MathAction SandBoxSurfaceComplex

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  - SandBoxSurfaceComplex <-- You are here.

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(1) -> )abbrev domain CMAP CellMap
++
CellMap(R,n) : Exports == Implementation where

  R: Join(Ring,Comparable)
  n: NonNegativeInteger

  X   ==> Expression R
  DP  ==> DirectProduct
  OF  ==> OutputForm
  NNI ==> NonNegativeInteger
  MAP ==> List X -> List X
  DOM ==> List(Segment X)

  Exports == Join(CoercibleTo OF,SetCategory,Evalable X) with

    _= : (%,%) -> Boolean
      ++ f1=f2 checks if two given cell maps are equal, that is if they have
      ++ the same domain D and the same mapping from D into X^n.
    cellMap : (DOM,MAP) -> %
      ++ cellMap(D,f) is the constructor. Usually one has to specify the
      ++ dimension of the target space. For example, let Q=[a..b,c..d], then
      ++ cellMap(Q,Z+->[sin(Z.1),cos(Z.2),Z.1*Z.2])$CMAP(INT,3) defines a
      ++ 2-surface in X^3.
    getDom : % -> DOM
      ++ getDom(f) extracts the domain of f.
    getMap : % -> MAP
      ++ getMap(f) extracts the map of f.
    faces  : % -> List List(%)
      ++ faces(f) returns the faces of f, that means the images of the boundary
      ++ of the domain. Note: the returned list contains pairs of faces 
      ++ corresponding to the endpoints of intervals.
    coords : (Symbol,PositiveInteger) -> List X
      ++ coords(s,m) provides a sample of coordinates s[1],..,s[m] as a list.
    coordSymbols : (Symbol,PositiveInteger) -> List Symbol
      ++ coordSymbols(s,m) provides a sample of coordinates s[1],..,s[m] as a
      ++ list of symbols.
    jacobianMatrix : % -> (List X -> Matrix X)
      ++ jacobianMatrix(f) returns the Jacobian matrix as a marix valued
      ++ function defined on the same cell as the cellMap.
    tangentSpace : % -> (List(X) -> List(Vector X))
      ++ tangentSpace(f) returns a 
    coerce : % -> OutputForm
      ++ coerce(f) gives the output representation.

  Implementation ==  add 

    Rep := Record(d:DOM,f:MAP)

    (x:% = y:%):Boolean == 
      l:NNI:=min(#(x.d),#(y.d))
      v:List X
      for j in 1..l repeat
        s:X:=subscript('z,[j::OF])::X
        v:=concat(v,s::X)
      x.d =y.d and (x.f) v = (y.f) v => true
      false

    cellMap(dd:DOM,ff:MAP):% == 
      #dd > n => error concat("#DOM > ",string n)
      v:List X:=[1::X for j in 1..#dd]
      ~test(#ff(v)=n) => error concat("#Range ~= ", string n) 
      construct(dd,ff)

    faceLoHi(x:%,i:NNI,lo:Boolean):% ==
      l:NNI:=#(x.d)
      v:List X
      for j in 1..l repeat
        if j=i then
          if lo then
            s : X := low((x.d)(i))
          else
            s: X := high((x.d)(i))
        else
          if j>i then
            s:X:=subscript('%,[(j-1)::OF])::X
          else
            s:X:=subscript('%,[j::OF])::X
        v:=concat(v,s::X)
      vv:=delete(v,i..i)
      dd:List(Segment X):=delete(x.d,i..i)
      ff:MAP:=vv+->(x.f) v
      cellMap(dd,ff)

    faces(x:%):List List(%) ==
      l:NNI:=#(x.d)
      [[faceLoHi(x,j,true), faceLoHi(x,j,false)] for j in 1..l]

    getDom(x) == x.d
    getMap(x) == x.f

    coordSymbols(s:Symbol,m:PositiveInteger):List Symbol == 
      [subscript(s,[j::OF]) for j in 1..m]

    coords(s:Symbol,m:PositiveInteger):List X ==
      xs:=[subscript(s,[j::OF]) for j in 1..m]
      [coerce(xs.j)$X for j in 1..#xs]

