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)abbrev domain RSPACE RealSpace
++ Author: kfp
++ Date Created: Thu Nov 06 03:51:56 CET 2014
++ License: BSD (same as Axiom)
++ Date Last Updated:
++ Basic Operations:
++ Related Domains:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ Examples:
++ References:
++
++ Description:
++
++
RealSpace(n) : Exports == Implementation where
NNI ==> NonNegativeInteger
PI  ==> PositiveInteger
I   ==> Integer
R   ==> Expression Integer -- Real
n:NNI
Exports == Join(VectorSpace R , CoercibleTo OutputForm) with
coerce : % -> OutputForm
coerce : List R -> %
coerce : Vector R -> %
coerce : List Symbol -> %
coerce : R -> %
coerce : % -> List R
coerce : % -> Vector R
dot : (%,%) -> R
dim : % -> NNI
"+" : (%,%) -> %
"-" : (%,%) -> %
"*" : (R,%) -> %
"*" : (NNI,%) -> %
"*" : (PI,%)  -> %
"*" : (I,%)   -> %
"/" : (%,R)   -> %
"#" : % -> NNI
"-" : % -> %
"=" : (%,%) -> Boolean
"~=": (%,%) -> Boolean
unitVector : PI -> %
elt : (%,I) -> R
eval: (%, Equation R) -> %
eval: (%, List Equation R) -> %
every?: (R -> Boolean, %) -> Boolean
map: (R -> R, %) -> %  -- strange, to check
member?: (R, %) -> Boolean
any?: (R -> Boolean, %) -> Boolean
copy: % -> %
D: (%, Symbol) -> %
D: (%, Symbol, NonNegativeInteger) -> %
D: (%, List Symbol) -> %
D: (%, List Symbol, List NonNegativeInteger) -> %
dist : (%,%) -> R
norm : % -> R
tprod : (%,%) -> Matrix(R)   -- tensor prod as Matrix Real
-- _*_* : (%,%) -> RealSpace(n^2)
cross : List(%) -> %
Rep := DirectProduct(n,R)
coerce(l:List R):% ==
#l=n => directProduct((vector l)::Vector(R))$Rep coerce(l:Vector R):% == #l=n => directProduct(l)$Rep
coerce(l:List Symbol):% ==
#l=n => coerce([r::R for r in l])
coerce(x:R):% == x*unitVector(1)$Rep coerce(v:%):List R == [v(j) for j in 1..#v] coerce(v:%):Vector R == vector [v(j) for j in 1..#v] dim(x:%):NNI == #(x::List R) -- n dist(x:%,y:%):R == sqrt dot(x-y,x-y) norm(x:%):R == sqrt dot(x,x) tprod(x:%,y:%):Matrix(R) == matrix [[x.i*(y.j)::R for j in 1..n] for i in 1..n] --wprod(x:%,y:%):Matrix(R) == (tprod(x,y)-tprod(y,x))/2::R --_*_*(x:%,y:%):RealSpace(n^2) == -- X:List Real:=[] -- for i in 1..n repeat -- X:=append(X,[x.i*(y.j)::Real for j in 1..n]) -- X::RealSpace(n^2) -- local ; better coerce in ??? vec2list(v:Vector(R)):List(R) == [v.j for j in 1..#v] cross(X:List %):% == #X ~= n-1 => error "Wrong number of arguments." L:List(List R):=[r::List(R) for r in X] M:Matrix(R):= transpose matrix L S:List(List R):=[vec2list(row(M,i)) for i in 1..n] LS:List(Matrix R):=[matrix([S.i for i in 1..n | i~=j]) for j in 1..n] r:List R:=[(-1)^(i-1)::R*determinant(LS.i) for i in 1..n] r::Rep )abbrev package INTVAL Interval ++ Author: kfp ++ Date Created: Thu Nov 06 02:46:09 CET 2014 ++ License: BSD (same as Axiom) ++ Date Last Updated: ++ Basic Operations: ++ Related Domains: ++ Also See: ++ AMS Classifications: ++ Keywords: ++ Examples: ++ References: ++ ++ Description: ++ ++ Interval(F:Field) : Exports == Implementation where OF ==> OutputForm PI ==> PositiveInteger NNI ==> NonNegativeInteger Q ==> Fraction Integer Exports == Join(CoercibleTo OutputForm) with coerce : Tuple F -> % coerce : List F -> % coerce : % -> List F coerce : % -> OutputForm closedInterval : (F,F) -> % openInterval : (F,F) -> % leftOpenInterval : (F,F) -> % rightOpenInterval : (F,F) -> % lb : % -> F ub : % -> F topology : % -> OutputForm measure : % -> F --"*" : (%,%) -> Cell(F,2) "*" : (F,%) -> % "+" : (F,%) -> % Implementation == add Rep:= Record(lb:F,ub:F,tp:String) --check(x:%):% == --if number?(x.lb) and number?