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SandBoxDFORM
  
last edited 6 years ago by pagani

fricas
(1) -> <spad>
fricas
)abbrev package DFORM DifferentialForms
++ Author: Kurt Pagani <nilqed@gmail.com>
++ Date Created: October 2014 
++ Revised: Sat Sep 17 17:10:00 CET 2016
++ License: FriCAS/BSD
++ Description: Basic differential form methods
++ Requirements: DeRhamComplex, FriCAS 1.3 or later.  
++ Documentation: Sphinx, folder doc and/or do 
++ - [pdf]latex dform.spad -> dvi or pdf file.
++ - run two times to get refs and labels correct.
++
DifferentialForms(R,v) : Exports == Implementation where

  R: Join(IntegralDomain,Comparable)
  v: List Symbol

  X   ==> Expression R
  OF  ==> OutputForm
  DRC ==> DeRhamComplex(R,v)
  CHC ==> ChainComplex(R,#v)
  BOP ==> BasicOperator
  NNI ==> NonNegativeInteger
  SMR ==> SquareMatrix(#v,X)
  EAB ==> ExtAlgBasis
  REA ==> Record(k : EAB, c : X)
  SGCF ==> SymmetricGroupCombinatoricFunctions

  Exports ==  with

    _* : (List X, List DRC) -> DRC
      ++ v*w computes the sum of the products v_i * w_i
      ++ where v_i is a scalar and w_i a differential form.
      ++ This is for convenience only, just to deal with
      ++ differential form valued vectors.
    _* : (List DRC, List DRC) -> DRC
      ++ w1*w2 computes the sum of the exterior products
      ++ w1_i * w2_i, where w1,w2 are differential forms.
    d : DRC -> DRC
      ++ d w computes the exterior derivative and is just an
      ++ abbreviation for the fucntion "exteriorDifferential"
      ++ defined in the domain "DeRhamComplex".
    one :  -> DRC
      ++ one() gives 1@DeRhamComplex, i.e. "1" as a differential
      ++ form. This is useful to intern elements of the function
      ++ ring (just multiply them by one()$DFORM).
    zero : -> DRC
      ++ zero() gives the zero form, i.e. 0@DRC.
    baseForms : () -> List DRC
      ++ baseForms()$M returns a list of all base forms in the
      ++ space M=DFORM(Ring,Coordinates).
    coordVector : () -> List X
      ++ coordVector()$M returns a list of the coordinates in
      ++ the space M=DFORM(Ring,Coordinates).
    coordSymbols : () -> List Symbol
      ++ coordSymbols()$M returns a list of the coordinates as
      ++ symbols. This is useful for example if the differential
      ++ operators "D" are going to be used.
    vectorField : Symbol -> List X
      ++ vectorField(V) creates a vector (actually a list) whose
      ++ components are given by V[j](x[1],...,x[n]), j=1..n, 
      ++ whereby "x" are the space coordinates (possibly not the
      ++ same symbol).
    scalarField : Symbol -> X
      ++ scalarField(s) creates a scalar function s(x[1],...,[n]),
      ++ whereby "x" are the space coordinates (possibly not the
      ++ same symbol).      
    covectorField : Symbol -> List DRC
      ++ covectorField(Y) creates a covector (actually a list)
      ++ whose components are given by w[j](x[1],...,x[n]), 
      ++ j=1..n.
    zeroForm : Symbol -> DRC
      ++ zeroForm(s) creates a zero form with symbol "s". This
      ++ is the same as scalarField(s)*one().
    volumeForm : SMR -> DRC
      ++ volumeForm(g) returns the volume form with respect to 
      ++ the (pseudo-) metric "g".
    conjBasisTerm : DRC -> DRC
      ++ Return the complement of a basis term w.r.t. volumeForm
    atomizeBasisTerm : DRC -> List DRC 
      ++ Given a basis term, e.g. dx*dy*du, atomizeBasisTerm  returns
      ++ a list of the generators (atoms), i.e. [dx,dy,du] for our example.
    monomials : NNI -> List DRC
      ++ List of all monomials of degree p (p in 1..n).
      ++ This is a basis for Lambda(n,p).
    hodgeStar : (SMR,DRC) -> DRC
      ++ computes the Hodge dual of the differential form % with respect
      ++ to a metric g.      
    dot : (SMR,DRC,DRC) -> X
      ++ computes the inner product of two differential forms w.r.t. g  
    proj : (NNI,DRC) -> DRC
      ++ projection to homogeneous terms of degree p 
    interiorProduct : (Vector(X),DRC) -> DRC
      ++ Calculates the interior product i_X(a) of the vector field X
      ++ with the differential form a.       
    lieDerivative : (Vector(X),DRC) -> DRC
      ++ Calculates the Lie derivative L_X(a) of the differential 
      ++ form a with respect to the vector field X.
    s : SMR -> X
      ++ s(g) determines the sign of determinant(g) and is related to the
      ++ signature of g (n=p+q,t=p-q,s=(-)^(n-t)/2 => s=(-)^q. 
    invHodgeStar : (SMR,DRC) -> DRC
      ++ invHodgeStar is the inverse of hodgeStar.
    codifferential : (SMR,DRC) -> DRC
      ++ codifferential(g,x), also known as "delta", computes the 
      ++ co-differential of a form.
    hodgeLaplacian : (SMR,DRC) -> DRC
      ++ hodgeLaplacian(g,x) also known as "Laplace-de Rham operator" is 
      ++ defined on any manifold equipped with a (pseudo-) Riemannian
      ++ metric and is given by d codifferential(g,x)+ codifferential(g, d x).
      ++ Note that in the Euclidean case hodgeLaplacian = - Laplacian.

  Implementation ==  add 

    -- error messages           
    err1:="Metric tensor must be symmetric"
    err2:="Degenerate metric"
    err3:="Index out of range" 
    err4:="Not implemented"
    err5:="Basis term expected"
    err6:="Zero is not a basis term"

    -- conversion functions (not very nice ;)
    eab2li(x:EAB):List(Integer) == x pretend List(Integer)
    drc2rea(x:DRC):List(REA) == x pretend List(REA)

    n:NNI:=#v
    d(f) == exteriorDifferential(f)
    one () == 1@DRC
    zero() == 0@DRC
    baseForms() == [generator(j)$DRC for j in 1..n]
    coordVector() == [s::X for s in v]
    coordSymbols() == v

    vectorField(s:Symbol):List X ==
      ls:=[subscript(s,[j::OF]) for j in 1..n]
      op:=[operator t for t in ls]
      x:=[a::X for a in v]
      [f x for f in op]

    scalarField(s:Symbol):X ==
      f:=operator s
      f [a::X for a in v]

    covectorField(s:Symbol):List DRC ==
      vf:=vectorField(s)
      [a*1@DRC for a in vf]

    zeroForm(s:Symbol):DRC == scalarField(s)*1@DRC

    (l:List(X) * x:List DRC):DRC == 
      l:List DRC:=[l.j * x.j for j in 1..n]
      reduce(_+,l)

    (x:List DRC * y:List DRC):DRC == 
      l:List DRC:=[x.j * y.j for j in 1..n]
      reduce(_+,l)

    volumeForm(g:SMR):DRC == 
      sqrt(abs(determinant(g)))*reduce(_*,baseForms())

    monomials(p:NNI):List DRC ==
      bf:=baseForms()
      p=0 => [1$DRC]
      p=1 => bf
      np:=[reverse subSet(n,p,i)$SGCF for i in 0..binomial(n,p)-1]
      [reduce(_*,[bf.(1+s.j) for j in 1..p]) for s in np]

    -- flip 0->1, 1->0 
    flip(b:EAB):EAB ==
      --bl := convert(b)$EAB
      bl:=eab2li(b)
      [(i+1) rem 2 for i in bl]::EAB

    -- list the positions of a's (a=0,1) in x
    pos(x:EAB, a:Integer):List(Integer) ==
      y:List(Integer):=eab2li(x) 
      [j for j in 1..#y | y.j=a]