    jacobianMatrix(S:%):List(X) -> Matrix(X) ==
      --xs:List Symbol:=v:=[subscript('x,[j::OF]) for j in 1..#(getDom S)]
      --x:List X:=[coerce(xs.j)$X for j in 1..#xs]
      xs:List Symbol:=coordSymbols('x,#(getDom S)::PositiveInteger)
      x:List X:=coords('x,#xs::PositiveInteger)
      F:List X:=(getMap S) x
      J:Matrix(X):=matrix [[D(ff,u) for u in xs] for ff in F]
      if Matrix(X) has Join(SetCategory,Evalable(X)) then
        (y:List X):Matrix(X)+-> eval(J,x,y)
      else
        (y:List X):Matrix(X)+-> J

    tangentSpace(S:%):List(X) -> List(Vector X) ==
      J:=jacobianMatrix(S)
      x:List X:=coords('x,#(getDom S)::PositiveInteger)
      if Vector(X) has Join(SetCategory,Evalable(X)) then
        if X has EuclideanDomain then 
          cs:List(Vector X):=columnSpace(J x)
          (y:List X):List Vector(X)+-> [eval(t,x,y) for t in cs]

    coerce(x) == 
      v:List X
      for j in 1..#(x.d) repeat
        s:X:=subscript('%,[j::OF])::X
        v:=concat(v,s::X)
      r:List X:=(x.f) v
      hconcat [message("|"), x.d::OF, message(" -> "), r::OF, message("|")]

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)abbrev domain SCMPLX SurfaceComplex
++
SurfaceComplex(R,n) : Exports == Implementation where

  NNI ==> NonNegativeInteger
  INT ==> Integer

  n : NNI
  R : Join(Ring,Comparable)

  CMAP ==> CellMap(R,n)
  CTOF ==> CoercibleTo OutputForm
  X    ==> Expression R
  OF   ==> OutputForm
  MAP  ==> List X -> List X
  DOM  ==> List(Segment X)

  Exports == Join(AbelianGroup ,CTOF, RetractableTo CMAP) with

    bdry : % -> %
      ++ bdry(S) computes the boundary of the surface complex S.
    size : % -> NNI
      ++ size(S) returns the number of "pieces" of the surface complex S.
    nthCoef : (%,Integer) -> Integer
      ++ nthCoef(x, n) returns the coefficient of the n^th term of x.
    nthFactor : (%,Integer) -> CMAP
      ++ nthFactor(x, n) returns the factor of the n^th term of x.
    zero? : % -> Boolean 
      ++ zero?(S) returns true if S is the empty surface complex.
    _= : (%,%) -> Boolean
      ++ S=S' checks if the surface complexes S and S' are equal.
    terms : % -> List(Record(gen: CMAP,exp: Integer))
      ++ terms(S) returns all terms of S as a record.
    mapGen : ((CMAP -> CMAP),%) -> %
      ++ mapGen(f, e1 a1 +...+ en an) returns
      ++ \spad{e1 f(a1) +...+ en f(an)}.
    mapCoef : ((Integer -> Integer),%) -> %
      ++ mapCoef(f, e1 a1 +...+ en an) returns
      ++ \spad{f(e1) a1 +...+ f(en) an}.
    construct : (DOM,MAP) -> %
      ++ construct(d,f) constructs a term (piece) of a k-surface, where
      ++ d is the domain (a k-cell) and f is a mapping from d to a vector
      ++ space of dimension n.