(x.ub) then --(x.ub)::Q < (x.lb)::Q => error "a>b" --x coerce(l:Tuple F):% == -- does not work ???? length l ~=2 => error "Form expected: (a,b)" [select(l,0), select(l,1),"open"]::Rep coerce(l:List F):% == #l ~=2 => error "Form expected: [a,b]" [l.1,l.2,"closed"]::Rep coerce(x:%):List F == [x.lb,x.ub] coerce(x:%):OF == x.tp = "closed" => hconcat ["[",x.lb::OF,",",x.ub::OF,"]"] x.tp = "open" => hconcat ["(",x.lb::OF,",",x.ub::OF,")"] x.tp = "leftOpen" => hconcat ["(",x.lb::OF,",",x.ub::OF,"]"] x.tp = "rightOpen" => hconcat ["[",x.lb::OF,",",x.ub::OF,")"] lb(x) == x.lb ub(x) == x.ub topology(x:%):OF == x.tp::Symbol::OF measure(x:%):F == x.ub - x.lb closedInterval(a,b) == [a,b,"closed"]::Rep openInterval(a,b) == [a,b,"open"]::Rep leftOpenInterval(a,b) == [a,b,"leftOpen"]::Rep rightOpenInterval(a,b) == [a,b,"rightOpen"]::Rep --x:% * y:% == --m:Matrix(F):=matrix([[lb x,ub x],[lb y,ub y]]) --m::Cell(F,2) a:F + x:% == [a+x.lb,a+x.ub,x.tp]::Rep a:F * x:% == [a*x.lb,a*x.ub,x.tp]::Rep )abbrev domain CELL Cell ++ Author: kfp ++ Date Created: Thu Nov 06 02:39:44 CET 2014 ++ License: BSD (same as Axiom) ++ Date Last Updated: ++ Basic Operations: ++ Related Domains: ++ Also See: ++ AMS Classifications: ++ Keywords: ++ Examples: ++ References: ++ ++ Description: ++ ++ Cell(k,F) : Exports == Implementation where NNI ==> NonNegativeInteger OF ==> OutputForm F: Field k: NNI Exports == Join(CoercibleTo OutputForm) with coerce : Matrix F -> % coerce : % -> List Interval F -- ex. c::List(Interval RR) coerce : List Interval F -> % coerce : % -> OutputForm measure : % -> F Implementation == Matrix F add Rep := Matrix F coerce(c:%):List Interval F == lc:List(List F):=listOfLists(c) #lc ~= k => error "Wrong dimension" [i::Interval(F) for i in lc] coerce(l:List Interval F):% == matrix [i::List(F) for i in l] coerce(m:Matrix F):% == ncols m ~= 2 => error "Expecting (k x 2) matrix" nrows m ~= k => error "Expecting (k x 2) matrix" m coerce(c:%):OF == (c::List Interval F)::OF measure(c:%):F == L:=c::List Interval F mL:=[measure(v) for v in L] determinant(diagonalMatrix(mL)) -- pi:=%pi$RR
)abbrev domain SURF Surface
++ Author: kfp
++ Date Created: Tue Nov 11 21:57:52 CET 2014
++ License: BSD (same as Axiom)
++ Date Last Updated:
++ Basic Operations:
++ Related Domains:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ Examples:
++ References:
++
++ Description:
++ k-surface
++
Surface(k,n) : Exports == Implementation where
PYI   ==> Polynomial(Integer)
NNI   ==> NonNegativeInteger
SYM   ==> Symbol
PI    ==> PositiveInteger
OF    ==> OutputForm
X     ==> RealSpace(k) -> RealSpace(n)
R     ==> Expression Integer
k:PI
n:PI
--V:Vectorspace(Real)
Exports == Join(CoercibleTo OutputForm) with
coerce : % -> OutputForm
coerce : X -> %
dim    : % -> PI
codim  : % -> PI
eval   : (%, List R) -> RealSpace(n)
eval   : (%, List Symbol) -> RealSpace(n)
JacobianMatrix : (%, List Symbol) -> Matrix(R)
TangentSpace : (%, List Symbol) -> List RealSpace(n)
CoTangentSpace : (%, List Symbol) -> List RealSpace(n)
normal : (%, List Symbol) -> RealSpace(n)
Rep := Record(kk:PI,nn:PI,ff:X)
coerce(f:X):% == n>k and k>0 => [k,n,f]::Rep
coerce(S:%):OF == hconcat [S.kk::OF,"-surface in R^"::SYM::OF,S.nn::OF]
dim(S:%):PI == S.kk
codim(S:%):PI == (S.nn - S.kk)::PI
eval(S:%,x:List R) == (S.ff)(x::RealSpace(k))
eval(S:%,x:List Symbol) == (S.ff)(x::RealSpace(k))
JacobianMatrix(S:%, x:List Symbol):Matrix(R) ==
transpose matrix [D((S.ff)(x::RealSpace(k)),u)::List(R) for u in x]
-- local ; better coerce in ???
vec2list(v:Vector(R)):List(R) == [v.j for j in 1..#v]
TangentSpace(S:%, x:List Symbol):List RealSpace(n) ==
J:=JacobianMatrix(S,x)
[vec2list(column(J,j))::RealSpace(n) for j in 1..ncols(J)]
CoTangentSpace(S:%, x:List Symbol):List RealSpace(n) ==
ts:=TangentSpace(S,x)
rd:=[[0::R for i in 1..n]::RealSpace(n) for j in k+1..n]
Q:=append(ts,rd)
QR:=[v::List R for v in Q]
M:=(matrix QR)::Matrix(R)
[vec2list(y)::RealSpace(n) for y in nullSpace(M)]
normal(S:%, x:List Symbol):RealSpace(n) ==
cs:=CoTangentSpace(S,x)
#cs ~= 1 => error "Not a hypersurface"
v:=cs.1
r:= sqrt(dot(v,v))
r = 0 => error "No normal; zero vector"
v/r
   Compiling FriCAS source code from file
using old system compiler.