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

    -- compute dot of singleton terms, general symmetric g
    dot2(g:SMR,r:REA, s:REA):X ==
      pr := pos(r.k,1) -- list positions of 1 in r
      ps := pos(s.k,1) -- list positions of 1 in s
      test(#pr ~= #ps) => 0::X  -- not same degree => 0
      pr = []  => r.c * s.c -- empty pr,ps => product of coefs
      G := inverse(g)::SMR -- compute the inverse of the metric g
      test(#pr = 1) => G(pr.1,ps.1)::X * r.c * s.c -- only one element
      M:Matrix(X) -- the minor
      M := matrix([[G(pr.i,ps.j)::X for j in 1..#ps] for i in 1..#pr])
      determinant(M)::X * r.c * s.c

    -- export
    dot(g:SMR,x:DRC,y:DRC):X ==
      not symmetric? g => error(err1)
      tx:List REA := drc2rea(x)
      ty:List REA := drc2rea(y)
      tx = [] or ty = [] => 0::X
      if diagonal? g then -- better performance 
        a:List(X):=concat[[dot1(g,tx.j,ty.l)::X for j in 1..#tx] for l in 1..#ty]
        reduce(_+,a)
      else
        b:List(X):=concat[[dot2(g,tx.j,ty.l)::X for j in 1..#tx] for l in 1..#ty]
        reduce(_+,b)

    proj(p,x) ==
      x=0 => x
      homogeneous? x and degree(x)=p => x
      a:=leadingBasisTerm(x)
      if degree(a)=p then
        leadingCoefficient(x)*a + proj(p, reductum x)
      else
        proj(p, reductum x)

    conjBasisTerm(x:DRC):DRC ==
      x=0$DRC => error(err6)
      x ~= leadingBasisTerm(x) => error(err5)
      t:EAB:=drc2rea(x).1.k
      l:List(Integer):=exponents(t)
      m:List(DRC):=[generator(i)$DRC for i in 1..#l | l.i=0]
      m=[] => 1$DRC
      reduce(_*,m)

    atomizeBasisTerm(x:DRC):List(DRC) ==
      x=0$DRC => error(err6)
      x ~= leadingBasisTerm(x) => error(err5)
      t:EAB:=drc2rea(x).1.k
      l:List(Integer):=exponents(t)
      [generator(i)$DRC for i in 1..#l | l.i=1] 

    intProdBasisTerm(w:Vector X, x:DRC):DRC ==
      x ~= leadingBasisTerm(x) => error(err5)
      degree(x)=0 => 0$DRC
      degree(x)=1 => w.position(x,baseForms()) * 1$DRC
      a:List(DRC):=atomizeBasisTerm(x)
      b:DRC:=reduce(_*,rest a)
      -- i_w is an antiderivative => 
      intProdBasisTerm(w,a.1)*b - a.1 * intProdBasisTerm(w,b)

    interiorProduct(w:Vector X, x:DRC):DRC ==
      x=0$DRC => x
      leadingCoefficient(x)*intProdBasisTerm(w,leadingBasisTerm(x)) + _
        interiorProduct(w, reductum(x))

    lieDerivative(w:Vector X,x:DRC):DRC ==
      a := exteriorDifferential(interiorProduct(w,x))
      b := interiorProduct(w, exteriorDifferential(x))
      a+b

    eps(x:DRC):X == leadingCoefficient(x*conjBasisTerm(x))

    hodgeStarBT(g:SMR,x:DRC):DRC ==
      q:=sqrt(abs(determinant(g)))
      p:=degree(x)
      J:=monomials(p)
      s:=[eps(y)*dot(g,y,x)*conjBasisTerm(y) for y in J]
      q*reduce(_+,s)

    hodgeStar(g:SMR,x:DRC):DRC ==
      x=0$DRC => x
      leadingCoefficient(x)*hodgeStarBT(g,leadingBasisTerm(x)) + _
        hodgeStar(g, reductum(x))

    s(g:SMR):X ==
      det:X:=determinant g
      sd:Union(Integer,"failed"):=sign(det)$ElementaryFunctionSign(Integer,X)
      sd case "failed" => 's?::X
      sd case Integer => coerce(sd)$X

    invHodgeStar(g:SMR,x:DRC):DRC ==
      x=0$DRC => x
      y:DRC:=leadingBasisTerm(x)
      k:X:=coerce(degree y)$X
      c:X:=s(g)*(-1)^(k*(coerce(n)$X-k))
      leadingCoefficient(x)*hodgeStarBT(g,c*y) + _
        invHodgeStar(g, reductum(x))

    codifferential(g:SMR,x:DRC):DRC ==
      x=0$DRC => 0
      y:DRC:=leadingBasisTerm(x)
      k:X:=coerce(degree y)$X
      c:X:=s(g)*(-1)^(coerce(n)$X*(k-1)+1)
      c*hodgeStar(g,d hodgeStar(g,leadingCoefficient(x)*y)) + _
        codifferential(g,reductum(x)) 

    hodgeLaplacian(g:SMR,x:DRC):DRC ==
      d codifferential(g,x)+ codifferential(g, d x)</spad>
fricas
Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/7780364366861049302-25px001.spad
      using old system compiler.
   DFORM abbreviates package DifferentialForms 
------------------------------------------------------------------------
   initializing NRLIB DFORM for DifferentialForms 
   compiling into NRLIB DFORM 
****** Domain: R already in scope
   compiling local eab2li : ExtAlgBasis -> List Integer
      DFORM;eab2li is replaced by x 
Time: 0.01 SEC.

   compiling local drc2rea : DeRhamComplex(R,v) -> List Record(k: ExtAlgBasis,c: Expression R)
      DFORM;drc2rea is replaced by x 
Time: 0 SEC.

   compiling exported d : DeRhamComplex(R,v) -> DeRhamComplex(R,v)
Time: 0 SEC.

   compiling exported one : () -> DeRhamComplex(R,v)
Time: 0 SEC.

   compiling exported zero : () -> DeRhamComplex(R,v)
Time: 0 SEC.

   compiling exported baseForms : () -> List DeRhamComplex(R,v)
Time: 0 SEC.

   compiling exported coordVector : () -> List Expression R
Time: 0 SEC.

   compiling exported coordSymbols : () -> List Symbol
Time: 0 SEC.

   compiling exported vectorField : Symbol -> List Expression R
Time: 0.01 SEC.

   compiling exported scalarField : Symbol -> Expression R
Time: 0 SEC.

   compiling exported covectorField : Symbol -> List DeRhamComplex(R,v)
Time: 0 SEC.

   compiling exported zeroForm : Symbol -> DeRhamComplex(R,v)
Time: 0 SEC.

   compiling exported * : (List Expression R,List DeRhamComplex(R,v)) -> DeRhamComplex(R,v)
Time: 0.01 SEC.

   compiling exported * : (List DeRhamComplex(R,v),List DeRhamComplex(R,v)) -> DeRhamComplex(R,v)
Time: 0.02 SEC.

   compiling exported volumeForm : SquareMatrix(# v,Expression R) -> DeRhamComplex(R,v)
Time: 0.03 SEC.

   compiling exported monomials : NonNegativeInteger -> List DeRhamComplex(R,v)
Time: 0 SEC.

   compiling local flip : ExtAlgBasis -> ExtAlgBasis
Time: 0 SEC.

   compiling local pos : (ExtAlgBasis,Integer) -> List Integer
Time: 0 SEC.

   compiling local dot1 : (SquareMatrix(# v,Expression R),Record(k: ExtAlgBasis,c: Expression R),Record(k: ExtAlgBasis,c: Expression R)) -> Expression R
Time: 0.14 SEC.

   compiling local dot2 : (SquareMatrix(# v,Expression R),Record(k: ExtAlgBasis,c: Expression R),Record(k: ExtAlgBasis,c: Expression R)) -> Expression R
Time: 0.35 SEC.