    --coerce : % -> OutputForm

  Implementation == FreeModule(Integer, CMAP) add

    Rep:=FreeModule(Integer, CMAP)

    bdry(c:%):% == 
      if size(c) = 1 then
        s:=nthFactor(c,1)
        l:=faces(s)
        fs:=[(a.2::Rep-a.1::Rep) for a in l]
        sgn:=(j:INT):INT+->if even? (j-1) then 1 else -1
        nthCoef(c,1)*reduce("+",[sgn(j)*fs.j::Rep for j in 1..#fs])
      else
        ct:=[(nthCoef(c,j)*nthFactor(c,j))::Rep for j in 1..size(c)]
        reduce("+",map(bdry,ct))

    construct(d:DOM,f:MAP):% == cellMap(d,f)$CMAP::%

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Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/9034321958150369551-25px.001.spad
      using old system compiler.
   CMAP abbreviates domain CellMap 
------------------------------------------------------------------------
   initializing NRLIB CMAP for CellMap 
   compiling into NRLIB CMAP 
****** Domain: R already in scope
   compiling exported = : (%,%) -> Boolean
Time: 0.05 SEC.

   compiling exported cellMap : (List Segment Expression R,List Expression R -> List Expression R) -> %
Time: 0 SEC.

   compiling local faceLoHi : (%,NonNegativeInteger,Boolean) -> %
Time: 0.01 SEC.

   compiling exported faces : % -> List List %
Time: 0 SEC.

   compiling exported getDom : % -> List Segment Expression R
      CMAP;getDom;%L;5 is replaced by QCAR 
Time: 0 SEC.

   compiling exported getMap : % -> List Expression R -> List Expression R
      CMAP;getMap;%M;6 is replaced by QCDR 
Time: 0 SEC.

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

   compiling exported coords : (Symbol,PositiveInteger) -> List Expression R
Time: 0.03 SEC.

   compiling exported jacobianMatrix : % -> List Expression R -> Matrix Expression R
****** Domain: (Matrix (Expression R)) already in scope
augmenting (Matrix (Expression R)): (Evalable (Expression R))
Time: 0.02 SEC.

   compiling exported tangentSpace : % -> List Expression R -> List Vector Expression R
****** Domain: (Vector (Expression R)) already in scope
augmenting (Vector (Expression R)): (Evalable (Expression R))
****** Domain: (Expression R) already in scope
augmenting (Expression R): (EuclideanDomain)
Time: 0.03 SEC.

   compiling exported coerce : % -> OutputForm
Time: 0 SEC.

(time taken in buildFunctor:  0)

;;;     ***       |CellMap| REDEFINED

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

   Warnings: 
      [1] =:  d has no value
      [2] =:  v has no value
      [3] =:  f has no value
      [4] faceLoHi:  d has no value
      [5] faceLoHi:  v has no value
      [6] faceLoHi:  f has no value
      [7] faces:  d has no value
      [8] getDom:  d has no value
      [9] getMap:  f has no value
      [10] coerce:  d has no value
      [11] coerce:  v has no value
      [12] coerce:  f has no value

   Cumulative Statistics for Constructor CellMap
      Time: 0.15 seconds

   finalizing NRLIB CMAP 
   Processing CellMap for Browser database:
--------constructor---------
--------(= ((Boolean) % %))---------
--------(cellMap (% (List (Segment (Expression R))) (Mapping (List (Expression R)) (List (Expression R)))))---------
--------(getDom ((List (Segment (Expression R))) %))---------
--------(getMap ((Mapping (List (Expression R)) (List (Expression R))) %))---------
--------(faces ((List (List %)) %))---------
--------(coords ((List (Expression R)) (Symbol) (PositiveInteger)))---------
--------(coordSymbols ((List (Symbol)) (Symbol) (PositiveInteger)))---------
--------(jacobianMatrix ((Mapping (Matrix (Expression R)) (List (Expression R))) %))---------
--------(tangentSpace ((Mapping (List (Vector (Expression R))) (List (Expression R))) %))---------
--------(coerce ((OutputForm) %))---------
; compiling file "/var/aw/var/LatexWiki/CMAP.NRLIB/CMAP.lsp" (written 28 DEC 2024 04:12:51 PM):

; wrote /var/aw/var/LatexWiki/CMAP.NRLIB/CMAP.fasl
; compilation finished in 0:00:00.040
------------------------------------------------------------------------
   CellMap is now explicitly exposed in frame initial 
   CellMap will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/CMAP.NRLIB/CMAP

