes
0
answers
49
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Defining a tensor in terms of its trace and other tensors
I'm starting to learn xAct for tensor operations. So I am dealing with 3 tensors: $T_{abc}$, $Q_{abc}$ and a metric tensor $g_{ab}$; with these tensors I have an equation like $$T_{abc} = Q_{abc} + ...
0
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0
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50
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Tensor product with zero
I am calculating the huge sums, that include tensor products and I noticed that Mathematica does not simplify the tensor products of the form $A \otimes 0 \otimes C$, and also it does not simplify the ...
0
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0
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41
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Perturbing a tensorial expression
I am new to Mathematica. I am trying to simplify an expression of the some form like:
$$
n_i \sigma_{ij} n_j - \gamma n_i \hat{\sigma_{ij}} n_j = 2 + v_i x_i + \kappa E_{ij} \chi_{ji}
$$
There are ...
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answers
64
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How to find linearly dependent eigenvectors of non-diagonalizable matrix (with exceptions)?
Here is the code to produce a $9\times9$ matrix:
...
2
votes
1
answer
59
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Handling different kind of indices in single tensor equation
I want to solve the following differential equation,
$$\partial_a (\sqrt{h} \, h^{a b} \partial_b X^{i}) = 0$$
where $$X^i \equiv \{X^1(\sigma^1, \sigma^2), X^2(\sigma^1, \sigma^2), X^3(\sigma^1, \...
3
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1
answer
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Make the ```*``` operation the Kronecker product by default for Kets?
So given two "Ket" objects,
u = Ket[{-1/2}];
d = Ket[{1/2}];
Is there a way to define multiplication, *, so that by ...
0
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1
answer
56
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Optimized and compact code for tensor product
I am calculating products like
in mathematica. In the given relations eta's are four vectors, $\gamma$'s are the Dirac matrices, $\sigma^{\mu\nu}$ is the anticommutation of the gamma matrices and ...
0
votes
1
answer
57
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Permute subsystems of a matrix?
Suppose I have a matrix $M$ acts on the space $\mathbb C^4\otimes \mathbb C^2\otimes \mathbb C^4\otimes \mathbb C^3$. Is there a method to permute the 2nd subsystem of $M$ and the 4th subsystem of $M$?...
0
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1
answer
114
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How to calculate this covariant derivative?
I try to calculate the covariant derivative:
$
\nabla_\beta \partial_\alpha~ \phi = \partial_\beta \partial_\alpha~ \phi
+ \Gamma^\sigma_{\beta\alpha} ~\partial_\sigma~ \phi
$
Where $\phi$ is a ...
3
votes
0
answers
139
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Using ChatGPT to write Mathematica code with the Ricci package [closed]
I generally use ChatGPT to write mathematica code for me.
However, now I want to use a Mathematica package called Ricci. Whenever I try to ask ChatGPT to write me mathematica code using that package, ...
1
vote
1
answer
76
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Metric tensor coordinate transformation with off-diagonal components
I know there is already an answer for this type of question given here: Computing the metric tensor under a coordinate transformation but the answer is not satisfactory as it is not clear to me how I ...
0
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1
answer
75
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Adding a symbolic tensor and a specific one
I need to add a symbolic tensor and a specific one, but it seems the symbolic tensor is always treated as a number and is added in very components of the specific tensor.
Firsts I give assumptions
<...
0
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1
answer
102
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A code to calculate Einstein tensor [duplicate]
I use the following MA code to calculate Einstein’s tensor. I’m asking about the zero component of the Einstein’s tensor, is it correct?
Because I think $G_{00}$ should contains the terms in the zero ...
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0
answers
53
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Calculating and collecting the terms of the zero component of the Einstein’s tensor
I try to calculate the $G_{00}$ of the Einstein tensor
$G_{\mu\nu}= R_{\mu\nu} -\frac{1}{2} g_{\mu\nu} R$
for the metric:
$g_{00}=-a^2(\tau)\left( 1+2 \phi^{(n)}\right),$
$g_{0i} = a^2(\tau)\left( \...