   compiling exported dot : (SquareMatrix(# v,Expression R),DeRhamComplex(R,v),DeRhamComplex(R,v)) -> Expression R
Time: 0 SEC.

   compiling exported proj : (NonNegativeInteger,DeRhamComplex(R,v)) -> DeRhamComplex(R,v)
Time: 0 SEC.

   compiling exported conjBasisTerm : DeRhamComplex(R,v) -> DeRhamComplex(R,v)
Time: 0.01 SEC.

   compiling exported atomizeBasisTerm : DeRhamComplex(R,v) -> List DeRhamComplex(R,v)
Time: 0 SEC.

   compiling local intProdBasisTerm : (Vector Expression R,DeRhamComplex(R,v)) -> DeRhamComplex(R,v)
Time: 0.03 SEC.

   compiling exported interiorProduct : (Vector Expression R,DeRhamComplex(R,v)) -> DeRhamComplex(R,v)
Time: 0 SEC.

   compiling exported lieDerivative : (Vector Expression R,DeRhamComplex(R,v)) -> DeRhamComplex(R,v)
Time: 0 SEC.

   compiling local eps : DeRhamComplex(R,v) -> Expression R
Time: 0 SEC.

   compiling local hodgeStarBT : (SquareMatrix(# v,Expression R),DeRhamComplex(R,v)) -> DeRhamComplex(R,v)
Time: 0.03 SEC.

   compiling exported hodgeStar : (SquareMatrix(# v,Expression R),DeRhamComplex(R,v)) -> DeRhamComplex(R,v)
Time: 0 SEC.

   compiling exported s : SquareMatrix(# v,Expression R) -> Expression R
Time: 0.02 SEC.

   compiling exported invHodgeStar : (SquareMatrix(# v,Expression R),DeRhamComplex(R,v)) -> DeRhamComplex(R,v)
Time: 0.09 SEC.

   compiling exported codifferential : (SquareMatrix(# v,Expression R),DeRhamComplex(R,v)) -> DeRhamComplex(R,v)
Time: 0.40 SEC.

   compiling exported hodgeLaplacian : (SquareMatrix(# v,Expression R),DeRhamComplex(R,v)) -> DeRhamComplex(R,v)
Time: 0 SEC.

(time taken in buildFunctor:  0)

;;;     ***       |DifferentialForms| REDEFINED

;;;     ***       |DifferentialForms| REDEFINED
Time: 0 SEC.

   Semantic Errors: 
      [1] invHodgeStar:  k is BOTH a variable and a literal
      [2] invHodgeStar:  c is BOTH a variable and a literal
      [3] codifferential:  k is BOTH a variable and a literal
      [4] codifferential:  c is BOTH a variable and a literal

   Warnings: 
      [1] dot1:  k has no value
      [2] dot1:  c has no value
      [3] dot2:  k has no value
      [4] dot2:  c has no value
      [5] conjBasisTerm:  k has no value
      [6] atomizeBasisTerm:  k has no value
      [7] s: not known that (AlgebraicallyClosedField) is of mode (CATEGORY domain (IF (has R (IntegralDomain)) (PROGN (ATTRIBUTE (AlgebraicallyClosedFunctionSpace R)) (ATTRIBUTE (TranscendentalFunctionCategory)) (ATTRIBUTE (CombinatorialOpsCategory)) (ATTRIBUTE (LiouvillianFunctionCategory)) (ATTRIBUTE (SpecialFunctionCategory)) (SIGNATURE reduce (% %)) (SIGNATURE number? ((Boolean) %)) (IF (has R (PolynomialFactorizationExplicit)) (ATTRIBUTE (PolynomialFactorizationExplicit)) noBranch) (SIGNATURE setSimplifyDenomsFlag ((Boolean) (Boolean))) (SIGNATURE getSimplifyDenomsFlag ((Boolean)))) noBranch))
      [8] s: not known that (TranscendentalFunctionCategory) is of mode (CATEGORY domain (IF (has R (IntegralDomain)) (PROGN (ATTRIBUTE (AlgebraicallyClosedFunctionSpace R)) (ATTRIBUTE (TranscendentalFunctionCategory)) (ATTRIBUTE (CombinatorialOpsCategory)) (ATTRIBUTE (LiouvillianFunctionCategory)) (ATTRIBUTE (SpecialFunctionCategory)) (SIGNATURE reduce (% %)) (SIGNATURE number? ((Boolean) %)) (IF (has R (PolynomialFactorizationExplicit)) (ATTRIBUTE (PolynomialFactorizationExplicit)) noBranch) (SIGNATURE setSimplifyDenomsFlag ((Boolean) (Boolean))) (SIGNATURE getSimplifyDenomsFlag ((Boolean)))) noBranch))
      [9] s: not known that (FunctionSpace (Integer)) is of mode (CATEGORY domain (IF (has R (IntegralDomain)) (PROGN (ATTRIBUTE (AlgebraicallyClosedFunctionSpace R)) (ATTRIBUTE (TranscendentalFunctionCategory)) (ATTRIBUTE (CombinatorialOpsCategory)) (ATTRIBUTE (LiouvillianFunctionCategory)) (ATTRIBUTE (SpecialFunctionCategory)) (SIGNATURE reduce (% %)) (SIGNATURE number? ((Boolean) %)) (IF (has R (PolynomialFactorizationExplicit)) (ATTRIBUTE (PolynomialFactorizationExplicit)) noBranch) (SIGNATURE setSimplifyDenomsFlag ((Boolean) (Boolean))) (SIGNATURE getSimplifyDenomsFlag ((Boolean)))) noBranch))