   SCMPLX abbreviates domain SurfaceComplex 
------------------------------------------------------------------------
   initializing NRLIB SCMPLX for SurfaceComplex 
   compiling into NRLIB SCMPLX 
****** Domain: R already in scope
Local variable Rep type redefined: (Join (BiModule (Integer) (Integer)) (FreeModuleCategory (Integer) (CellMap R n)) (CATEGORY domain (IF (has (Integer) (CommutativeRing)) (ATTRIBUTE (Module (Integer))) noBranch) (IF (has (Integer) (SemiRing)) (ATTRIBUTE (RetractableTo (CellMap R n))) noBranch) (IF (has (Integer) (Hashable)) (IF (has (CellMap R n) (Hashable)) (ATTRIBUTE (Hashable)) noBranch) noBranch) (IF (has (CellMap R n) (OrderedSet)) (PROGN (IF (has (Integer) (OrderedAbelianMonoid)) (ATTRIBUTE (OrderedAbelianMonoid)) noBranch) (IF (has (Integer) (OrderedAbelianMonoidSup)) (ATTRIBUTE (OrderedAbelianMonoidSup)) noBranch)) noBranch) (SIGNATURE * (% (Integer) (CellMap R n))) (SIGNATURE * (% (CellMap R n) (Integer))))) to (Join (SetCategory) (CATEGORY domain (SIGNATURE construct ((Record (: d (List (Segment (Expression R)))) (: f (Mapping (List (Expression R)) (List (Expression R))))) (List (Segment (Expression R))) (Mapping (List (Expression R)) (List (Expression R))))) (SIGNATURE ~= ((Boolean) (Record (: d (List (Segment (Expression R)))) (: f (Mapping (List (Expression R)) (List (Expression R))))) (Record (: d (List (Segment (Expression R)))) (: f (Mapping (List (Expression R)) (List (Expression R))))))) (SIGNATURE coerce ((OutputForm) (Record (: d (List (Segment (Expression R)))) (: f (Mapping (List (Expression R)) (List (Expression R))))))) (SIGNATURE elt ((List (Segment (Expression R))) (Record (: d (List (Segment (Expression R)))) (: f (Mapping (List (Expression R)) (List (Expression R))))) d)) (SIGNATURE elt ((Mapping (List (Expression R)) (List (Expression R))) (Record (: d (List (Segment (Expression R)))) (: f (Mapping (List (Expression R)) (List (Expression R))))) f)) (SIGNATURE setelt! ((List (Segment (Expression R))) (Record (: d (List (Segment (Expression R)))) (: f (Mapping (List (Expression R)) (List (Expression R))))) d (List (Segment (Expression R))))) (SIGNATURE setelt! ((Mapping (List (Expression R)) (List (Expression R))) (Record (: d (List (Segment (Expression R)))) (: f (Mapping (List (Expression R)) (List (Expression R))))) f (Mapping (List (Expression R)) (List (Expression R))))) (SIGNATURE copy ((Record (: d (List (Segment (Expression R)))) (: f (Mapping (List (Expression R)) (List (Expression R))))) (Record (: d (List (Segment (Expression R)))) (: f (Mapping (List (Expression R)) (List (Expression R)))))))))
   compiling exported bdry : % -> %
Time: 0.02 SEC.

   compiling exported construct : (List Segment Expression R,List Expression R -> List Expression R) -> %
Time: 0 SEC.

(time taken in buildFunctor:  0)

