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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11
votes
1answer
122 views

Nearest Kronecker Product

Some people (see The ubiquitous Kronecker product by Van Loan) have worked on finding two matrices $\mathbf A$,$\mathbf B$ of specified size whose tensor product $\mathbf A\otimes\mathbf B$ is closest ...
5
votes
3answers
185 views

How to simplify symbolic expressions with KroneckerProduct

Having $X,Y$ being symbols for matrices, I was wondering if there is a way to simplify expressions like KroneckerProduct[X, X] + KroneckerProduct[-X, X] to ...
4
votes
2answers
89 views

The fastest way of raising indices of high rank tensor

I have some high dimensional high rank tensors, let's say $$F_{ijkl}$$ and I need to find $$F^{abcd}=g^{ai}g^{bj}g^{ck}g^{dl}F_{ijkl}.$$ Here $g^{ij}$ is the contravariant metric. Simple summation ...
9
votes
4answers
1k views

Ways to compute inner products of tensors

One way to evaluate the following sums is combining Table and Sum: $u_{abcd} = \sum_{e=1}^3 v_{aeb}w_{ced}$ $q_{ab} = \sum_{d,e=1}^3 v_{d e a}w_{deb}$ It will look like ...
0
votes
1answer
50 views

Issue with Differentiation over a tensor

I am new user in Mathematica. I am working with liquid crystal theory, and one of the notation there is the tensor: $ \Pi_{ij} = \frac {\partial {F_{el}}} { \partial ( \partial {n_i} / \partial { ...
0
votes
1answer
43 views

Is there a package that can calculate the Ricci tensor from a numerically given metric?

There are many packages about general relativity or differential geometry, and they can calculate the Ricci tensor from a symbolically given metric, for example, $g_{tt}=-f(r)$, $g_{rr}=h(r)$, etc. ...
24
votes
4answers
8k views

How to calculate scalar curvature Ricci tensor and Christoffel symbols in Mathematica?

I am seeking a convenient and effective way to calculate such geometric quantities. I've used packages like TensoriaCalc, but they don't work at all time. ...
1
vote
1answer
54 views

How to swap tensor indices without permutation?

Say I have a tensor of rank R, How to swap $i^{th}$ and $j^{th}$ tensor index? I do not want to use TensorTranspose because it requires to write down the entire permutation, i.e. ...
8
votes
2answers
231 views

How to read off coefficients of tensor-like expression in a speedy way?

I am considering identities involving t[a, b, c, d, ...], where number of indices is fixed. t has the cyclic property so that ...
2
votes
1answer
83 views

Tensors with mixed symmetry

I want to work with tensors of mixed symmetry, for example with a rank 3 tensor $A_{ijk}$ that is symmetric under the exchange of $i$ and $j$ and antisymmetric with respect to $i$ and $k$. I was ...
1
vote
0answers
57 views

Fast construction and reshaping of large tensors

I'm trying to construct a large matrix which is derived from some higher rank tensor (the rank of interest to me changes case by case, so it needs to be a general method). Currently, the process of ...
4
votes
2answers
72 views

Using LeviCivitaTensor

I have 2 questions about using LeviCivitaTensor, based on the following session: The signs for the cross product seems to come out "opposite" of what I expected. Why is that? Did I miss anything? ...
2
votes
1answer
76 views

Abstract iterator in Do

I'm trying to write a module. Its input is a matrix (tensor), the module should return new tensor with increased rank with 1. The new tensor is defined as $$ O \Rightarrow N_{\alpha \beta \ldots ...
3
votes
0answers
137 views

Rule transformations for compositions of tensor products [closed]

The tensor product satisfies far commutativity: $(f \circ g) \otimes (h \circ k) =(f \otimes h) \circ (g \otimes k)$. I have some replacement rules I'd like to apply to an expression containing ...
0
votes
0answers
56 views

Atlas 2 For Mathematica. Need to calculate Riemann tensor, etc [duplicate]

Is anyone familiar with Atlas 2 for Mathematica to calculate the Riemann Tensor, Ricci Tensor, and scalar I have a metric that I need to calculate these things for. Can anyone help? I'm not too up on ...
5
votes
1answer
166 views