   Cumulative Statistics for Constructor DifferentialForms
      Time: 1.24 seconds

   finalizing NRLIB DFORM 
   Processing DifferentialForms for Browser database:
--------constructor---------
--------(* ((DeRhamComplex R v) (List (Expression R)) (List (DeRhamComplex R v))))---------
--------(* ((DeRhamComplex R v) (List (DeRhamComplex R v)) (List (DeRhamComplex R v))))---------
--------(d ((DeRhamComplex R v) (DeRhamComplex R v)))---------
--------(one ((DeRhamComplex R v)))---------
--------(zero ((DeRhamComplex R v)))---------
--------(baseForms ((List (DeRhamComplex R v))))---------
--------(coordVector ((List (Expression R))))---------
--------(coordSymbols ((List (Symbol))))---------
--------(vectorField ((List (Expression R)) (Symbol)))---------
--------(scalarField ((Expression R) (Symbol)))---------
--------(covectorField ((List (DeRhamComplex R v)) (Symbol)))---------
--------(zeroForm ((DeRhamComplex R v) (Symbol)))---------
--------(volumeForm ((DeRhamComplex R v) (SquareMatrix (# v) (Expression R))))---------
--------(conjBasisTerm ((DeRhamComplex R v) (DeRhamComplex R v)))---------
--->-->DifferentialForms((conjBasisTerm ((DeRhamComplex R v) (DeRhamComplex R v)))): Improper first word in comments: Return
"Return the complement of a basis term \\spad{w}.\\spad{r}.\\spad{t}. volumeForm"
--------(atomizeBasisTerm ((List (DeRhamComplex R v)) (DeRhamComplex R v)))---------
--->-->DifferentialForms((atomizeBasisTerm ((List (DeRhamComplex R v)) (DeRhamComplex R v)))): Improper first word in comments: Given
"Given a basis term,{} \\spadignore{e.g.} dx*dy*du,{} atomizeBasisTerm returns a list of the generators (atoms),{} \\spadignore{i.e.} [\\spad{dx},{}dy,{}du] for our example."
--------(monomials ((List (DeRhamComplex R v)) (NonNegativeInteger)))---------
--->-->DifferentialForms((monomials ((List (DeRhamComplex R v)) (NonNegativeInteger)))): Improper first word in comments: List
"List of all monomials of degree \\spad{p} (\\spad{p} in 1..\\spad{n}). This is a basis for Lambda(\\spad{n},{}\\spad{p})."
--------(hodgeStar ((DeRhamComplex R v) (SquareMatrix (# v) (Expression R)) (DeRhamComplex R v)))---------
--->-->DifferentialForms((hodgeStar ((DeRhamComplex R v) (SquareMatrix (# v) (Expression R)) (DeRhamComplex R v)))): Improper first word in comments: computes
"computes the Hodge dual of the differential form \\% with respect to a metric \\spad{g}."
--------(dot ((Expression R) (SquareMatrix (# v) (Expression R)) (DeRhamComplex R v) (DeRhamComplex R v)))---------
--->-->DifferentialForms((dot ((Expression R) (SquareMatrix (# v) (Expression R)) (DeRhamComplex R v) (DeRhamComplex R v)))): Improper first word in comments: computes
"computes the inner product of two differential forms \\spad{w}.\\spad{r}.\\spad{t}. \\spad{g}"
--------(proj ((DeRhamComplex R v) (NonNegativeInteger) (DeRhamComplex R v)))---------
--->-->DifferentialForms((proj ((DeRhamComplex R v) (NonNegativeInteger) (DeRhamComplex R v)))): Improper first word in comments: projection
"projection to homogeneous terms of degree \\spad{p}"
--------(interiorProduct ((DeRhamComplex R v) (Vector (Expression R)) (DeRhamComplex R v)))---------
--->-->DifferentialForms((interiorProduct ((DeRhamComplex R v) (Vector (Expression R)) (DeRhamComplex R v)))): Improper first word in comments: Calculates
"Calculates the interior product i_X(a) of the vector field \\spad{X} with the differential form a."
--------(lieDerivative ((DeRhamComplex R v) (Vector (Expression R)) (DeRhamComplex R v)))---------
--->-->DifferentialForms((lieDerivative ((DeRhamComplex R v) (Vector (Expression R)) (DeRhamComplex R v)))): Improper first word in comments: Calculates
"Calculates the Lie derivative \\spad{L_X}(a) of the differential form a with respect to the vector field \\spad{X}."
--------(s ((Expression R) (SquareMatrix (# v) (Expression R))))---------
--->-->DifferentialForms((s ((Expression R) (SquareMatrix (# v) (Expression R))))): Missing right pren
"\\spad{s(g)} determines the sign of determinant(\\spad{g}) and is related to the signature of \\spad{g} (n=p+q,{}\\spad{t=p}-\\spad{q},{}\\spad{s=}(-)^(\\spad{n}-\\spad{t})\\spad{/2} \\spad{=>} \\spad{s=}(-)\\spad{^q}."
--------(invHodgeStar ((DeRhamComplex R v) (SquareMatrix (# v) (Expression R)) (DeRhamComplex R v)))---------
--->-->DifferentialForms((invHodgeStar ((DeRhamComplex R v) (SquareMatrix (# v) (Expression R)) (DeRhamComplex R v)))): Improper initial operator in comments: is
"invHodgeStar is the inverse of hodgeStar."
--------(codifferential ((DeRhamComplex R v) (SquareMatrix (# v) (Expression R)) (DeRhamComplex R v)))---------
--------(hodgeLaplacian ((DeRhamComplex R v) (SquareMatrix (# v) (Expression R)) (DeRhamComplex R v)))---------
; compiling file "/var/aw/var/LatexWiki/DFORM.NRLIB/DFORM.lsp" (written 07 OCT 2024 05:50:13 PM):

; wrote /var/aw/var/LatexWiki/DFORM.NRLIB/DFORM.fasl
; compilation finished in 0:00:00.120
------------------------------------------------------------------------
   DifferentialForms is now explicitly exposed in frame initial 
   DifferentialForms will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/DFORM.NRLIB/DFORM

Test flavours

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-- =======================================
DFORM tests (package DifferentialForms) -- ======================================= -- Requires: -- .... DeRhamComplex (src/derham.spad) or FriCAS 1.3 or later. -- .... DifferentialForms (dform.spad). -- Version 1.1: 09-ARP-2016 -- Version 1.2: 17-DEC-2016 -- -- FriCAS Computer Algebra System -- Version: FriCAS 1.3.0 -- Timestamp: Don Sep 29 21:26:01 CEST 2016
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)set break resume
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)expose UnittestCount UnittestAux Unittest

   UnittestCount is now explicitly exposed in frame initial 
   UnittestAux is now explicitly exposed in frame initial 
   Unittest is now explicitly exposed in frame initial
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)expose DFORM

   DifferentialForms is already explicitly exposed in frame initial 

-----------------
testsuite "dform"

   All user variables and function definitions have been cleared.
Type: Void
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testcase  "all"

   All user variables and function definitions have been cleared.
Type: Void
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-----------------