;;;     ***       |SurfaceComplex| REDEFINED

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

   Cumulative Statistics for Constructor SurfaceComplex
      Time: 0.03 seconds

--------------non extending category----------------------
.. SurfaceComplex(#1,#2) of cat 
(|Join| (|AbelianGroup|) (|CoercibleTo| (|OutputForm|))
        (|RetractableTo| (|CellMap| |#1| |#2|))
        (CATEGORY |domain| (SIGNATURE |bdry| (% %))
         (SIGNATURE |size| ((|NonNegativeInteger|) %))
         (SIGNATURE |nthCoef| ((|Integer|) % (|Integer|)))
         (SIGNATURE |nthFactor| ((|CellMap| |#1| |#2|) % (|Integer|)))
         (SIGNATURE |zero?| ((|Boolean|) %)) (SIGNATURE = ((|Boolean|) % %))
         (SIGNATURE |terms|
          ((|List|
            (|Record| (|:| |gen| (|CellMap| |#1| |#2|))
                      (|:| |exp| (|Integer|))))
           %))
         (SIGNATURE |mapGen|
          (% (|Mapping| (|CellMap| |#1| |#2|) (|CellMap| |#1| |#2|)) %))
         (SIGNATURE |mapCoef| (% (|Mapping| (|Integer|) (|Integer|)) %))
         (SIGNATURE |construct|
          (% (|List| (|Segment| (|Expression| |#1|)))
           (|Mapping| (|List| (|Expression| |#1|))
                      (|List| (|Expression| |#1|)))))))   has no 
(|BiModule| (|Integer|) (|Integer|))    finalizing NRLIB SCMPLX 
   Processing SurfaceComplex for Browser database:
--------constructor---------
--------(bdry (% %))---------
--------(size ((NonNegativeInteger) %))---------
--------(nthCoef ((Integer) % (Integer)))---------
--------(nthFactor ((CellMap R n) % (Integer)))---------
--------(zero? ((Boolean) %))---------
--------(= ((Boolean) % %))---------
--------(terms ((List (Record (: gen (CellMap R n)) (: exp (Integer)))) %))---------
--------(mapGen (% (Mapping (CellMap R n) (CellMap R n)) %))---------
--------(mapCoef (% (Mapping (Integer) (Integer)) %))---------
--------(construct (% (List (Segment (Expression R))) (Mapping (List (Expression R)) (List (Expression R)))))---------
; compiling file "/var/aw/var/LatexWiki/SCMPLX.NRLIB/SCMPLX.lsp" (written 28 DEC 2024 04:12:51 PM):

; wrote /var/aw/var/LatexWiki/SCMPLX.NRLIB/SCMPLX.fasl
; compilation finished in 0:00:00.016
------------------------------------------------------------------------
   SurfaceComplex is now explicitly exposed in frame initial 
   SurfaceComplex will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/SCMPLX.NRLIB/SCMPLX

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)clear all

   All user variables and function definitions have been cleared.
R ==> EXPR INT

Type: Void

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OF ==> OutputForm

Type: Void

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-- Cell map
R2 ==> CellMap(INT,2)

Type: Void

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R3 ==> CellMap(INT,3)

Type: Void

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R4 ==> CellMap(INT,4)

Type: Void

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Q2 ==> [0..1,0..1::R]

Type: Void

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Q3 ==> concat(Q2,[0..1::R])

Type: Void

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--xs:List Symbol:=coordSymbols('x,4)$R4

----------------------------------------------------------------
-- https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant
----------------------------------------------------------------
-- Example 1
F1:=cellMap(Q2,X+->[X.1^2*X.2,5*X.1+sin(X.2)])$R2

$\label{eq1}|{\left[{0..1}, \:{0..1}\right]}\mbox{\rm - >}{\left[{{{\%_{1}}^{2}}\ {\%_{2}}}, \:{{\sin \left({\%_{2}}\right)}+{5 \ {\%_{1}}}}\right]}|$

(1)

Type: CellMap?(Integer,2)

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J:=jacobianMatrix(F1)

$\label{eq2}\mbox{theMap (...)}$

(2)

Type: (List(Expression(Integer)) -> Matrix(Expression(Integer)))

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x:=coords('x,2)$R2

$\label{eq3}\left[{x_{1}}, \:{x_{2}}\right]$

(3)

Type: List(Expression(Integer))

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J x

$\label{eq4}\left[ \begin{array}{cc} {2 \ {x_{1}}\ {x_{2}}}&{{x_{1}}^{2}} \ 5 &{\cos \left({x_{2}}\right)}$

(4)

Type: Matrix(Expression(Integer))

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determinant(J x)