RSPACE abbreviates domain RealSpace
------------------------------------------------------------------------
initializing NRLIB RSPACE for RealSpace
compiling into NRLIB RSPACE
compiling exported coerce : List Expression Integer -> $Time: 0.05 SEC. compiling exported coerce : Vector Expression Integer ->$
Time: 0 SEC.
compiling exported coerce : List Symbol -> $Time: 0.01 SEC. compiling exported coerce : Expression Integer ->$
Time: 0.02 SEC.
compiling exported coerce : $-> List Expression Integer Time: 0.01 SEC. compiling exported coerce :$ -> Vector Expression Integer
Time: 0 SEC.
compiling exported dim : $-> NonNegativeInteger Time: 0.01 SEC. compiling exported dist : ($,$) -> Expression Integer Time: 0 SEC. compiling exported norm :$ -> Expression Integer
Time: 0 SEC.
compiling exported tprod : ($,$) -> Matrix Expression Integer
Time: 0.02 SEC.
compiling local vec2list : Vector Expression Integer -> List Expression Integer
Time: 0 SEC.
compiling exported cross : List $->$
Time: 0.06 SEC.
(time taken in buildFunctor:  0)
;;;     ***       |RealSpace| REDEFINED
;;;     ***       |RealSpace| REDEFINED
Time: 0.04 SEC.
Cumulative Statistics for Constructor RealSpace
Time: 0.22 seconds
--------------non extending category----------------------
.. RealSpace(#1) of cat
(|Join| (|VectorSpace| (|Expression| (|Integer|)))
(|CoercibleTo| (|OutputForm|))
(CATEGORY |domain| (SIGNATURE |coerce| ((|OutputForm|) $)) (SIGNATURE |coerce| ($ (|List| (|Expression| (|Integer|)))))
(SIGNATURE |coerce| ($(|Vector| (|Expression| (|Integer|))))) (SIGNATURE |coerce| ($ (|List| (|Symbol|))))
(SIGNATURE |coerce| ($(|Expression| (|Integer|)))) (SIGNATURE |coerce| ((|List| (|Expression| (|Integer|)))$))
(SIGNATURE |coerce| ((|Vector| (|Expression| (|Integer|))) $)) (SIGNATURE |dot| ((|Expression| (|Integer|))$ $)) (SIGNATURE |dim| ((|NonNegativeInteger|)$)) (SIGNATURE + ( $)) (SIGNATURE - ($ )) (SIGNATURE * ($(|Expression| (|Integer|))$))
(SIGNATURE * ($(|NonNegativeInteger|)$))
(SIGNATURE * ($(|PositiveInteger|)$))
(SIGNATURE * ($(|Integer|)$))
(SIGNATURE / ( (|Expression| (|Integer|))))
(SIGNATURE |#| ((|NonNegativeInteger|) $)) (SIGNATURE - ($ $)) (SIGNATURE = ((|Boolean|)$ $)) (SIGNATURE ~= ((|Boolean|)$ $)) (SIGNATURE |unitVector| ($ (|PositiveInteger|)))
(SIGNATURE |elt| ((|Expression| (|Integer|)) $(|Integer|))) (SIGNATURE |eval| ($ $(|Equation| (|Expression| (|Integer|))))) (SIGNATURE |eval| ($ $(|List| (|Equation| (|Expression| (|Integer|)))))) (SIGNATURE |every?| ((|Boolean|) (|Mapping| (|Boolean|) (|Expression| (|Integer|)))$))
(SIGNATURE |map|
($(|Mapping| (|Expression| (|Integer|)) (|Expression| (|Integer|)))$))
(SIGNATURE |member?| ((|Boolean|) (|Expression| (|Integer|)) $)) (SIGNATURE |any?| ((|Boolean|) (|Mapping| (|Boolean|) (|Expression| (|Integer|)))$))
(SIGNATURE |copy| ()) (SIGNATURE D ( (|Symbol|)))
(SIGNATURE D ( (|Symbol|) (|NonNegativeInteger|)))
(SIGNATURE D ( (|List| (|Symbol|))))
(SIGNATURE D
( (|List| (|Symbol|)) (|List| (|NonNegativeInteger|))))
(SIGNATURE |dist| ((|Expression| (|Integer|)) ))
(SIGNATURE |norm| ((|Expression| (|Integer|)) $)) (SIGNATURE |tprod| ((|Matrix| (|Expression| (|Integer|)))$ $)) (SIGNATURE |cross| ($ (|List| $))))) has no (|DirectProductCategory| |#1| (|Expression| (|Integer|))) finalizing NRLIB RSPACE Processing RealSpace for Browser database: --------constructor--------- --->-->RealSpace((coerce ((OutputForm) %))): Not documented!!!! --->-->RealSpace((coerce (% (List (Expression (Integer)))))): Not documented!!!! --->-->RealSpace((coerce (% (Vector (Expression (Integer)))))): Not documented!!!! --->-->RealSpace((coerce (% (List (Symbol))))): Not documented!!!! --->-->RealSpace((coerce (% (Expression (Integer))))): Not documented!!!! --->-->RealSpace((coerce ((List (Expression (Integer))) %))): Not documented!!!! --->-->RealSpace((coerce ((Vector (Expression (Integer))) %))): Not documented!!!! --->-->RealSpace((dot ((Expression (Integer)) % %))): Not documented!!!! --->-->RealSpace((dim ((NonNegativeInteger) %))): Not