-----------
-- Setup --
-----------

n:=7 -- dim of base space (n>=2 !), may change in course
\label{eq1}7(1)
Type: PositiveInteger?
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N:=91 -- number of tests
\label{eq2}91(2)
Type: PositiveInteger?
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O ==> OutputForm
Type: Void
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-- HodgeStar package for DERHAM(R,v)
R:=Integer  -- Ring
\label{eq3}\hbox{\axiomType{Integer}\ }(3)
Type: Type
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v:=[subscript(x,[j::OutputForm]) for j in 1..n] -- (x_1,..,x_n)
\label{eq4}\left[{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}, \:{x_{5}}, \:{x_{6}}, \:{x_{7}}\right](4)
Type: List(Symbol)
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M:=DFORM(R,v)
\label{eq5}\hbox{\axiomType{DifferentialForms}\ } \left({\hbox{\axiomType{Integer}\ } , \:{\left[{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}, \:{x_{5}}, \:{x_{6}}, \:{x_{7}}\right]}}\right)(5)
Type: Type
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-- basis 1-forms and coordinate vector
dx:=baseForms()$M     -- [dx[1],...,dx[n]]
\label{eq6}\left[{dx_{1}}, \:{dx_{2}}, \:{dx_{3}}, \:{dx_{4}}, \:{dx_{5}}, \:{dx_{6}}, \:{dx_{7}}\right](6)
Type: List(DeRhamComplex?(Integer,[x[1],x[2],x[3],x[4],x[5],x[6],x[7]]))
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x:=coordVector()$M    -- [x[1],...,x[n]]
\label{eq7}\left[{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}, \:{x_{5}}, \:{x_{6}}, \:{x_{7}}\right](7)
Type: List(Expression(Integer))
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xs:=coordSymbols()$M  -- as above but as List Symbol (for differentiate, D)
\label{eq8}\left[{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}, \:{x_{5}}, \:{x_{6}}, \:{x_{7}}\right](8)
Type: List(Symbol)
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-- operator, vector field, scalar field, symbol
a:=operator 'a         -- operator
\label{eq9}a(9)
Type: BasicOperator?
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b:=vectorField(b)$M    -- generic vector field [b1(x1..xn),...,bn(x1..xn)]
\label{eq10}\begin{array}{@{}l} \displaystyle \left[{{b_{1}}\left({{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}, \:{x_{5}}, \:{x_{6}}, \:{x_{7}}}\right)}, \:{{b_{2}}\left({{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}, \:{x_{5}}, \:{x_{6}}, \:{x_{7}}}\right)}, \: \right. \ \ \displaystyle \left.{{b_{3}}\left({{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}, \:{x_{5}}, \:{x_{6}}, \:{x_{7}}}\right)}, \:{{b_{4}}\left({{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}, \:{x_{5}}, \:{x_{6}}, \:{x_{7}}}\right)}, \: \right. \ \ \displaystyle \left.{{b_{5}}\left({{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}, \:{x_{5}}, \:{x_{6}}, \:{x_{7}}}\right)}, \:{{b_{6}}\left({{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}, \:{x_{5}}, \:{x_{6}}, \:{x_{7}}}\right)}, \: \right. \ \ \displaystyle \left.{{b_{7}}\left({{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}, \:{x_{5}}, \:{x_{6}}, \:{x_{7}}}\right)}\right] (10)
Type: List(Expression(Integer))
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c:=vectorField(c)$M
\label{eq11}\begin{array}{@{}l} \displaystyle \left[{{c_{1}}\left({{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}, \:{x_{5}}, \:{x_{6}}, \:{x_{7}}}\right)}, \:{{c_{2}}\left({{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}, \:{x_{5}}, \:{x_{6}}, \:{x_{7}}}\right)}, \: \right. \ \ \displaystyle \left.{{c_{3}}\left({{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}, \:{x_{5}}, \:{x_{6}}, \:{x_{7}}}\right)}, \:{{c_{4}}\left({{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}, \:{x_{5}}, \:{x_{6}}, \:{x_{7}}}\right)}, \: \right. \ \ \displaystyle \left.{{c_{5}}\left({{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}, \:{x_{5}}, \:{x_{6}}, \:{x_{7}}}\right)}, \:{{c_{6}}\left({{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}, \:{x_{5}}, \:{x_{6}}, \:{x_{7}}}\right)}, \: \right. \ \ \displaystyle \left.{{c_{7}}\left({{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}, \:{x_{5}}, \:{x_{6}}, \:{x_{7}}}\right)}\right] (11)
Type: List(Expression(Integer))
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P:=scalarField(P)$M    -- scalar field P(x1,..,xn)
\label{eq12}P \left({{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}, \:{x_{5}}, \:{x_{6}}, \:{x_{7}}}\right)(12)
Type: Expression(Integer)
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-- (pseudo-) random form & zeroForm
rd:=reduce(_*,[dx.j for j in 1..random(n)$NNI+1])
\label{eq13}{dx_{1}}\ {dx_{2}}\ {dx_{3}}\ {dx_{4}}\ {dx_{5}}(13)
Type: DeRhamComplex?(Integer,[x[1],x[2],x[3],x[4],x[5],x[6],x[7]])
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nf:=zeroForm(nf)$M
\label{eq14}nf \left({{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}, \:{x_{5}}, \:{x_{6}}, \:{x_{7}}}\right)(14)
Type: DeRhamComplex?(Integer,[x[1],x[2],x[3],x[4],x[5],x[6],x[7]])
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-- metric
g:=diagonalMatrix([1 for i in 1..n])$SquareMatrix(n,EXPR R)  -- Euclidean
\label{eq15}\left[ \begin{array}{ccccccc} 1 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 1 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 1 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 1 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 1 (15)
Type: SquareMatrix?(7,Expression(Integer))
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h:=diagonalMatrix(c)$SquareMatrix(n,EXPR R)
\label{eq16}\left[ \begin{array}{ccccccc} {{c_{1}}\left({{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}, \:{x_{5}}, \:{x_{6}}, \:{x_{7}}}\right)}& 0 & 0 & 0 & 0 & 0 & 0 \ 0 &{{c_{2}}\left({{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}, \:{x_{5}}, \:{x_{6}}, \:{x_{7}}}\right)}& 0 & 0 & 0 & 0 & 0 \ 0 & 0 &{{c_{3}}\left({{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}, \:{x_{5}}, \:{x_{6}}, \:{x_{7}}}\right)}& 0 & 0 & 0 & 0 \ 0 & 0 & 0 &{{c_{4}}\left({{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}, \:{x_{5}}, \:{x_{6}}, \:{x_{7}}}\right)}& 0 & 0 & 0 \ 0 & 0 & 0 & 0 &{{c_{5}}\left({{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}, \:{x_{5}}, \:{x_{6}}, \:{x_{7}}}\right)}& 0 & 0 \ 0 & 0 & 0 & 0 & 0 &{{c_{6}}\left({{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}, \:{x_{5}}, \:{x_{6}}, \:{x_{7}}}\right)}& 0 \ 0 & 0 & 0 & 0 & 0 & 0 &{{c_{7}}\left({{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}, \:{x_{5}}, \:{x_{6}}, \:{x_{7}}}\right)} (16)
Type: SquareMatrix?(7,Expression(Integer))
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-- vector field
vf:=vector b
\label{eq17}\begin{array}{@{}l} \displaystyle \left[{{b_{1}}\left({{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}, \:{x_{5}}, \:{x_{6}}, \:{x_{7}}}\right)}, \:{{b_{2}}\left({{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}, \:{x_{5}}, \:{x_{6}}, \:{x_{7}}}\right)}, \: \right. \ \ \displaystyle \left.{{b_{3}}\left({{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}, \:{x_{5}}, \:{x_{6}}, \:{x_{7}}}\right)}, \:{{b_{4}}\left({{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}, \:{x_{5}}, \:{x_{6}}, \:{x_{7}}}\right)}, \: \right. \ \ \displaystyle \left.{{b_{5}}\left({{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}, \:{x_{5}}, \:{x_{6}}, \:{x_{7}}}\right)}, \:{{b_{6}}\left({{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}, \:{x_{5}}, \:{x_{6}}, \:{x_{7}}}\right)}, \: \right. \ \ \displaystyle \left.{{b_{7}}\left({{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}, \:{x_{5}}, \:{x_{6}}, \:{x_{7}}}\right)}\right] (17)
Type: Vector(Expression(Integer))
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-- Result list  
res:List(Boolean):=[false for j in 1..N]
\label{eq18}\begin{array}{@{}l} \displaystyle \left[ \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \right. \ \ \displaystyle \left. \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \right. \ \ \displaystyle \left. \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \right. \ \ \displaystyle \left. \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \right. \ \ \displaystyle \left. \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \right. \ \ \displaystyle \left. \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \right. \ \ \displaystyle \left. \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \right. \ \ \displaystyle \left. \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \right. \ \ \displaystyle \left. \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \right. \ \ \displaystyle \left. \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \mbox{\rm false} , \: \right. \ \ \displaystyle \left. \mbox{\rm false} \right] (18)
Type: List(Boolean)
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-- 
res.1 := test (dx.1=d(x.1*one()$M))
\label{eq19} \mbox{\rm true} (19)
Type: Boolean
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res.2 := test (#dx = n)
\label{eq20} \mbox{\rm true} (20)
Type: Boolean
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res.3 := test (#x = n)
\label{eq21} \mbox{\rm true} (21)
Type: Boolean
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res.4 := test (a x = a(x))
\label{eq22} \mbox{\rm true} (22)
Type: Boolean
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res.5 := test (#b = n)
\label{eq23} \mbox{\rm true} (23)
Type: Boolean
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-- null Form (i.e. degree 0)
res.6 := test (zeroForm('a)$M = a(x)*one()$M)
\label{eq24} \mbox{\rm true} (24)
Type: Boolean
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res.7 := test (d zeroForm('a)$M = reduce(_+,[D(a(x),xs.i)*dx.i for i in 1..n]))
\label{eq25} \mbox{\rm true} (25)
Type: Boolean
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-- products (à la Flanders, vector forms)
res.8 := test ( x*dx = reduce(_+,[x.i*dx.i for i in 1..n]))
\label{eq26} \mbox{\rm true} (26)
Type: Boolean
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res.9 := test ( dx * dx = 0)
\label{eq27} \mbox{\rm true} (27)
Type: Boolean
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res.10 := test ( (x*dx)*dx = -dx*(x*dx))
\label{eq28} \mbox{\rm true} (28)
Type: Boolean
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-- (co-)vector field
res.11 := test (vectorField(W)$M * dx = dx * vectorField(W)$M)
\label{eq29} \mbox{\rm true} (29)
Type: Boolean
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res.12 := test (typeOf(vectorField(S)$M)::O=List(Expression(R))::O)
\label{eq30} \mbox{\rm true} (30)
Type: Boolean
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res.13 := test (typeOf(covectorField(K)$M)::O=List(DERHAM(R,v))::O)
\label{eq31} \mbox{\rm true} (31)
Type: Boolean
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res.14 := test (typeOf(vectorField(W)$M * covectorField(K)$M)::O=DERHAM(R,v)::O)
\label{eq32} \mbox{\rm true} (32)
Type: Boolean
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-- dot products w.r.t. g
res.15 := reduce(_and,[test ( dot(g,dx.j,dx.j)$M = 1) for j in 1..n])
\label{eq33} \mbox{\rm true} (33)
Type: Boolean
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res.16 := reduce(_and,[test ( dot(g,dx.j,dx.(j+1))$M = 0) for j in 1..n-1])
\label{eq34} \mbox{\rm true} (34)
Type: Boolean
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res.17 := reduce(_and,[test ( dot(g,dx.1*dx.j,dx.1*dx.j)$M = 1) for j in 2..n])
\label{eq35} \mbox{\rm true} (35)
Type: Boolean
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res.18 := reduce(_and,[test ( dot(h,dx.j,dx.j)$M = 1/c.j) for j in 1..n])
\label{eq36} \mbox{\rm true} (36)
Type: Boolean
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res.19 := reduce(_and,[test ( dot(h,dx.j,dx.(j+1))$M = 0) for j in 1..n-1])
\label{eq37} \mbox{\rm true} (37)
Type: Boolean
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res.20 := reduce(_and,[test ( dot(h,dx.1*dx.j,dx.1*dx.j)$M = 1/(c.1*c.j)) _
                                  for j in 2..n])
\label{eq38} \mbox{\rm true} (38)
Type: Boolean
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-- Hodge star
res.21 := test (one()$M*hodgeStar(g,1)$M = dot(g,1,1)$M * volumeForm(g)$M)
\label{eq39} \mbox{\rm true} (39)
Type: Boolean
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res.22 := test (rd*hodgeStar(g,rd)$M = dot(g,rd,rd)$M * volumeForm(g)$M)
\label{eq40} \mbox{\rm true} (40)
Type: Boolean
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res.23 := test (degree(hodgeStar(g,dx.n)$M) = n-1)
\label{eq41} \mbox{\rm true} (41)
Type: Boolean
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res.24 := test (one()$M*hodgeStar(h,1)$M = dot(h,1,1)$M * volumeForm(h)$M)
\label{eq42} \mbox{\rm true} (42)
Type: Boolean
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res.25 := test (rd*hodgeStar(h,rd)$M = dot(h,rd,rd)$M * volumeForm(h)$M)
\label{eq43} \mbox{\rm true} (43)
Type: Boolean
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res.26 := test (degree(hodgeStar(h,dx.n)$M) = n-1)
\label{eq44} \mbox{\rm true} (44)
Type: Boolean
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-- Projections
res.27 := test (proj(0,nf+b*dx+d(b*dx))$M = nf)
\label{eq45} \mbox{\rm true} (45)
Type: Boolean
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res.28 := test (proj(1,nf+b*dx+d(b*dx))$M = b*dx)
\label{eq46} \mbox{\rm true} (46)
Type: Boolean
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res.29 := test (proj(2,nf+b*dx+d(b*dx))$M = d(b*dx))
\label{eq47} \mbox{\rm true} (47)
Type: Boolean
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res.30 := test (proj(n,volumeForm(g)$M)$M = volumeForm(g)$M)
\label{eq48} \mbox{\rm true} (48)
Type: Boolean
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res.31 := test (proj(random(n)$NNI,volumeForm(g)$M)$M = 0)
\label{eq49} \mbox{\rm true} (49)
Type: Boolean
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-- Interior product
res.32 := test (interiorProduct(vf,dx.1)$M = b.1)
\label{eq50} \mbox{\rm true} (50)
Type: Boolean
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res.33 := test (interiorProduct(vf,dx.n)$M = b.n)
\label{eq51} \mbox{\rm true} (51)
Type: Boolean
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res.34 := test (d interiorProduct(vf, volumeForm(g)$M)$M =  _
                 reduce(_+,[D(b.j,xs.j) for j in 1..n])*volumeForm(g)$M)
\label{eq52} \mbox{\rm true} (52)
Type: Boolean
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-- Lie derivative
res.35 := test ( d interiorProduct(vf,b*dx)$M + _
                interiorProduct(vf,d(b*dx))$M = lieDerivative(vf,b*dx)$M)
\label{eq53} \mbox{\rm true} (53)
Type: Boolean
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-- Version 1.2
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)clear values n R v M x xs dx a b c p g h P vf 