$\label{eq5}{2 \ {x_{1}}\ {x_{2}}\ {\cos \left({x_{2}}\right)}}-{5 \ {{x_{1}}^{2}}}$

(5)

Type: Expression(Integer)

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test(J x = matrix [[2*x.1*x.2,x.1^2],[5,cos(x.2)]])

$\label{eq6} \mbox{\rm true}$

(6)

Type: Boolean

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test(determinant(J x) = 2*x.1*x.2*cos(x.2)-5*x.1^2)

$\label{eq7} \mbox{\rm true}$

(7)

Type: Boolean

fricas

-- Example 2
F2:=cellMap(Q2,X+->[X.1*cos(X.2),X.1*sin(X.2)])$R2

$\label{eq8}|{\left[{0..1}, \:{0..1}\right]}\mbox{\rm - >}{\left[{{\%_{1}}\ {\cos \left({\%_{2}}\right)}}, \:{{\%_{1}}\ {\sin \left({\%_{2}}\right)}}\right]}|$

(8)

Type: CellMap?(Integer,2)

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J:=jacobianMatrix(F2)

$\label{eq9}\mbox{theMap (...)}$

(9)

Type: (List(Expression(Integer)) -> Matrix(Expression(Integer)))

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x:=[r::R,phi::R]

$\label{eq10}\left[ r , \: phi \right]$

(10)

Type: List(Expression(Integer))

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(getMap F2) x

$\label{eq11}\left[{r \ {\cos \left({phi}\right)}}, \:{r \ {\sin \left({phi}\right)}}\right]$

(11)

Type: List(Expression(Integer))

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J x

$\label{eq12}\left[ \begin{array}{cc} {\cos \left({phi}\right)}& -{r \ {\sin \left({phi}\right)}} \ {\sin \left({phi}\right)}&{r \ {\cos \left({phi}\right)}}$

(12)

Type: Matrix(Expression(Integer))

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determinant(J x)

$\label{eq13}{r \ {{\sin \left({phi}\right)}^{2}}}+{r \ {{\cos \left({phi}\right)}^{2}}}$

(13)

Type: Expression(Integer)

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test( J x = matrix [[cos(x.2),-x.1*sin(x.2)],[sin(x.2),x.1*cos(x.2)]])

$\label{eq14} \mbox{\rm true}$

(14)

Type: Boolean

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test( normalize determinant(J x) = x.1)

$\label{eq15} \mbox{\rm true}$

(15)

Type: Boolean

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-- Example 3
F3:=cellMap(Q3,Z+->[Z.1*sin(Z.2)*cos(Z.3),Z.1*sin(Z.2)*sin(Z.3),Z.1*cos(Z.2)])$R3

$\label{eq16}|{\left[{0..1}, \:{0..1}, \:{0..1}\right]}\mbox{\rm - >}{\left[{{\%_{1}}\ {\cos \left({\%_{3}}\right)}\ {\sin \left({\%_{2}}\right)}}, \:{{\%_{1}}\ {\sin \left({\%_{2}}\right)}\ {\sin \left({\%_{3}}\right)}}, \:{{\%_{1}}\ {\cos \left({\%_{2}}\right)}}\right]}|$

(16)

Type: CellMap?(Integer,3)

fricas

J:=jacobianMatrix(F3)

$\label{eq17}\mbox{theMap (...)}$

(17)

Type: (List(Expression(Integer)) -> Matrix(Expression(Integer)))

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z:=[r::R,th::R,phi::R]

$\label{eq18}\left[ r , \: th , \: phi \right]$

(18)

Type: List(Expression(Integer))

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(getMap F3) z

$\label{eq19}\left[{r \ {\cos \left({phi}\right)}\ {\sin \left({th}\right)}}, \:{r \ {\sin \left({phi}\right)}\ {\sin \left({th}\right)}}, \:{r \ {\cos \left({th}\right)}}\right]$

(19)

Type: List(Expression(Integer))

