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fricas
)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:=lo(x.d.i) else s:X:=hi(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 ["|",x.d::OF," -> ",r::OF,"|"]
fricas
)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 == FreeAbelianGroup(CMAP) add
Rep:=FreeAbelianGroup(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::%
fricas
Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/9220523246559970704-25px.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: 0.10 SEC.
compiling exported cellMap : (List Segment Expression R,List Expression R -> List Expression R) -> $ Time: 0.01 SEC.
compiling local faceLoHi : ($,NonNegativeInteger,Boolean) -> $ Time: 0.02 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: 0.01 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: 0.07 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: 0.06 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: 0.09 SEC.
compiling exported coerce : $ -> OutputForm Time: 0.01 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: 0.37 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 23 DEC 2016 03:21:17 AM):
; /var/aw/var/LatexWiki/CMAP.NRLIB/CMAP.fasl written ; compilation finished in 0:00:00.084 ------------------------------------------------------------------------ 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 (AbelianGroup) (Module (Integer)) (FreeAbelianMonoidCategory (CellMap R n) (Integer)) (CATEGORY package (IF (has (CellMap R n) (OrderedSet)) (ATTRIBUTE (OrderedSet)) noBranch))) 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: 0.05 SEC.
compiling exported construct : (List Segment Expression R,List Expression R -> List Expression R) -> $ Time: 0.01 SEC.
(time taken in buildFunctor: 0)
;;; *** |SurfaceComplex| REDEFINED
;;; *** |SurfaceComplex| REDEFINED Time: 0 SEC.
Cumulative Statistics for Constructor SurfaceComplex Time: 0.06 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 (|Module| (|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 23 DEC 2016 03:21:18 AM):
; /var/aw/var/LatexWiki/SCMPLX.NRLIB/SCMPLX.fasl written ; compilation finished in 0:00:00.032 ------------------------------------------------------------------------ SurfaceComplex is now explicitly exposed in frame initial SurfaceComplex will be automatically loaded when needed from /var/aw/var/LatexWiki/SCMPLX.NRLIB/SCMPLX

fricas
)clear all
All user variables and function definitions have been cleared. R ==> EXPR INT
Type: Void
fricas
OF ==> OutputForm
Type: Void
fricas
-- Cell map
R2 ==> CellMap(INT,2)
Type: Void
fricas
R3 ==> CellMap(INT,3)
Type: Void
fricas
R4 ==> CellMap(INT,4)
Type: Void
fricas
Q2 ==> [0..1,0..1::R]
Type: Void
fricas
Q3 ==> concat(Q2,[0..1::R])
Type: Void
fricas
--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)
fricas
J:=jacobianMatrix(F1)

\label{eq2}\mbox{theMap (...)}(2)
Type: (List(Expression(Integer)) -> Matrix(Expression(Integer)))
fricas
x:=coords('x,2)$R2

\label{eq3}\left[{x_{1}}, \:{x_{2}}\right](3)
Type: List(Expression(Integer))
fricas
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))
fricas
determinant(J x)

\label{eq5}{2 \ {x_{1}}\ {x_{2}}\ {\cos \left({x_{2}}\right)}}-{5 \ {{x_{1}}^{2}}}(5)
Type: Expression(Integer)
fricas
test(J x = matrix [[2*x.1*x.2,x.1^2],[5,cos(x.2)]])

\label{eq6} \mbox{\rm true} (6)
Type: Boolean
fricas
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)
fricas
J:=jacobianMatrix(F2)

\label{eq9}\mbox{theMap (...)}(9)
Type: (List(Expression(Integer)) -> Matrix(Expression(Integer)))
fricas
x:=[r::R,phi::R]

\label{eq10}\left[ r , \: phi \right](10)
Type: List(Expression(Integer))
fricas
(getMap F2) x

\label{eq11}\left[{r \ {\cos \left({phi}\right)}}, \:{r \ {\sin \left({phi}\right)}}\right](11)
Type: List(Expression(Integer))
fricas
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))
fricas
determinant(J x)

\label{eq13}{r \ {{\sin \left({phi}\right)}^{2}}}+{r \ {{\cos \left({phi}\right)}^{2}}}(13)
Type: Expression(Integer)
fricas
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
fricas
test( normalize determinant(J x) = x.1)

\label{eq15} \mbox{\rm true} (15)
Type: Boolean
fricas
-- 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)))
fricas
z:=[r::R,th::R,phi::R]

\label{eq18}\left[ r , \: th , \: phi \right](18)
Type: List(Expression(Integer))
fricas
(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))
fricas
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))
fricas
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)
fricas
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)))
fricas
test( J z = matrix M)

\label{eq23} \mbox{\rm true} (23)
Type: Boolean
fricas
test( simplify determinant(J z) = z.1^2*sin(z.2) )

\label{eq24} \mbox{\rm true} (24)
Type: Boolean
fricas
-- 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)
fricas
J:=jacobianMatrix(F4)

\label{eq26}\mbox{theMap (...)}(26)
Type: (List(Expression(Integer)) -> Matrix(Expression(Integer)))
fricas
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)
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] 
(41)
Type: List(List(Expression(Integer)))
fricas
T:=tangentSpace(F5)$R3

\label{eq42}\mbox{theMap (...)}(42)
Type: (List(Expression(Integer)) -> List(Vector(Expression(Integer))))
fricas
test(J x = matrix M)

\label{eq43} \mbox{\rm true} (43)
Type: Boolean
fricas
test(determinant (J x) = -40*x.1*x.2)

\label{eq44} \mbox{\rm true} (44)
Type: Boolean
fricas
test( J x = transpose matrix (T x))

\label{eq45} \mbox{\rm true} (45)
Type: Boolean




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