Skip to main content

tensors - Solving antisymmetric tensorial equation


Assume we have the following Tensor objects: \begin{equation} F_{i}{}^{j}\;and\;S_{ij}{}^{k}, \end{equation} where the components of $F$ are known, and we would like to solve for the components of $S$ if they satisfy the following equation \begin{equation} F^{l}{}_{i}S_{jl}{}^{k}-F^{l}{}_{j}S_{il}{}^{k}=0. \end{equation} $l$ is summed over, all the indices run from 1 to 4, and $S$ is symmetric in the lower indices.


Can you please help in writing a Mathematica code for this.


My attempt:



First, suppose we know all the components of $F$, and they are given by \begin{equation} F= \begin{matrix} a & b & c & d\\ e & f & g & h \\ i & j & k & l \\ m & n & o & p \end{matrix} \end{equation}


Then I defined the components of $S$ by:


S[i_, j_, k_] := S[i, j, k]

The first term of the equation I defined it as:


SF[i_, j_, k_] := SF[i, j, k] = S[1, i, j].F[k, 1] +
                                V[2, i, j].F[k, 2] + 
                                V[3, i, j].F[k, 3] + 
                                V[4, i, j].F[k, 4];


As for the second term in the equation, I think it can be found using Transpose


FS[i, j, k] = Transpose[SF, {i, k}]

Then for example:


Solve[SF==FS,{S[i,j,k]},{i,4},{j,4},{k,4}]

is not working. I'm sure there is something wrong in my commands, but I can't figure out what it is. The functions $a$,$b$,$c$,... in the expression of $F$ are some complicated scalar functions of space coordinates.



Answer



Perhaps


f = RandomInteger[{-1, 1}, {4, 4}]; 

Solve[
 And @@ Join[
   Thread[Equal[Flatten[Table[
       Sum[f[[l, i]] s[j, l, k] - f[[l, j]] s[i, l, k], {l, 4}], 
   {i, 4}, {j, 4}, {k, 4}], 2], 0]], 
   Flatten@Table[s[i, j, k] == s[j, i, k], {i, 4}, {j, 4}, {k, 4}]]]

Comments

Post a Comment

Popular posts from this blog

functions - Get leading series expansion term?

Given a function f[x] , I would like to have a function leadingSeries that returns just the leading term in the series around x=0 . For example: leadingSeries[(1/x + 2)/(4 + 1/x^2 + x)] x and leadingSeries[(1/x + 2 + (1 - 1/x^3)/4)/(4 + x)] -(1/(16 x^3)) Is there such a function in Mathematica? Or maybe one can implement it efficiently? EDIT I finally went with the following implementation, based on Carl Woll 's answer: lds[ex_,x_]:=( (ex/.x->(x+O[x]^2))/.SeriesData[U_,Z_,L_List,Mi_,Ma_,De_]:>SeriesData[U,Z,{L[[1]]},Mi,Mi+1,De]//Quiet//Normal) The advantage is, that this one also properly works with functions whose leading term is a constant: lds[Exp[x],x] 1 Answer Update 1 Updated to eliminate SeriesData and to not return additional terms Perhaps you could use: leadingSeries[expr_, x_] := Normal[expr /. x->(x+O[x]^2) /. a_List :> Take[a, 1]] Then for your examples: leadingSeries[(1/x + 2)/(4 + 1/x^2 + x), x] leadingSeries[Exp[x], x] leadingSeries[(1/x + 2 + (1 - 1/x...
Post a Comment
Read more

How to thread a list

I have data in format data = {{a1, a2}, {b1, b2}, {c1, c2}, {d1, d2}} Tableform: I want to thread it to : tdata = {{{a1, b1}, {a2, b2}}, {{a1, c1}, {a2, c2}}, {{a1, d1}, {a2, d2}}} Tableform: And I would like to do better then pseudofunction[n_] := Transpose[{data2[[1]], data2[[n]]}]; SetAttributes[pseudofunction, Listable]; Range[2, 4] // pseudofunction Here is my benchmark data, where data3 is normal sample of real data. data3 = Drop[ExcelWorkBook[[Column1 ;; Column4]], None, 1]; data2 = {a #, b #, c #, d #} & /@ Range[1, 10^5]; data = RandomReal[{0, 1}, {10^6, 4}]; Here is my benchmark code kptnw[list_] := Transpose[{Table[First@#, {Length@# - 1}], Rest@#}, {3, 1, 2}] &@list kptnw2[list_] := Transpose[{ConstantArray[First@#, Length@# - 1], Rest@#}, {3, 1, 2}] &@list OleksandrR[list_] := Flatten[Outer[List, List@First[list], Rest[list], 1], {{2}, {1, 4}}] paradox2[list_] := Partition[Riffle[list[[1]], #], 2] & /@ Drop[list, 1] RM[list_] := FoldList[Transpose[{First@li...
Post a Comment
Read more

front end - keyboard shortcut to invoke Insert new matrix

I frequently need to type in some matrices, and the menu command Insert > Table/Matrix > New... allows matrices with lines drawn between columns and rows, which is very helpful. I would like to make a keyboard shortcut for it, but cannot find the relevant frontend token command (4209405) for it. Since the FullForm[] and InputForm[] of matrices with lines drawn between rows and columns is the same as those without lines, it's hard to do this via 3rd party system-wide text expanders (e.g. autohotkey or atext on mac). How does one assign a keyboard shortcut for the menu item Insert > Table/Matrix > New... , preferably using only mathematica? Thanks! Answer In the MenuSetup.tr (for linux located in the $InstallationDirectory/SystemFiles/FrontEnd/TextResources/X/ directory), I changed the line MenuItem["&New...", "CreateGridBoxDialog"] to read MenuItem["&New...", "CreateGridBoxDialog", MenuKey["m", Modifiers-...
Post a Comment
Read more