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J z

$\label{eq20}\left[ \begin{array}{ccc} {{\cos \left({phi}\right)}\ {\sin \left({th}\right)}}&{r \ {\cos \left({phi}\right)}\ {\cos \left({th}\right)}}& -{r \ {\sin \left({phi}\right)}\ {\sin \left({th}\right)}} \ {{\sin \left({phi}\right)}\ {\sin \left({th}\right)}}&{r \ {\cos \left({th}\right)}\ {\sin \left({phi}\right)}}&{r \ {\cos \left({phi}\right)}\ {\sin \left({th}\right)}} \ {\cos \left({th}\right)}& -{r \ {\sin \left({th}\right)}}& 0$

(20)

Type: Matrix(Expression(Integer))

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determinant(J z)

$\label{eq21}\begin{array}{@{}l} \displaystyle {{\left({{{r}^{2}}\ {{\sin \left({phi}\right)}^{2}}}+{{{r}^{2}}\ {{\cos \left({phi}\right)}^{2}}}\right)}\ {{\sin \left({th}\right)}^{3}}}+ \ \ \displaystyle {{\left({{{r}^{2}}\ {{\cos \left({th}\right)}^{2}}\ {{\sin \left({phi}\right)}^{2}}}+{{{r}^{2}}\ {{\cos \left({phi}\right)}^{2}}\ {{\cos \left({th}\right)}^{2}}}\right)}\ {\sin \left({th}\right)}}$

(21)

Type: Expression(Integer)

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M:=[[sin(z.2)*cos(z.3),z.1*cos(z.2)*cos(z.3),-z.1*sin(z.2)*sin(z.3)],_
    [sin(z.2)*sin(z.3),z.1*cos(z.2)*sin(z.3),z.1*sin(z.2)*cos(z.3)],_
    [cos(z.2),-z.1*sin(z.2),0]]

$\label{eq22}\begin{array}{@{}l} \displaystyle \left[{\left[{{\cos \left({phi}\right)}\ {\sin \left({th}\right)}}, \:{r \ {\cos \left({phi}\right)}\ {\cos \left({th}\right)}}, \: -{r \ {\sin \left({phi}\right)}\ {\sin \left({th}\right)}}\right]}, \: \right. \ \ \displaystyle \left.{\left[{{\sin \left({phi}\right)}\ {\sin \left({th}\right)}}, \:{r \ {\cos \left({th}\right)}\ {\sin \left({phi}\right)}}, \:{r \ {\cos \left({phi}\right)}\ {\sin \left({th}\right)}}\right]}, \: \right. \ \ \displaystyle \left.{\left[{\cos \left({th}\right)}, \: -{r \ {\sin \left({th}\right)}}, \: 0 \right]}\right]$

(22)

Type: List(List(Expression(Integer)))

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test( J z = matrix M)

$\label{eq23} \mbox{\rm true}$

(23)

Type: Boolean

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test( simplify determinant(J z) = z.1^2*sin(z.2) )

$\label{eq24} \mbox{\rm true}$

(24)

Type: Boolean

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-- Example 4
F4:=cellMap(Q3,X+->[X.1,5*X.3,4*X.2^2-2*X.3,X.3*sin(X.1)])$R4

$\label{eq25}|{\left[{0..1}, \:{0..1}, \:{0..1}\right]}\mbox{\rm - >}{\left[{\%_{1}}, \:{5 \ {\%_{3}}}, \:{-{2 \ {\%_{3}}}+{4 \ {{\%_{2}}^{2}}}}, \:{{\%_{3}}\ {\sin \left({\%_{1}}\right)}}\right]}|$

(25)

Type: CellMap?(Integer,4)

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J:=jacobianMatrix(F4)

$\label{eq26}\mbox{theMap (...)}$

(26)

Type: (List(Expression(Integer)) -> Matrix(Expression(Integer)))

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x:=coords('x,4)$R4

$\label{eq27}\left[{x_{1}}, \:{x_{2}}, \:{x_{3}}, \:{x_{4}}\right]$

(27)