0
votes
0
answers
71
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How to calculate Einstein tensor components for this metric?
I try to calculate the Einstein tensor compenents from the eqution:
$
G_{\alpha\beta} = \frac{\nabla_\beta (\partial_\alpha \phi)}{\phi} - \frac{1}{2\phi^2} \left[ \frac{\partial_4 \phi \partial_4 g_{\...
2
votes
1
answer
86
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How to create a vector as a tensor object for different euclidean bases?
The components of a tensor are always displayed with respect to one or multiple basis vectors.
For a tensor of rank 1, a vector, in 3D-euclidean space, we resort to three orthonormal basis vectors.
...
2
votes
0
answers
60
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Calculating the strength tensor of a vector field
I'm trying to calculate
$$T_{ab} = g_{ab}F_{gd}F^{gd} - F_a^g F_{bg},$$
where
$$F_{ab} = \partial_a A_b-\partial_b A_a$$
So I define $F_{ab}$ by:
...
0
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1
answer
117
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Is there a good package for solving the Einstein field equations given a metric tensor?
I have a very simple metric tensor. I've built a notebook to calculate the EFE solutions based on this metric. Does anyone know of a good package in the Mathematica library that takes a metric tensor ...
0
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0
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102
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Metric pertubation in xAct
I start to learn xAct. Following this thread:
expanding-the-riemann-tensor-perturbation I noticed that xAct set a default perturbation to the metric by:
...
1
vote
1
answer
103
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Which Slots Are Which? Tensors
I'm trying to verify a calculation in Mathematica and I'm confused about how Mathematica arranges tensor "slots" for contractions.
I have the matrix
$$J_1 = \begin{pmatrix} 0 & -ih & ...
0
votes
0
answers
149
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Simplifying the Einstein tensor in case of a perturbed FRW metric
I use the code in this thread's answer:
(Calculating Einstein tensor components in Kaluza-Klein model)
to get the Einstein tensor components of a four-dimensional Kaluza Klein model.
But instead of ...
0
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0
answers
36
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Cannot symmetrize products involving tensors with specified symmetry?
I am trying to check some manipulations with tensors with Mathematica (13.3 on Windows), but was stuck on an error from TensorTranspose. I isolated my problem to the following simple case:
...
0
votes
1
answer
127
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How would you find the metric tensor for this formula?
I have a metric formula that does some interesting things for me. It's excellent at predicting the luminosity of Sne 1a. I'd like to see what the EFE solutions are, but I need to convert it from ...
1
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1
answer
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Calculating Einstein tensor components in Kaluza-Klein model
I try to calculate the Einstein tensor of Kaluza-Klein model from this paper. It is given by Equation (55)
$
G_{\alpha\beta} = \frac{\nabla_\beta (\partial_\alpha \phi)}{\phi} - \frac{1}{2\phi^2} \...
4
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1
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What are some Mathematica packages used for general relativity?
Could you suggest any Mathematica packages that are used for General relativity calculations? Id like to write code to solve the Schwarzchild Lagrangian equation.
0
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0
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Tensor Algebra of Symmetric Algebra with Symoblic Tensors
Given
$V=\mathbb{R}^3,$
I would like to work symbolically with elements of
$\mathrm{T}^{(k_1,\dotsc,k_n)}(\mathrm{S} V),$
i.e. vectors of the form
$$\phi = \phi_1 \otimes \cdots \otimes \phi_n$$
with
$...
2
votes
2
answers
185
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Lorentz indices contraction
I have a complex expression involving second-rank tensors and Kronecker Deltas. How can I instruct Mathematica to utilize the Kronecker Deltas and perform proper index replacements? For example, ...
1
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1
answer
92
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Multiplying abstract tensors in Mathematica
I want to do an operation like
$$ AA \otimes BB \times AA' \otimes BB' $$
and have
$$AA \times AA' \otimes BB \times BB' $$
without specifying the elements of $AA$s and $BB$s (and so are their primes)....