documented!!!! --->-->RealSpace((+ (% % %))): Not documented!!!! --->-->RealSpace((- (% % %))): Not documented!!!! --->-->RealSpace((* (% (Expression (Integer)) %))): Not documented!!!! --->-->RealSpace((* (% (NonNegativeInteger) %))): Not documented!!!! --->-->RealSpace((* (% (PositiveInteger) %))): Not documented!!!! --->-->RealSpace((* (% (Integer) %))): Not documented!!!! --->-->RealSpace((/ (% % (Expression (Integer))))): Not documented!!!! --->-->RealSpace((# ((NonNegativeInteger) %))): Not documented!!!! --->-->RealSpace((- (% %))): Not documented!!!! --->-->RealSpace((= ((Boolean) % %))): Not documented!!!! --->-->RealSpace((~= ((Boolean) % %))): Not documented!!!! --->-->RealSpace((unitVector (% (PositiveInteger)))): Not documented!!!! --->-->RealSpace((elt ((Expression (Integer)) % (Integer)))): Not documented!!!! --->-->RealSpace((eval (% % (Equation (Expression (Integer)))))): Not documented!!!! --->-->RealSpace((eval (% % (List (Equation (Expression (Integer))))))): Not documented!!!! --->-->RealSpace((every? ((Boolean) (Mapping (Boolean) (Expression (Integer))) %))): Not documented!!!! --->-->RealSpace((map (% (Mapping (Expression (Integer)) (Expression (Integer))) %))): Not documented!!!! --->-->RealSpace((member? ((Boolean) (Expression (Integer)) %))): Not documented!!!! --->-->RealSpace((any? ((Boolean) (Mapping (Boolean) (Expression (Integer))) %))): Not documented!!!! --->-->RealSpace((copy (% %))): Not documented!!!! --->-->RealSpace((D (% % (Symbol)))): Not documented!!!! --->-->RealSpace((D (% % (Symbol) (NonNegativeInteger)))): Not documented!!!! --->-->RealSpace((D (% % (List (Symbol))))): Not documented!!!! --->-->RealSpace((D (% % (List (Symbol)) (List (NonNegativeInteger))))): Not documented!!!! --->-->RealSpace((dist ((Expression (Integer)) % %))): Not documented!!!! --->-->RealSpace((norm ((Expression (Integer)) %))): Not documented!!!! --->-->RealSpace((tprod ((Matrix (Expression (Integer))) % %))): Not documented!!!! --->-->RealSpace((cross (% (List %)))): Not documented!!!! ; compiling file "/var/aw/var/LatexWiki/RSPACE.NRLIB/RSPACE.lsp" (written 23 NOV 2014 02:37:55 AM): ; /var/aw/var/LatexWiki/RSPACE.NRLIB/RSPACE.fasl written ; compilation finished in 0:00:00.071 ------------------------------------------------------------------------ RealSpace is now explicitly exposed in frame initial RealSpace will be automatically loaded when needed from /var/aw/var/LatexWiki/RSPACE.NRLIB/RSPACE INTVAL abbreviates package Interval ------------------------------------------------------------------------ initializing NRLIB INTVAL for Interval compiling into NRLIB INTVAL Local variable Rep type redefined: (Join (SetCategory) (CATEGORY domain (SIGNATURE construct ((Record (: lb F) (: ub F) (: tp (String))) F F (String))) (SIGNATURE ~= ((Boolean) (Record (: lb F) (: ub F) (: tp (String))) (Record (: lb F) (: ub F) (: tp (String))))) (SIGNATURE coerce ((OutputForm) (Record (: lb F) (: ub F) (: tp (String))))) (SIGNATURE elt (F (Record (: lb F) (: ub F) (: tp (String))) lb)) (SIGNATURE elt (F (Record (: lb F) (: ub F) (: tp (String))) ub)) (SIGNATURE elt ((String) (Record (: lb F) (: ub F) (: tp (String))) tp)) (SIGNATURE setelt! (F (Record (: lb F) (: ub F) (: tp (String))) lb F)) (SIGNATURE setelt! (F (Record (: lb F) (: ub F) (: tp (String))) ub F)) (SIGNATURE setelt! ((String) (Record (: lb F) (: ub F) (: tp (String))) tp (String))) (SIGNATURE copy ((Record (: lb F) (: ub F) (: tp (String))) (Record (: lb F) (: ub F) (: tp (String))))))) to (DirectProductCategory n (Expression (Integer))) compiling exported coerce : Tuple F ->$
Time: 0.01 SEC.
compiling exported coerce : List F -> $Time: 0 SEC. compiling exported coerce :$ -> List F
Time: 0 SEC.
compiling exported coerce : $-> OutputForm Time: 0.01 SEC. compiling exported lb :$ -> F
INTVAL;lb;$F;5 is replaced by QVELTx0 Time: 0 SEC. compiling exported ub :$ -> F
INTVAL;ub;$F;6 is replaced by QVELTx1 Time: 0 SEC. compiling exported topology :$ -> OutputForm
Time: 0.01 SEC.