n:=4 -- dim of base space (n>=2 !)
\label{eq54}4(54)
Type: PositiveInteger?
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O ==> OutputForm
Type: Void
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-- HodgeStar package for DERHAM(R,v)
R:=Integer  -- Ring
\label{eq55}\hbox{\axiomType{Integer}\ }(55)
Type: Type
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v:=[subscript(x,[j::OutputForm]) for j in 1..n] -- (x_1,..,x_n)
\label{eq56}\left[{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}\right](56)
Type: List(Symbol)
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M:=DFORM(R,v)
\label{eq57}\hbox{\axiomType{DifferentialForms}\ } \left({\hbox{\axiomType{Integer}\ } , \:{\left[{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}\right]}}\right)(57)
Type: Type
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-- basis 1-forms and coordinate vector
dx:=baseForms()$M     -- [dx[1],...,dx[n]]
\label{eq58}\left[{dx_{1}}, \:{dx_{2}}, \:{dx_{3}}, \:{dx_{4}}\right](58)
Type: List(DeRhamComplex?(Integer,[x[1],x[2],x[3],x[4]]))
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x:=coordVector()$M    -- [x[1],...,x[n]]
\label{eq59}\left[{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}\right](59)
Type: List(Expression(Integer))
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xs:=coordSymbols()$M  -- as above but as List Symbol (for differentiate, D)
\label{eq60}\left[{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}\right](60)
Type: List(Symbol)
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-- operator, vector field, scalar field, symbol
a:=operator 'a         -- operator
\label{eq61}a(61)
Type: BasicOperator?
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b:=vectorField(b)$M    -- generic vector field [b1(x1..xn),...,bn(x1..xn)]
\label{eq62}\begin{array}{@{}l} \displaystyle \left[{{b_{1}}\left({{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}}\right)}, \:{{b_{2}}\left({{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}}\right)}, \:{{b_{3}}\left({{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}}\right)}, \: \right. \ \ \displaystyle \left.{{b_{4}}\left({{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}}\right)}\right] (62)
Type: List(Expression(Integer))
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c:=vectorField(c)$M
\label{eq63}\begin{array}{@{}l} \displaystyle \left[{{c_{1}}\left({{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}}\right)}, \:{{c_{2}}\left({{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}}\right)}, \:{{c_{3}}\left({{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}}\right)}, \: \right. \ \ \displaystyle \left.{{c_{4}}\left({{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}}\right)}\right] (63)
Type: List(Expression(Integer))
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P:=scalarField(P)$M -- scalar field P(x1,..,xn)
\label{eq64}P \left({{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}}\right)(64)
Type: Expression(Integer)
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-- metric
g:=diagonalMatrix([1 for i in 1..n])$SquareMatrix(n,EXPR R)  -- Euclidean
\label{eq65}\left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1 (65)
Type: SquareMatrix?(4,Expression(Integer))
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h:=diagonalMatrix(c)$SquareMatrix(n,EXPR R)
\label{eq66}\left[ \begin{array}{cccc} {{c_{1}}\left({{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}}\right)}& 0 & 0 & 0 \ 0 &{{c_{2}}\left({{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}}\right)}& 0 & 0 \ 0 & 0 &{{c_{3}}\left({{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}}\right)}& 0 \ 0 & 0 & 0 &{{c_{4}}\left({{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}}\right)} (66)
Type: SquareMatrix?(4,Expression(Integer))
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eta:=diagonalMatrix(concat(1,[-1 for i in 2..n]))$SquareMatrix(n,EXPR R)
\label{eq67}\left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \ 0 & - 1 & 0 & 0 \ 0 & 0 & - 1 & 0 \ 0 & 0 & 0 & - 1 (67)
Type: SquareMatrix?(4,Expression(Integer))
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-- vector field
vf:=vector b
\label{eq68}\begin{array}{@{}l} \displaystyle \left[{{b_{1}}\left({{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}}\right)}, \:{{b_{2}}\left({{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}}\right)}, \:{{b_{3}}\left({{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}}\right)}, \: \right. \ \ \displaystyle \left.{{b_{4}}\left({{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}}\right)}\right] (68)
Type: Vector(Expression(Integer))
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-- macros
dV(g) ==> volumeForm(g)$M
Type: Void
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i(X,w) ==> interiorProduct(X,w)$M
Type: Void
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L(X,w) ==> lieDerivative(X,w)$M
Type: Void
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** w ==> hodgeStar(g,w)$M
Type: Void
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---