0
votes
0
answers
69
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Variation of Lagrangian with respect to components of inverse of metric tensor
I am super new to Mathematica, so I apologize if the question is trivial.
I have defined metric tensor (4x4 matrix), inverse metric and Lagrangian:
...
0
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0
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28
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Symmetric expression not recognized
Why the following symmetric tensor is not recognized by TensorSymmetry?
...
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0
answers
72
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Operations on symbolic tensors
I have recently discovered the set of functions related to symbolic tensors, namely TensorProduct, TensorContract, ...
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3
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137
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Product involving Kronecker Deltas
I am trying to do some products involving some objects made out of Kronecker deltas. For example, taking an object like $x_{abcd}=\delta_{ab}\delta_{cd}$, where all the indices run from $1$ to $N$, I ...
2
votes
0
answers
139
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How to define a Kerr-Schild metric in xAct?
A Kerr-Schild can be expressed as $g_{ab}=\eta_{ab}+\phi k_a k_b$, where $\eta_{ab}$ is Minkowski metric, $\phi$ is a scalar function and $k_a$ is a (co)vector field which is null and geodetic with ...
1
vote
0
answers
77
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Finding Petrov type of a metric
Is there a subpackage in xAct (or another Mathematica package for general relativity calculations) to determine the Petrov type of a metric?
I understand that xAct has a lot of subpackages for ...
0
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0
answers
136
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How to write the metric tensor in n-dimensions using xAct?
I am trying to reproduce some calculations from a paper, where they express the angular part of the metric as a solid angle in (d-2) dimensions:
where
I know how to calculate the metric tensor using ...
2
votes
1
answer
184
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Efficient way to MapApply for Tensor
I need to Map a function with certain arguments $f(\alpha,\beta,\gamma,\delta)$ onto a whole tensor.
My current approach looks like
...
5
votes
3
answers
121
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Implementing symmetry assumptions in FullSimplify
I want to symmetrise a long expression, M, that involves a function of 4 arguments, f[u1,u2,d1,d2], and its products (for ...
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1
answer
78
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Calculating the orthonormal frame of a metric in Mathematica
Let us have a given a general metric (like say Kerr metric) of which I want to find the orthonormal coordinates by developing a general code in Mathematica. One of the reliable method to do this (by ...
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1
answer
85
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What is the right way to simplify a tensor expression (with many indices, but no derivatives) in Mathematica?
After not finding the desired capabilities in base Mathematica, I am trying to use xAct`xTensor package. It appears that I do not need most of it capabilities ...
0
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0
answers
115
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Simplification of Momentums in FeynCalc
I have a little problem involving the following contraction in FeynCalc:
$$\frac{k^{\delta}k^{\sigma}}{k^2}$$
By hand one would lower $\sigma$ and contract with the other momentum to cancel $k^2$, ...
8
votes
4
answers
614
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How can I "multiply" nested lists?
Given two nested lists
alist={{a,b,c},{d,e,f}}
blist={{r,s,t},{x,y,z}}
How can I get
...
1
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0
answers
68
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How to multiply five tensors of different ranks (contravariant, covariant and mixed) in Mathematica?
How to compute the following multiplication of five different tensors of different ranks in Mathematica
...
2
votes
2
answers
122
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Symbolic matrix tensor an identity without specifying the dimension?
I want to calculate an expression like $\left( M_1\otimes I+I\otimes M_2 \right) ^l$ with $M_i$'s symbolic matrices and $I$ the identity matrix with Mathematica. $M_i$'s are of the same dimension and ...
1
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1
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124
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TensoriaCalc does not display the correct output
I am trying to use TensoriaCalc to calculate the components of the Ricci and the Riemann tensor of the following metric: $R^{2} \left(d\theta^{2} + \sin^{2}\left(\theta \right)d\phi^{2} \right)$;
...
1
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0
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122
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Mathematica, ML and TensorFlow
I am currently studying a specialization on Coursera in Machine Learning and am investigating various tools to help me out with the maths and with visualisations and so on.
Although I have many ...