compiling exported measure : $-> F Time: 0 SEC. compiling exported closedInterval : (F,F) ->$
INTVAL;closedInterval;2F$;9 is replaced by VECTORabclosed Time: 0 SEC. compiling exported openInterval : (F,F) ->$
INTVAL;openInterval;2F$;10 is replaced by VECTORabopen Time: 0 SEC. compiling exported leftOpenInterval : (F,F) ->$
INTVAL;leftOpenInterval;2F$;11 is replaced by VECTORableftOpen Time: 0 SEC. compiling exported rightOpenInterval : (F,F) ->$
INTVAL;rightOpenInterval;2F$;12 is replaced by VECTORabrightOpen Time: 0 SEC. compiling exported + : (F,$) -> $Time: 0 SEC. compiling exported * : (F,$) -> $Time: 0 SEC. (time taken in buildFunctor: 0) ;;; *** |Interval| REDEFINED ;;; *** |Interval| REDEFINED Time: 0 SEC. Warnings: [1] coerce: tp has no value [2] topology: tp has no value [3] +: tp has no value [4] *: tp has no value Cumulative Statistics for Constructor Interval Time: 0.03 seconds finalizing NRLIB INTVAL Processing Interval for Browser database: --------constructor--------- --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((coerce ((OutputForm) %))): Not documented!!!! --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((coerce (% (List (Expression (Integer)))))): Not documented!!!! --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((coerce (% (Vector (Expression (Integer)))))): Not documented!!!! --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((coerce (% (List (Symbol))))): Not documented!!!! --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((coerce (% (Expression (Integer))))): Not documented!!!! --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((coerce ((List (Expression (Integer))) %))): Not documented!!!! --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((coerce ((Vector (Expression (Integer))) %))): Not documented!!!! --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((dot ((Expression (Integer)) % %))): Not documented!!!! --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((dim ((NonNegativeInteger) %))): Not documented!!!! --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((+ (% % %))): Not documented!!!! --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((- (% % %))): Not documented!!!! --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((* (% (Expression (Integer)) %))): Not documented!!!! --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((* (% (NonNegativeInteger) %))): Not documented!!!! --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((* (% (PositiveInteger) %))): Not documented!!!! --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((* (% (Integer) %))): Not documented!!!! --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((/ (% % (Expression (Integer))))): Not documented!!!! --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((# ((NonNegativeInteger) %))): Not documented!!!! --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((- (% %))): Not documented!!!! --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((= ((Boolean) % %))): Not documented!!!! --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((~= ((Boolean) % %))): Not documented!!!! --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((unitVector (% (PositiveInteger)))): Not documented!!!! --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((elt ((Expression (Integer)) % (Integer)))): Not documented!!!! --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((eval (% % (Equation (Expression (Integer)))))): Not documented!!!! --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((eval (% % (List (Equation (Expression (Integer))))))): Not documented!!!! --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((every? ((Boolean) (Mapping (Boolean) (Expression (Integer))) %))): Not documented!!!! --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((map (% (Mapping (Expression (Integer)) (Expression (Integer))) %))): Not documented!!!! --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((member? ((Boolean) (Expression (Integer)) %))): Not documented!!!! --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((any? ((Boolean) (Mapping (Boolean) (Expression (Integer))) %))): Not documented!!!! --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((copy (% %))): Not documented!!!! --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((D (% % (Symbol)))): Not documented!!!! --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((D (% % (Symbol) (NonNegativeInteger)))): Not documented!!!! --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((D (% % (List (Symbol))))): Not documented!!!! --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((D (% % (List (Symbol)) (List (NonNegativeInteger))))): Not documented!!!! --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((dist ((Expression (Integer)) % %))): Not documented!!!! --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((norm ((Expression (Integer)) %))): Not documented!!!! --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((tprod ((Matrix (Expression (Integer))) % %))): Not documented!!!! --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((cross (% (List %)))): Not documented!!!! --------constructor--------- --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((coerce (% (Tuple F)))): Not documented!!!! --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((coerce (% (List F)))): Not documented!!!! --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((coerce ((List F) %))): Not documented!!!! --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((coerce ((OutputForm) %))): Not documented!!!! --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((closedInterval (% F F))): Not documented!!!! --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((openInterval (% F F))): Not documented!!!! --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((leftOpenInterval (% F F))): Not documented!!!! --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((rightOpenInterval (% F F))): Not documented!!!! --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((lb (F %))): Not documented!!!! --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((ub (F %))): Not documented!!!! --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((topology ((OutputForm) %))): Not documented!!!! --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((measure (F %))): Not documented!!!! --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((* (% F %))): Not documented!!!! --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval((+ (% F %))): Not documented!!!! --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/INTRVL.spad-->Interval(): Spurious comments: \blankline \blankline ; compiling file "/var/aw/var/LatexWiki/INTVAL.NRLIB/INTVAL.lsp" (written 23 NOV 2014 02:37:55 AM): ; /var/aw/var/LatexWiki/INTVAL.NRLIB/INTVAL.fasl written ; compilation finished in 0:00:00.043 ------------------------------------------------------------------------ Interval is now explicitly exposed in frame initial Interval will be automatically loaded when needed from /var/aw/var/LatexWiki/INTVAL.NRLIB/INTVAL CELL abbreviates domain Cell ------------------------------------------------------------------------ initializing NRLIB CELL for Cell compiling into NRLIB CELL Local variable Rep type redefined: (Join (MatrixCategory F (Vector F) (Vector F)) (CATEGORY domain (SIGNATURE diagonalMatrix ($ (Vector F))) (IF (has F (ConvertibleTo (InputForm))) (ATTRIBUTE (ConvertibleTo (InputForm))) noBranch) (IF (has F (Field)) (SIGNATURE inverse ((Union $failed)$)) noBranch))) to (Join (SetCategory) (CATEGORY domain (SIGNATURE construct ((Record (: lb F) (: ub F) (: tp (String))) F F (String))) (SIGNATURE ~= ((Boolean) (Record (: lb F) (: ub F) (: tp (String))) (Record (: lb F) (: ub F) (: tp (String))))) (SIGNATURE coerce ((OutputForm) (Record (: lb F) (: ub F) (: tp (String))))) (SIGNATURE elt (F (Record (: lb F) (: ub F) (: tp (String))) lb)) (SIGNATURE elt (F (Record (: lb F) (: ub F) (: tp (String))) ub)) (SIGNATURE elt ((String) (Record (: lb F) (: ub F) (: tp (String))) tp)) (SIGNATURE setelt! (F (Record (: lb F) (: ub F) (: tp (String))) lb F)) (SIGNATURE setelt! (F (Record (: lb F) (: ub F) (: tp (String))) ub F)) (SIGNATURE setelt! ((String) (Record (: lb F) (: ub F) (: tp (String))) tp (String))) (SIGNATURE copy ((Record (: lb F) (: ub F) (: tp (String))) (Record (: lb F) (: ub F) (: tp (String)))))))
compiling exported coerce : $-> List Interval F Time: 0 SEC. compiling exported coerce : List Interval F ->$
Time: 0.01 SEC.
