Edit detail for SandBoxDifferentialGeometry revision 3 of 10

Subject: DeRhamComplex?

On 7 October 2014 12:13 Kurt Pagani wrote to fricas-devel@googlegroups.com:

If we were to create a new domain that allows metric to be part of the definition, what should it be called?

I have no idea. You tell me? I can't yet overview how the final module(s) should look like. What I have in mind is first to complete this by functions like proj (projection on homogeneous parts, subspaces), push forward/pull back, interior product i_X and Lie derivative L_X which is possible which little effort, second to have simplicial homology (e.g. a type Simplex) then using FreeAbelianGroup?(Simplex), defining boundaryOperator and so on such that one can deal with integration of differential forms over simplicial complexes (Stokes theorem, Hodge pairing and so on you know). It's almost all there but one has to organize it. It should be a teamwork to becoming really useful, so any suggestions welcome.

)abbrev domain DIFGEOM DifferentialGeometry

DifferentialGeometry(CoefRing, listIndVar : List Symbol, g:SMR) : Export == Implement where
  CoefRing :  Join(IntegralDomain, Comparable)
  ASY     ==> AntiSymm(R, listIndVar)
  DERHAM ==> DeRhamComplex(CoefRing, listIndVar)
  DIFRING ==> DifferentialRing
  LALG    ==> LeftAlgebra
  FMR     ==> FreeMod(R, EAB)
  I       ==> Integer
  L       ==> List
  EAB     ==> ExtAlgBasis  -- these are exponents of basis elements in order
  NNI     ==> NonNegativeInteger
  O       ==> OutputForm
  R       ==> Expression(CoefRing)
  SMR     ==> SquareMatrix(#listIndVar,R)   -- 

  Export == Join(LALG(R), RetractableTo(R)) with
      leadingCoefficient : %           -> R
        ++ leadingCoefficient(df) returns the leading
        ++ coefficient of differential form df.
      leadingBasisTerm   : %           -> %
        ++ leadingBasisTerm(df) returns the leading
        ++ basis term of differential form df.
      reductum           : %           -> %
        ++ reductum(df), where df is a differential form,
        ++ returns df minus the leading
        ++ term of df if df has two or more terms, and
        ++ 0 otherwise.
      coefficient        : (%, %)     -> R
        ++ coefficient(df, u), where df is a differential form,
        ++ returns the coefficient of df containing the basis term u
        ++ if such a term exists, and 0 otherwise.
      generator          : NNI         -> %
        ++ generator(n) returns the nth basis term for a differential form.
      homogeneous?       : %           -> Boolean
        ++  homogeneous?(df) tests if all of the terms of
        ++  differential form df have the same degree.
      retractable?       : %           -> Boolean
        ++  retractable?(df) tests if differential form df is a 0-form,
        ++  i.e., if degree(df) = 0.
      degree             : %           -> NNI
        ++  changed from I to NNI , then works
        ++  degree(df) returns the homogeneous degree of differential form df.
      map                : (R -> R, %) -> %
        ++  map(f, df) replaces each coefficient x of differential
        ++  form df by \spad{f(x)}.
      totalDifferential    : R -> %
        ++  totalDifferential(x) returns the total differential
        ++  (gradient) form for element x.
      exteriorDifferential : % -> %
        ++  exteriorDifferential(df) returns the exterior
        ++  derivative (gradient, curl, divergence, ...) of
        ++  the differential form df.

--=-=-=-=-=-==-=-==-=-==-=-==-=-==-=-==-=-==-=-==-=-==-=-==-=-==-=-==-=-==-=-=       
      dim : % -> NNI
        ++ dimension of underlying space
        ++ that is dim ExtAlg = 2^dim

      hodgeStar : (%) -> %
        ++ computes the Hodge dual of the differential form % with respect
        ++ to a metric g.

      dot : (%,%) -> R
        ++ computes the inner product of two differential forms w.r.t. g

--=-=-=-=-=-==-=-==-=-==-=-==-=-==-=-==-=-==-=-==-=-==-=-==-=-==-=-==-=-==-=-=

  Implement == DERHAM add
      Rep := DERHAM

      dim := #listIndVar

      terms : % -> List Record(k : EAB, c : R)
      terms(a) ==
        -- it is the case that there are at least two terms in a
        a pretend List Record(k : EAB, c : R)

      -- error messages
      err2:="Not implemented"
      err3:="Degenerate metric"