w:=x.1*dx.2-x.2*dx.1
\label{eq69}{{x_{1}}\ {dx_{2}}}-{{x_{2}}\ {dx_{1}}}(69)
Type: DeRhamComplex?(Integer,[x[1],x[2],x[3],x[4]])
fricas
res.36 := test(d w = 2*dx.1*dx.2)
\label{eq70} \mbox{\rm true} (70)
Type: Boolean
fricas
res.37 := test(w*w = zero()$M)
\label{eq71} \mbox{\rm true} (71)
Type: Boolean
fricas
res.38 := test(i(vf,w) = x.1*b.2-x.2*b.1)
\label{eq72} \mbox{\rm true} (72)
Type: Boolean
fricas
res.39 := test(L(vf,w) = d i(vf,w) + i(vf,d w))
\label{eq73} \mbox{\rm true} (73)
Type: Boolean
fricas
res.40 := test(d i(vf,w) + i(vf,d w) - L(vf,w) = zero()$M)
\label{eq74} \mbox{\rm true} (74)
Type: Boolean
fricas
res.41 := test(dot(g,w,w)$M = x.1^2+x.2^2)
\label{eq75} \mbox{\rm true} (75)
Type: Boolean
fricas
-- div(b) dV
res.41 := test(d i(vf,dV(g)) = reduce(_+,[D(b.j,xs.j) for j in 1..n])*dV(g))
\label{eq76} \mbox{\rm true} (76)
Type: Boolean
fricas
res.42 := test(d (P*one()$M) = reduce(_+,[D(P,xs.j)*dx.j for j in 1..n]))
\label{eq77} \mbox{\rm true} (77)
Type: Boolean
fricas
res.43 := test(i(vf,d (P*one()$M))= reduce(_+,[D(P,xs.j)*b.j for j in 1..n])*one()$M)
\label{eq78} \mbox{\rm true} (78)
Type: Boolean
fricas
res.44 := test(1/dot(g,w,w)$M*w =  w*(1/(x.1^2+x.2^2)))
\label{eq79} \mbox{\rm true} (79)
Type: Boolean
fricas
res.45 := test(d (1/dot(g,w,w)$M*w) = zero()$M)
\label{eq80} \mbox{\rm true} (80)
Type: Boolean
fricas
---
s:=zeroForm('s)$M
\label{eq81}s \left({{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}}\right)(81)
Type: DeRhamComplex?(Integer,[x[1],x[2],x[3],x[4]])
fricas
res.46 := test(d s =  totalDifferential(retract s)$DeRhamComplex(Integer,v))
\label{eq82} \mbox{\rm true} (82)
Type: Boolean
fricas
res.47 := test(d s =  totalDifferential(retract s)$typeOf(s))
\label{eq83} \mbox{\rm true} (83)
Type: Boolean
fricas
res.48 := test(d (** s) = 0$typeOf(s))
\label{eq84} \mbox{\rm true} (84)
Type: Boolean
fricas
res.49 := test(dot(g,** ( d s),w*dx.2*dx.3)$M = x.2*D(retract s, xs.4))
\label{eq85} \mbox{\rm true} (85)
Type: Boolean
fricas
res.50 := test(d (** ( d s)) = reduce(_+,[D(retract s,xs.j,2) for j in 1..n])*dV(g))
\label{eq86} \mbox{\rm true} (86)
Type: Boolean
fricas
r:=sin(x.1*x.2)*one()$M
\label{eq87}\sin \left({{x_{1}}\ {x_{2}}}\right)(87)
Type: DeRhamComplex?(Integer,[x[1],x[2],x[3],x[4]])
fricas
res.51 := test(d r = x.1*cos(x.1*x.2)*dx.2+x.2*cos(x.1*x.2)*dx.1)
\label{eq88} \mbox{\rm true} (88)
Type: Boolean
fricas
res.52 := test(d (** ( d r)) = -(x.1^2+x.2^2)*sin(x.1*x.2)*dV(g))
\label{eq89} \mbox{\rm true} (89)
Type: Boolean
fricas
res.53 := test(** (d (** ( d r))) = -(x.1^2+x.2^2)*sin(x.1*x.2))
\label{eq90} \mbox{\rm true} (90)
Type: Boolean
fricas
res.53 := test(** (d (** ( d r)))::EXPR INT = retract (** (d (** ( d r)))))
\label{eq91} \mbox{\rm true} (91)
Type: Boolean
fricas
res.54 := test(eval(** (d (** ( d r)))::EXPR INT,xs.1=%pi) = (-%pi^2-x.2^2)*sin(%pi*x.2))
\label{eq92} \mbox{\rm true} (92)
Type: Boolean
fricas
res.55 := test(eval(eval(** (d (** ( d r)))::EXPR INT,xs.1=%pi) ,xs.2=%pi)=-2*%pi^2*sin(%pi^2))
\label{eq93} \mbox{\rm true} (93)
Type: Boolean
fricas
a(P)*one()$M
\label{eq94}a \left({P \left({{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}}\right)}\right)(94)
Type: DeRhamComplex?(Integer,[x[1],x[2],x[3],x[4]])
fricas
-- chain diff
res.56 := test(d (a(P)*one()$M) =  eval(D(a(t),'t),t=P)*d (P*one()$M))
\label{eq95} \mbox{\rm true} (95)
Type: Boolean
fricas
res.57 := test(** invHodgeStar(g,w)$M = w)
\label{eq96} \mbox{\rm true} (96)
Type: Boolean
fricas
res.58 := test(invHodgeStar(g,** w)$M = w)
\label{eq97} \mbox{\rm true} (97)
Type: Boolean
fricas
res.59 := test(** invHodgeStar(g,** w + dx.1)$M = ** w + dx.1)
\label{eq98} \mbox{\rm true} (98)
Type: Boolean
fricas
res.60 := test( ** dV(g) = invHodgeStar(g,dV(g))$M)
\label{eq99} \mbox{\rm true} (99)
Type: Boolean
fricas
res.61 := test(** dV(h) ~= invHodgeStar(h,dV(g))$M)
\label{eq100} \mbox{\rm true} (100)
Type: Boolean
fricas
res.62 := test( dot(h,w,w)$M = (c.2*x.2^2+c.1*x.1^2)/(c.1*c.2))
\label{eq101} \mbox{\rm true} (101)
Type: Boolean
fricas
res.63 := test( s(g)$M = 1)
\label{eq102} \mbox{\rm true} (102)
Type: Boolean
fricas
res.64 := test( s(eta)$M = -1)
\label{eq103} \mbox{\rm true} (103)
Type: Boolean
fricas
res.65 := test( s(h)$M = 's? )
\label{eq104} \mbox{\rm true} (104)
Type: Boolean
fricas
-- https://en.wikipedia.org/wiki/Hodge_dual
-- Four dimensions