Type: List(Expression(Integer))

fricas

J x

$\label{eq28}\left[ \begin{array}{ccc} 1 & 0 & 0 \ 0 & 0 & 5 \ 0 &{8 \ {x_{2}}}& - 2 \ {{x_{3}}\ {\cos \left({x_{1}}\right)}}& 0 &{\sin \left({x_{1}}\right)}$

(28)

Type: Matrix(Expression(Integer))

fricas

nullSpace (J x)

$\label{eq29}\left[ \right]$

(29)

Type: List(Vector(Expression(Integer)))

fricas

rank (J x)

$\label{eq30}3$

(30)

Type: PositiveInteger?

fricas

T:=tangentSpace(F4)$R4

$\label{eq31}\mbox{theMap (...)}$

(31)

Type: (List(Expression(Integer)) -> List(Vector(Expression(Integer))))

fricas

T x

$\label{eq32}\left[{\left[ 1, \: 0, \: 0, \:{{x_{3}}\ {\cos \left({x_{1}}\right)}}\right]}, \:{\left[ 0, \: 0, \:{8 \ {x_{2}}}, \: 0 \right]}, \:{\left[ 0, \: 5, \: - 2, \:{\sin \left({x_{1}}\right)}\right]}\right]$

(32)

Type: List(Vector(Expression(Integer)))

fricas

test(J x = matrix [[1,0,0],[0,0,5],[0,8*x.2,-2],[x.3*cos(x.1),0,sin(x.1)]])

$\label{eq33} \mbox{\rm true}$

(33)

Type: Boolean

fricas

test( rank (J x) = 3)

$\label{eq34} \mbox{\rm true}$

(34)

Type: Boolean

fricas

test( J x = transpose matrix (T x))

$\label{eq35} \mbox{\rm true}$

(35)

Type: Boolean

fricas

-- Example 5
F5:=cellMap(Q3,X+->[5*X.2,4*X.1^2-2*sin(X.2*X.3),X.2*X.3])$R3

$\label{eq36}|{\left[{0..1}, \:{0..1}, \:{0..1}\right]}\mbox{\rm - >}{\left[{5 \ {\%_{2}}}, \:{-{2 \ {\sin \left({{\%_{2}}\ {\%_{3}}}\right)}}+{4 \ {{\%_{1}}^{2}}}}, \:{{\%_{2}}\ {\%_{3}}}\right]}|$

(36)

Type: CellMap?(Integer,3)

fricas

J:=jacobianMatrix(F5)

$\label{eq37}\mbox{theMap (...)}$

(37)

Type: (List(Expression(Integer)) -> Matrix(Expression(Integer)))

fricas

x:=coords('x,3)$R3

$\label{eq38}\left[{x_{1}}, \:{x_{2}}, \:{x_{3}}\right]$

(38)

Type: List(Expression(Integer))

fricas

J x

$\label{eq39}\left[ \begin{array}{ccc} 0 & 5 & 0 \ {8 \ {x_{1}}}& -{2 \ {x_{3}}\ {\cos \left({{x_{2}}\ {x_{3}}}\right)}}& -{2 \ {x_{2}}\ {\cos \left({{x_{2}}\ {x_{3}}}\right)}} \ 0 &{x_{3}}&{x_{2}}$

(39)

Type: Matrix(Expression(Integer))

fricas

determinant (J x)

$\label{eq40}-{{40}\ {x_{1}}\ {x_{2}}}$

(40)

Type: Expression(Integer)

fricas

M:=[[0,5,0],[8*x.1,-2*x.3*cos(x.2*x.3),-2*x.2*cos(x.2*x.3)],[0,x.3,x.2]]

$\label{eq41}\begin{array}{@{}l} \displaystyle \left[{\left[ 0, \: 5, \: 0 \right]}, \:{\left[{8 \ {x_{1}}}, \: -{2 \ {x_{3}}\ {\cos \left({{x_{2}}\ {x_{3}}}\right)}}, \: -{2 \ {x_{2}}\ {\cos \left({{x_{2}}\ {x_{3}}}\right)}}\right]}, \: \right. \ \ \displaystyle \left.{\left[ 0, \:{x_{3}}, \:{x_{2}}\right]}\right]$