compiling exported coerce : Matrix F -> $Time: 0 SEC. compiling exported coerce :$ -> OutputForm
Time: 0 SEC.
compiling exported measure : $-> F Time: 0 SEC. (time taken in buildFunctor: 0) ;;; *** |Cell| REDEFINED ;;; *** |Cell| REDEFINED Time: 0.01 SEC. Cumulative Statistics for Constructor Cell Time: 0.02 seconds --------------non extending category---------------------- .. Cell(#1,#2) of cat (|Join| (|CoercibleTo| (|OutputForm|)) (CATEGORY |domain| (SIGNATURE |coerce| ($ (|Matrix| |#2|)))
(SIGNATURE |coerce| ((|List| (|Interval| |#2|)) $)) (SIGNATURE |coerce| ($ (|List| (|Interval| |#2|))))
(SIGNATURE |coerce| ((|OutputForm|) $)) (SIGNATURE |measure| (|#2|$))))   has no
(|MatrixCategory| |#2| (|Vector| |#2|) (|Vector| |#2|))    finalizing NRLIB CELL
Processing Cell for Browser database:
--------constructor---------
--->-->Cell((coerce ((OutputForm) %))): Not documented!!!!
--->-->Cell((coerce (% (List (Expression (Integer)))))): Not documented!!!!
--->-->Cell((coerce (% (Vector (Expression (Integer)))))): Not documented!!!!
--->-->Cell((coerce (% (List (Symbol))))): Not documented!!!!
--->-->Cell((coerce (% (Expression (Integer))))): Not documented!!!!
--->-->Cell((coerce ((List (Expression (Integer))) %))): Not documented!!!!
--->-->Cell((coerce ((Vector (Expression (Integer))) %))): Not documented!!!!
--->-->Cell((dot ((Expression (Integer)) % %))): Not documented!!!!
--->-->Cell((dim ((NonNegativeInteger) %))): Not documented!!!!
--->-->Cell((+ (% % %))): Not documented!!!!
--->-->Cell((- (% % %))): Not documented!!!!
--->-->Cell((* (% (Expression (Integer)) %))): Not documented!!!!
--->-->Cell((* (% (NonNegativeInteger) %))): Not documented!!!!
--->-->Cell((* (% (PositiveInteger) %))): Not documented!!!!
--->-->Cell((* (% (Integer) %))): Not documented!!!!
--->-->Cell((/ (% % (Expression (Integer))))): Not documented!!!!
--->-->Cell((# ((NonNegativeInteger) %))): Not documented!!!!
--->-->Cell((- (% %))): Not documented!!!!
--->-->Cell((= ((Boolean) % %))): Not documented!!!!
--->-->Cell((~= ((Boolean) % %))): Not documented!!!!
--->-->Cell((unitVector (% (PositiveInteger)))): Not documented!!!!
--->-->Cell((elt ((Expression (Integer)) % (Integer)))): Not documented!!!!
--->-->Cell((eval (% % (Equation (Expression (Integer)))))): Not documented!!!!
--->-->Cell((eval (% % (List (Equation (Expression (Integer))))))): Not documented!!!!
--->-->Cell((every? ((Boolean) (Mapping (Boolean) (Expression (Integer))) %))): Not documented!!!!
--->-->Cell((map (% (Mapping (Expression (Integer)) (Expression (Integer))) %))): Not documented!!!!
--->-->Cell((member? ((Boolean) (Expression (Integer)) %))): Not documented!!!!
--->-->Cell((any? ((Boolean) (Mapping (Boolean) (Expression (Integer))) %))): Not documented!!!!
--->-->Cell((copy (% %))): Not documented!!!!
--->-->Cell((D (% % (Symbol)))): Not documented!!!!
--->-->Cell((D (% % (Symbol) (NonNegativeInteger)))): Not documented!!!!
--->-->Cell((D (% % (List (Symbol))))): Not documented!!!!
--->-->Cell((D (% % (List (Symbol)) (List (NonNegativeInteger))))): Not documented!!!!