FindFit returns “tensors have incompatible shapes”

I'm trying to fit experimental data with FindFit and implicitly calculated function. It seems, that function works fine and satisfactorily approximates data when parameters are fed manually. However ...
2
votes
0answers
42 views

Finding basis of isotropic tensors of rank r

I am looking for a way to obtain a basis of isotropic tensors of rank $r$. Actually I am mostly interested in rank $8$ isotropic tensors but maybe you know already a simple algorithm in order to ...
2
votes
1answer
91 views

Efficient tensor product followed by contraction [duplicate]

Say I want to efficiently evaluate $\sum_{kl}A_{ikjl}B_{kl}$ where $A$, $B$ are numerical tensors. This has been discussed before but with no focus on efficiency. A straightforward way as mentioned ...
1
vote
1answer
33 views

Inverse KroneckerProduct given other arguments

I'm having problems working out how to do the inverse of a two argument Kronecker Product, given one of the decomposed column vectors. For example, say I have used the Kronecker Product on two ...
11
votes
1answer
244 views

How can I define or use a new coordinate system in Mathematica?

I want to use the dipole coordinate system as defined in this paper: http://arxiv.org/abs/physics/0606044 I see that Mathematica can do all kinds of vector analysis using different kinds of ...
7
votes
1answer
165 views

How to define an orthogonal basis in the right way?

I am trying to work with the vector notation without defining vector components explicitly. $Assumptions = (x | y | z) \[Element] Vectors[3] The vectors ...
4
votes
1answer
207 views

How to put the tensor product of two operators onto two variables?

I am trying to make an intuitive Mathemtaica program to show the principle of quantum walking. The idea of quantum walking is very simple and direct, but when I try to transfer it into Mathemitca ...
7
votes
1answer
2k views

Tensor analysis

Has anybody used tensors in Mathematica? How to properly work with them? I find Mathematica not very friendly in this field, as I am defining my own functions for lowering & raising indices, ...
0
votes
0answers
43 views

Tensor sum contraction [duplicate]

Table[Sum[ KroneckerDelta[[i, j]] LeviCivitaTensor[3][[i, j, k]], {i, 1, 3}, {j, 1, 3}], {k, 1}] Why this doesn't work? how i do to make that sum?
4
votes
1answer
78 views

TensorReduce not simplifying transpose for symmetric matrix?

I know Mathematica is not very good with symbolic manipulation of matrix expressions, but I was surprised to find that it can't reduce the following expression: ...
0
votes
0answers
128 views

Writing rules for Einstein summation

I'm trying to write a list of rules for tensor manipulations and in particular, Einstein summation convention. What I've tried, so far is to write something that would take a generic functions with ...
0
votes
0answers
107 views

Symbolic Tensor Algebra

I need to perform basic tensor algebra in order to double check some very complicated simplification. It's nothing fancy, it just has so many factors by the end that it's hard to tell if an error has ...
0
votes
0answers
110 views

Solving for a coordinate transformation for a rank 2 tensor

I am working on a project in general relativity, and I have a metric or rank 2 tensor $g_{ab}$ of the form, $$g =\left( \begin{array}{ccccc} f(x^a) & 0 & 0 & 0 & 0\\ 0 & -f(x^a) ...
0
votes
0answers
82 views

How to implement a tensor system using indices and Einstein notation?

There are several function like TensorProduct and TensorContract doing basic tensor operations by regarding tensors as lists ...
2
votes
0answers
37 views

Nested `TensorContract` bug

When nesting TensorContract the result I get is wrong. Input: TensorContract[TensorContract[A, {{1, 3}, {2, 4}}], {{1, 2}}] ...
4
votes
1answer
417 views

Perturbation theory in general relativity using xAct

I'm trying to use the xAct Mathematica package for manipulating tensors, and I'd like to plug in a metric into the perturbation equations to first order in general relativity, and have everything ...
1
vote
0answers
87 views

Derivative with respect to a tensor

I am trying to differentiate a tensor with respect to another one in Mathematica, but I can't do it. Down below is the code I am using: ...
1
vote
0answers
24 views