      -- coord space dimension 
      dim(f) == dim

      -- flip 0->1, 1->0 
      flip(b:ExtAlgBasis):ExtAlgBasis ==
        bl := b pretend List(NNI)
        [(i+1) rem 2 for i in bl] pretend ExtAlgBasis

      -- list the positions of a's (a=0,1) in x
      pos(x:EAB, a:NNI):List(NNI) ==
        y:= x pretend List(NNI) 
        [j for j in 1..#y | y.j=a]

      -- compute factors for hodgeStar  
      facs(cc:Record(k : EAB, c : R)):R ==
        not diagonal? g => error(err2)
        G := reduce("*", [g(j,j) for j in 1..dim]::List(R)) 
        G = 0 => error(err3)
        --
        idx := pos(cc.k,0) -- pos of 0 since already flipped
        eps := concat(pos(cc.k,1),pos(cc.k,0))::List(NNI)
        dom := [j for j in 1..dim]::List(NNI)
        sgn := sign(coercePreimagesImages([dom,eps])::Permutation(NNI))::R
        if idx ~= [] then
          fg:R:= sgn*reduce("*",[1/g(j,j) for j in idx]::List(R))
        else
          fg:R:=sgn
        fg * sqrt(abs(G)) * cc.c

      -- export
      hodgeStar(x) ==
        t := terms(x)
        s := [copy(r) for r in t] -- we need a copy of x!
        for j in 1..#t repeat
          s.j.k := flip(s.j.k)
          s.j.c := facs(s.j)  -- builtin g
        s pretend %

      -- compute dot of singletons
      dot1(r:Record(k : EAB, c : R),s:Record(k : EAB, c : R)):R ==
        test(r.k ~= s.k) => 0::R
        idx := pos(r.k,1)
        reduce("*",[1/g(j,j) for j in idx]::List(R)) * r.c * s.c

      -- export
      dot(x,y) ==
        tx := terms(x)
        ty := terms(y)
        reduce("+",[dot1(tx.j,ty.j) for j in 1..#tx])

   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/7409824215686792721-25px001.spad
      using old system compiler.
   DIFGEOM abbreviates domain DifferentialGeometry 
------------------------------------------------------------------------
   initializing NRLIB DIFGEOM for DifferentialGeometry 
   compiling into NRLIB DIFGEOM 
****** Domain: CoefRing already in scope
   compiling local terms : $ -> List Record(k: ExtAlgBasis,c: Expression CoefRing)
      DIFGEOM;terms is replaced by a 
Time: 0.05 SEC.

   compiling exported dim : $ -> NonNegativeInteger
Time: 0 SEC.

   compiling local flip : ExtAlgBasis -> ExtAlgBasis
Time: 0.02 SEC.

   compiling local pos : (ExtAlgBasis,NonNegativeInteger) -> List NonNegativeInteger
Time: 0.03 SEC.

   compiling local facs : Record(k: ExtAlgBasis,c: Expression CoefRing) -> Expression CoefRing
Time: 0.11 SEC.

   compiling exported hodgeStar : $ -> $
Time: 0.01 SEC.

   compiling local dot1 : (Record(k: ExtAlgBasis,c: Expression CoefRing),Record(k: ExtAlgBasis,c: Expression CoefRing)) -> Expression CoefRing
Time: 0.62 SEC.

   compiling exported dot : ($,$) -> Expression CoefRing
Time: 0 SEC.

(time taken in buildFunctor:  0)

;;;     ***       |DifferentialGeometry| REDEFINED

;;;     ***       |DifferentialGeometry| REDEFINED
Time: 0.01 SEC.

   Warnings: 
      [1] facs:  k has no value
      [2] facs:  c has no value
      [3] dot1:  k has no value
      [4] dot1:  c has no value

   Cumulative Statistics for Constructor DifferentialGeometry
      Time: 0.85 seconds