--->-->Cell((dist ((Expression (Integer)) % %))): Not documented!!!!
--->-->Cell((norm ((Expression (Integer)) %))): Not documented!!!!
--->-->Cell((tprod ((Matrix (Expression (Integer))) % %))): Not documented!!!!
--->-->Cell((cross (% (List %)))): Not documented!!!!
--------constructor---------
--->-->Cell((coerce (% (Tuple F)))): Not documented!!!!
--->-->Cell((coerce (% (List F)))): Not documented!!!!
--->-->Cell((coerce ((List F) %))): Not documented!!!!
--->-->Cell((coerce ((OutputForm) %))): Not documented!!!!
--->-->Cell((closedInterval (% F F))): Not documented!!!!
--->-->Cell((openInterval (% F F))): Not documented!!!!
--->-->Cell((leftOpenInterval (% F F))): Not documented!!!!
--->-->Cell((rightOpenInterval (% F F))): Not documented!!!!
--->-->Cell((lb (F %))): Not documented!!!!
--->-->Cell((ub (F %))): Not documented!!!!
--->-->Cell((topology ((OutputForm) %))): Not documented!!!!
--->-->Cell((measure (F %))): Not documented!!!!
--->-->Cell((* (% F %))): Not documented!!!!
--->-->Cell((+ (% F %))): Not documented!!!!
--------constructor---------
--->-->Cell((coerce (% (Matrix F)))): Not documented!!!!
--->-->Cell((coerce ((List (Interval F)) %))): Not documented!!!!
--->-->Cell((coerce (% (List (Interval F))))): Not documented!!!!
--->-->Cell((coerce ((OutputForm) %))): Not documented!!!!
--->-->Cell((measure (F %))): Not documented!!!!
; compiling file "/var/aw/var/LatexWiki/CELL.NRLIB/CELL.lsp" (written 23 NOV 2014 02:37:55 AM):
; /var/aw/var/LatexWiki/CELL.NRLIB/CELL.fasl written
; compilation finished in 0:00:00.024
------------------------------------------------------------------------
Cell is now explicitly exposed in frame initial
Cell will be automatically loaded when needed from
/var/aw/var/LatexWiki/CELL.NRLIB/CELL
SURF abbreviates domain Surface
------------------------------------------------------------------------
initializing NRLIB SURF for Surface
compiling into NRLIB SURF
Local variable Rep type redefined: (Join (SetCategory) (CATEGORY domain (SIGNATURE construct ((Record (: kk (PositiveInteger)) (: nn (PositiveInteger)) (: ff (Mapping (RealSpace n) (RealSpace k)))) (PositiveInteger) (PositiveInteger) (Mapping (RealSpace n) (RealSpace k)))) (SIGNATURE ~= ((Boolean) (Record (: kk (PositiveInteger)) (: nn (PositiveInteger)) (: ff (Mapping (RealSpace n) (RealSpace k)))) (Record (: kk (PositiveInteger)) (: nn (PositiveInteger)) (: ff (Mapping (RealSpace n) (RealSpace k)))))) (SIGNATURE coerce ((OutputForm) (Record (: kk (PositiveInteger)) (: nn (PositiveInteger)) (: ff (Mapping (RealSpace n) (RealSpace k)))))) (SIGNATURE elt ((PositiveInteger) (Record (: kk (PositiveInteger)) (: nn (PositiveInteger)) (: ff (Mapping (RealSpace n) (RealSpace k)))) kk)) (SIGNATURE elt ((PositiveInteger) (Record (: kk (PositiveInteger)) (: nn (PositiveInteger)) (: ff (Mapping (RealSpace n) (RealSpace k)))) nn)) (SIGNATURE elt ((Mapping (RealSpace n) (RealSpace k)) (Record (: kk (PositiveInteger)) (: nn (PositiveInteger)) (: ff (Mapping (RealSpace n) (RealSpace k)))) ff)) (SIGNATURE setelt! ((PositiveInteger) (Record (: kk (PositiveInteger)) (: nn (PositiveInteger)) (: ff (Mapping (RealSpace n) (RealSpace k)))) kk (PositiveInteger))) (SIGNATURE setelt! ((PositiveInteger) (Record (: kk (PositiveInteger)) (: nn (PositiveInteger)) (: ff (Mapping (RealSpace n) (RealSpace k)))) nn (PositiveInteger))) (SIGNATURE setelt! ((Mapping (RealSpace n) (RealSpace k)) (Record (: kk (PositiveInteger)) (: nn (PositiveInteger)) (: ff (Mapping (RealSpace n) (RealSpace k)))) ff (Mapping (RealSpace n) (RealSpace k)))) (SIGNATURE copy ((Record (: kk (PositiveInteger)) (: nn (PositiveInteger)) (: ff (Mapping (RealSpace n) (RealSpace k)))) (Record (: kk (PositiveInteger)) (: nn (PositiveInteger)) (: ff (Mapping (RealSpace n) (RealSpace k)))))))) to (Join (MatrixCategory F (Vector F) (Vector F)) (CATEGORY domain (SIGNATURE diagonalMatrix ($(Vector F))) (IF (has F (ConvertibleTo (InputForm))) (ATTRIBUTE (ConvertibleTo (InputForm))) noBranch) (IF (has F (Field)) (SIGNATURE inverse ((Union$ failed) $)) noBranch))) compiling exported coerce : RealSpace k -> RealSpace n ->$
Time: 0 SEC.