res.66 := test( hodgeStar(eta,dx.1)$M = dx.2*dx.3*dx.4)
\label{eq105} \mbox{\rm true} (105)
Type: Boolean
fricas
res.67 := test( hodgeStar(eta,dx.2)$M = dx.1*dx.3*dx.4)
\label{eq106} \mbox{\rm true} (106)
Type: Boolean
fricas
res.68 := test( hodgeStar(eta,dx.3)$M = dx.1*dx.4*dx.2)
\label{eq107} \mbox{\rm true} (107)
Type: Boolean
fricas
res.69 := test( hodgeStar(eta,dx.4)$M = dx.1*dx.2*dx.3)
\label{eq108} \mbox{\rm true} (108)
Type: Boolean
fricas
res.70 := test( hodgeStar(eta,dx.1*dx.2)$M = dx.4*dx.3)
\label{eq109} \mbox{\rm true} (109)
Type: Boolean
fricas
res.71 := test( hodgeStar(eta,dx.1*dx.3)$M = dx.2*dx.4)
\label{eq110} \mbox{\rm true} (110)
Type: Boolean
fricas
res.72 := test( hodgeStar(eta,dx.1*dx.4)$M = dx.3*dx.2)
\label{eq111} \mbox{\rm true} (111)
Type: Boolean
fricas
res.73 := test( hodgeStar(eta,dx.2*dx.3)$M = dx.1*dx.4)
\label{eq112} \mbox{\rm true} (112)
Type: Boolean
fricas
res.74 := test( hodgeStar(eta,dx.2*dx.4)$M = dx.3*dx.1)
\label{eq113} \mbox{\rm true} (113)
Type: Boolean
fricas
res.75 := test( hodgeStar(eta,dx.3*dx.4)$M = dx.1*dx.2)
\label{eq114} \mbox{\rm true} (114)
Type: Boolean
fricas
-- codifferential: (delta=(-)^degree(x)*invHodgeStar(g, d hodgeStar(g,x))
res.76 := test(codifferential(g,s*dx.1)$M = -D(retract s,xs.1)*one()$M)
\label{eq115} \mbox{\rm true} (115)
Type: Boolean
fricas
res.77 := test(codifferential(g,P*dx.2)$M = -D(P,xs.2)*one()$M)
\label{eq116} \mbox{\rm true} (116)
Type: Boolean
fricas
res.78 := test(codifferential(g,s*P*dx.1*dx.2)$M = _
    (-P*D(retract s,xs.1)-s*D(P,xs.1))*dx.2+_
    (P*D(retract s,xs.2)+s*D(P,xs.2))*dx.1)
\label{eq117} \mbox{\rm true} (117)
Type: Boolean
fricas
res.79 := test(codifferential(g,P*w)$M = x.2*D(P,xs.1)*one()$M -_
     x.1*D(P,xs.2)*one()$M)
\label{eq118} \mbox{\rm true} (118)
Type: Boolean
fricas
res.80 := test(codifferential(g,P*w)$M = _
    (-1)^degree(w)*invHodgeStar(g, d hodgeStar(g,P*w)$M)$M)
\label{eq119} \mbox{\rm true} (119)
Type: Boolean
fricas
res.81 := test(codifferential(eta,P*w)$M = _
    (-1)^degree(w)*invHodgeStar(eta, d hodgeStar(eta,P*w)$M)$M)
\label{eq120} \mbox{\rm true} (120)
Type: Boolean
fricas
res.82 := test(codifferential(h,P*w)$M = _
    (-1)^degree(w)*invHodgeStar(h, d hodgeStar(h,P*w)$M)$M)
\label{eq121} \mbox{\rm true} (121)
Type: Boolean
fricas
res.83 := test(codifferential(h,P*w*dx.3)$M = _
    (-1)^degree(w*dx.3)*invHodgeStar(h, d hodgeStar(h,P*w*dx.3)$M)$M)
\label{eq122} \mbox{\rm true} (122)
Type: Boolean
fricas
-- the codifferential \delta is (sort of) the adjoint to the differential.
-- This doesn’t quite hold, but we can show that it does hold “up to homology”.
-- We can calculate their difference times the canonical volume form:
res.84 := test((dot(g,d w, dx.1*dx.2)$M -_
    dot(g,w,codifferential(g,dx.1*dx.2)$M)$M)*dV(g)=
    d (w*hodgeStar(g,dx.1*dx.2)$M))
\label{eq123} \mbox{\rm true} (123)
Type: Boolean
fricas
res.85 := test((dot(eta,d w, dx.1*dx.2)$M -_
    dot(eta,w,codifferential(eta,dx.1*dx.2)$M)$M)*dV(eta)=
    d (w*hodgeStar(eta,dx.1*dx.2)$M))
\label{eq124} \mbox{\rm true} (124)
Type: Boolean
fricas
res.86 := test(hodgeLaplacian(g,w*(** w))$M = -4 * dV(g))
\label{eq125} \mbox{\rm true} (125)
Type: Boolean
fricas
res.87 := test(hodgeLaplacian(eta,w*(hodgeStar(eta,w)$M))$M = 4 * dV(eta))
\label{eq126} \mbox{\rm true} (126)
Type: Boolean
fricas
res.88 := test(hodgeLaplacian(h,w*(hodgeStar(h,w)$M))$M =_
    d codifferential(h,w*hodgeStar(h,w)$M)$M)
\label{eq127} \mbox{\rm true} (127)
Type: Boolean
fricas
res.89 := test(d(vf*dx) *  (** d(vf*dx))=dot(g,d(vf*dx),d(vf*dx))$M*dV(g))
\label{eq128} \mbox{\rm true} (128)
Type: Boolean
fricas
-- next is inequality because ** = hodgeStar w.r.t to metric g!
res.90 := test(d(vf*dx) *  (** d(vf*dx))~=dot(eta,d(vf*dx),d(vf*dx))$M*dV(eta))
\label{eq129} \mbox{\rm true} (129)
Type: Boolean
fricas
-- correct is:
res.91 := test(d(vf*dx)*hodgeStar(eta,d(vf*dx))$M=_
    dot(eta,d(vf*dx),d(vf*dx))$M*dV(eta))
\label{eq130} \mbox{\rm true} (130)
Type: Boolean
fricas
-------------
-- Results --
-------------
fricas
)version

"FriCAS 1.3.10 compiled at Wed 10 Jan 02:19:45 CET 2024"
fricas
)lisp (lisp-implementation-type)

   Your user access level is compiler and this command is therefore not
      available. See the )set userlevel command for more information.
fricas
)lisp (lisp-implementation-version)

   Your user access level is compiler and this command is therefore not
      available. See the )set userlevel command for more information.

res
\label{eq131}\begin{array}{@{}l} \displaystyle \left[ \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \right. \ \ \displaystyle \left.\: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \right. \ \ \displaystyle \left.\: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \right. \ \ \displaystyle \left.\: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \right. \ \ \displaystyle \left.\: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \right. \ \ \displaystyle \left.\: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \right. \ \ \displaystyle \left.\: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \right. \ \ \displaystyle \left.\: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} , \right. \ \ \displaystyle \left.\: \mbox{\rm true} , \: \mbox{\rm true} , \: \mbox{\rm true} \right] (131)
Type: List(Boolean)
fricas
reduce(_and,res)
\label{eq132} \mbox{\rm true} (132)
Type: Boolean
fricas
-- dummy statistics
[testEquals("res." string(j),"true") for j in 1..N];
Type: List(Void)
fricas
statistics()

   =============================================================================
   General WARNINGS:
   * do not use ')clear completely' before having used 'statistics()'
     It clears the statistics without warning!
   * do not forget to pass the arguments of the testXxxx functions as Strings!
     Otherwise, the test will fail and statistics() will not notice!

   =============================================================================
   Testsuite: dform
     failed (total): 0 (1)

   =============================================================================
   testsuite | testcases: failed (total) | tests: failed (total)
   dform                       0     (1)               0    (91)
   =============================================================================
   File summary.
   unexpected failures: 0
   expected failures: 0
   unexpected passes: 0
   total tests: 91
Type: Void

Some changes necessary: use vectors instead of lists, ...



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