--------------non extending category----------------------
.. DifferentialGeometry(#1,#2,#3) of cat 
(|Join| (|LeftAlgebra| (|Expression| |#1|))
        (|RetractableTo| (|Expression| |#1|))
        (CATEGORY |domain|
         (SIGNATURE |leadingCoefficient| ((|Expression| |#1|) $))
         (SIGNATURE |leadingBasisTerm| ($ $)) (SIGNATURE |reductum| ($ $))
         (SIGNATURE |coefficient| ((|Expression| |#1|) $ $))
         (SIGNATURE |generator| ($ (|NonNegativeInteger|)))
         (SIGNATURE |homogeneous?| ((|Boolean|) $))
         (SIGNATURE |retractable?| ((|Boolean|) $))
         (SIGNATURE |degree| ((|NonNegativeInteger|) $))
         (SIGNATURE |map|
          ($ (|Mapping| (|Expression| |#1|) (|Expression| |#1|)) $))
         (SIGNATURE |totalDifferential| ($ (|Expression| |#1|)))
         (SIGNATURE |exteriorDifferential| ($ $))
         (SIGNATURE |dim| ((|NonNegativeInteger|) $))
         (SIGNATURE |hodgeStar| ($ $))
         (SIGNATURE |dot| ((|Expression| |#1|) $ $))))    has no  degree : % -> Integer 
   finalizing NRLIB DIFGEOM 
   Processing DifferentialGeometry for Browser database:
--->-->DifferentialGeometry(constructor): Not documented!!!!
--------(leadingCoefficient ((Expression CoefRing) %))---------
--------(leadingBasisTerm (% %))---------
--------(reductum (% %))---------
--------(coefficient ((Expression CoefRing) % %))---------
--------(generator (% (NonNegativeInteger)))---------
--------(homogeneous? ((Boolean) %))---------
--------(retractable? ((Boolean) %))---------
--------(degree ((NonNegativeInteger) %))---------
--->-->DifferentialGeometry((degree ((NonNegativeInteger) %))): Improper first word in comments: changed
"changed from \\spad{I} to NNI ,{} then works degree(\\spad{df}) returns the homogeneous degree of differential form \\spad{df}."
--------(map (% (Mapping (Expression CoefRing) (Expression CoefRing)) %))---------
--------(totalDifferential (% (Expression CoefRing)))---------
--------(exteriorDifferential (% %))---------
--------(dim ((NonNegativeInteger) %))---------
--->-->DifferentialGeometry((dim ((NonNegativeInteger) %))): Improper first word in comments: dimension
"dimension of underlying space that is dim ExtAlg = 2^dim"
--------(hodgeStar (% %))---------
--->-->DifferentialGeometry((hodgeStar (% %))): Improper first word in comments: computes
"computes the Hodge dual of the differential form \\% with respect to a metric \\spad{g}."
--------(dot ((Expression CoefRing) % %))---------
--->-->DifferentialGeometry((dot ((Expression CoefRing) % %))): Improper first word in comments: computes
"computes the inner product of two differential forms \\spad{w}.\\spad{r}.\\spad{t}. \\spad{g}"
--->-->DifferentialGeometry(): Missing Description
; compiling file "/var/aw/var/LatexWiki/DIFGEOM.NRLIB/DIFGEOM.lsp" (written 08 OCT 2014 01:30:48 AM):

; /var/aw/var/LatexWiki/DIFGEOM.NRLIB/DIFGEOM.fasl written
; compilation finished in 0:00:00.074
------------------------------------------------------------------------
   DifferentialGeometry is now explicitly exposed in frame initial 
   DifferentialGeometry will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/DIFGEOM.NRLIB/DIFGEOM

)set output tex off
 

)set output algebra on
 

)set output abbrev on

G:=diagonalMatrix([1,1,1])

        +1  0  0+
        |       |
   (1)  |0  1  0|
        |       |
        +0  0  1+

X := DIFGEOM(Integer,[x,y,z],G)

   (2)
  DIFGEOM(INT,[x,y,z],#2A((((0 . 1) . #1=(0 . 1)) #2=((0 . 0) . #1#) #2#)
    (
  #2# ((0 . 1) . #1#) #2#)
    (#2# #2# ((0 . 1) . #1#))))

[dx,dy,dz] := [generator(i)$X for i in 1..3]

   (3)  [dx,dy,dz]

(#2# ((0 . 1) . #1#) #2#)
    (#2# #2# ((0 . 1) . #1#)))))
f : BOP := operator('f);

g : BOP := operator('g);

h : BOP := operator('h);

sigma := f(x,y,z) * dx + g(x,y,z) * dy + h(x,y,z) * dz

   (7)  h(x,y,z)dz + g(x,y,z)dy + f(x,y,z)dx

(#2# ((0 . 1) . #1#) #2#)
    (#2# #2# ((0 . 1) . #1#))))
dot(sigma,sigma)

                2           2           2
   (8)  h(x,y,z)  + g(x,y,z)  + f(x,y,z)

hodgeStar(sigma)

   (9)  h(x,y,z)dx dy - g(x,y,z)dx dz + f(x,y,z)dy dz

(#2# ((0 . 1) . #1#) #2#)
    (#2# #2# ((0 . 1) . #1#))))