Assign a symbolic tensor with determinant greater than zero in Mathematica

I can create a symbolic tensor in Mathematica: $Assumptions = F \[Element] Arrays[{3, 3}, Reals] Is there anyway that I can tell Mathematica, that the ...
5
votes
1answer
93 views

Error messages from TensorContract and TensorReduce

I am struggling with a few errors when using symbolic tensors. I am using mathematica 9.0.1.0, linux x86. The following code generates what seems to me an incorrect tensor, this is the smallest ...
0
votes
0answers
42 views

Permute indices of a large sparse array

I'm starting with a very large rank 3 tensor, F, that is stored as a SparseArray, and I need to permute its indices. I first tried using ArrayRules to get the positions of the non-zero elements, to ...
2
votes
0answers
73 views

SymmetrizedArray of stiffness/compliance tensor

The stiffness and compliance tensors of a material are 3x3x3x3 tensors relating stress and strain. ...
1
vote
1answer
97 views

How to take derivative of a tensor leading to kronecker delta? [duplicate]

I'm trying to write a Mathematica program to find derivatives such as the following simple example: Could you please help?
1
vote
2answers
99 views

Change of basis for a rank 3 Cartesian tensor

I have a Cartesian tensor $\chi_{ijk}$ and I want to express the elements in terms of a new basis to get $\chi_{ijk}^\prime$. The transformation is represented using $a_{ij}$. The tensor transforms ...
1
vote
1answer
157 views

Solving a System/Matrix of Equations in Matrix Form

Ok, I am trying to solve an equation involving matrices (well, tensors actually), which is of the form: $\mathbf{e}^{T} \cdot \mathbf{M} \cdot \mathbf{e}$ = $\mathbf{N}$, where $\mathbf{e}$ is an ...
2
votes
3answers
480 views

Sum with Levi-Civita [duplicate]

I'm trying to write the expression $$\sum_{\alpha,\beta = 1}^{4}\epsilon_{\mu \nu\alpha\beta}a^{\nu} b^{\alpha} c^{\beta}$$ in Mathematica, where $\epsilon$ is the Levi-Civita symbol and $a$, $b$, $c$ ...
7
votes
1answer
198 views

Symbolic tensor simplifications and the identity matrix

How can I get Mathematica to simplify the following expressions, with $Assumptions including Element[n,Integers], n > 0, and ...
1
vote
1answer
149 views

How to apply a tensor to a list of arguments

The problem I have is the following: Let C be a list of coordinates, say, C = {x1, x2, ..., xn} and ...
3
votes
1answer
343 views

Covariant derivative for symbolic tensors

I want to define a "prefix" (D_i) covariant derivative operator CD[] for symbolic tensors in form of a function, i.e. for ...
3
votes
1answer
328 views

Linearized Einstein Equations with Mathematica

I need to compute the linearised Einstein Equations around a fixed metric $g_{\mu \nu}$ which is not the flat metric. Someone knows some Mathematica package or some review that can help me?
4
votes
1answer
268 views

Mathematica package for supergravity and superstring theory

I am looking for a Mathematica package that can manipulate tensors for supergravity, string theory or M-theory. I am particularly looking for a package that can do spinor and Clifford algebra ...
7
votes
0answers
117 views

Can I use symbolic tensors for simple linear algebra and calculus?

As an example, I'd like to calculate this symbolic derivative: $\frac{\partial}{\partial b}(x-A.b)^{\mathsf{T}}.(x-A.b)$ where $x$ and $b$ are vectors and $A$ is a matrix. What I tried is this: ...
0
votes
0answers
51 views

Tensor product involving Levi Civita symbol [duplicate]

I'm working with xAct package. I've defined the tensor component values and my issue is that I want an expression involving Levi Civita symbols and tensor components, I don't know how to deal with it. ...
0
votes
0answers
50 views

Get rid of external library dependency

I am trying to generate C code for some functions, which I do not post in full because they are a few hundred lines long. The functions do nothing too fancy: a bunch of dot products, powers and roots. ...
6
votes
2answers
240 views

How to get rid of nested matrices

If I type into Mathematica TensorProduct[IdentityMatrix[2],IdentityMatrix[2]] It gives me a result that has nested matrices. How do I turn that into a normal ...