compiling exported coerce : $-> OutputForm Time: 0 SEC. compiling exported dim :$ -> PositiveInteger
SURF;dim;$Pi;3 is replaced by QVELTS0 Time: 0.01 SEC. compiling exported codim :$ -> PositiveInteger
Time: 0 SEC.
compiling exported eval : ($,List Expression Integer) -> RealSpace n Time: 0 SEC. compiling exported eval : ($,List Symbol) -> RealSpace n
Time: 0 SEC.
compiling exported JacobianMatrix : ($,List Symbol) -> Matrix Expression Integer Time: 0.02 SEC. compiling local vec2list : Vector Expression Integer -> List Expression Integer Time: 0 SEC. compiling exported TangentSpace : ($,List Symbol) -> List RealSpace n
Time: 0.01 SEC.
compiling exported CoTangentSpace : ($,List Symbol) -> List RealSpace n Time: 0.04 SEC. compiling exported normal : ($,List Symbol) -> RealSpace n
Time: 0.01 SEC.
(time taken in buildFunctor:  0)
;;;     ***       |Surface| REDEFINED
;;;     ***       |Surface| REDEFINED
Time: 0 SEC.
Warnings:
[1] coerce:  kk has no value
[2] coerce:  nn has no value
[3] dim:  kk has no value
[4] codim:  nn has no value
[5] codim:  kk has no value
[6] eval:  ff has no value
[7] JacobianMatrix:  ff has no value
Cumulative Statistics for Constructor Surface
Time: 0.09 seconds
finalizing NRLIB SURF
Processing Surface for Browser database:
--------constructor---------
--->-->Surface((coerce ((OutputForm) %))): Not documented!!!!
--->-->Surface((coerce (% (List (Expression (Integer)))))): Not documented!!!!
--->-->Surface((coerce (% (Vector (Expression (Integer)))))): Not documented!!!!
--->-->Surface((coerce (% (List (Symbol))))): Not documented!!!!
--->-->Surface((coerce (% (Expression (Integer))))): Not documented!!!!
--->-->Surface((coerce ((List (Expression (Integer))) %))): Not documented!!!!
--->-->Surface((coerce ((Vector (Expression (Integer))) %))): Not documented!!!!
--->-->Surface((dot ((Expression (Integer)) % %))): Not documented!!!!
--->-->Surface((dim ((NonNegativeInteger) %))): Not documented!!!!
--->-->Surface((+ (% % %))): Not documented!!!!
--->-->Surface((- (% % %))): Not documented!!!!
--->-->Surface((* (% (Expression (Integer)) %))): Not documented!!!!
--->-->Surface((* (% (NonNegativeInteger) %))): Not documented!!!!
--->-->Surface((* (% (PositiveInteger) %))): Not documented!!!!
--->-->Surface((* (% (Integer) %))): Not documented!!!!
--->-->Surface((/ (% % (Expression (Integer))))): Not documented!!!!
--->-->Surface((# ((NonNegativeInteger) %))): Not documented!!!!
--->-->Surface((- (% %))): Not documented!!!!
--->-->Surface((= ((Boolean) % %))): Not documented!!!!
--->-->Surface((~= ((Boolean) % %))): Not documented!!!!
--->-->Surface((unitVector (% (PositiveInteger)))): Not documented!!!!
--->-->Surface((elt ((Expression (Integer)) % (Integer)))): Not documented!!!!
--->-->Surface((eval (% % (Equation (Expression (Integer)))))): Not documented!!!!
--->-->Surface((eval (% % (List (Equation (Expression (Integer))))))): Not documented!!!!
--->-->Surface((every? ((Boolean) (Mapping (Boolean) (Expression (Integer))) %))): Not documented!!!!
--->-->Surface((map (% (Mapping (Expression (Integer)) (Expression (Integer))) %))): Not documented!!!!
--->-->Surface((member? ((Boolean) (Expression (Integer)) %))): Not documented!!!!
--->-->Surface((any? ((Boolean) (Mapping (Boolean) (Expression (Integer))) %))): Not documented!!!!
--->-->Surface((copy (% %))): Not documented!!!!
--->-->Surface((D (% % (Symbol)))): Not documented!!!!
--->-->Surface((D (% % (Symbol) (NonNegativeInteger)))): Not documented!!!!
--->-->Surface((D (% % (List (Symbol))))): Not documented!!!!
--->-->Surface((D (% % (List (Symbol)) (List (NonNegativeInteger))))): Not documented!!!!
--->-->Surface((dist ((Expression (Integer)) % %))): Not documented!!!!
--->-->Surface((norm ((Expression (Integer)) %))): Not documented!!!!
--->-->Surface((tprod ((Matrix (Expression (Integer))) % %))): Not documented!!!!
--->-->Surface((cross (% (List %)))): Not documented!!!!
--------constructor---------
--->-->Surface((coerce (% (Tuple F)))): Not documented!!!!
--->-->Surface((coerce (% (List F)))): Not documented!!!!
--->-->Surface((coerce ((List F) %))): Not documented!!!!
--->-->Surface((coerce ((OutputForm) %))): Not documented!!!!
--->-->Surface((closedInterval (% F F))): Not documented!!!!
--->-->Surface((openInterval (% F F))): Not documented!!!!
--->-->Surface((leftOpenInterval (% F F))): Not documented!!!!
--->-->Surface((rightOpenInterval (% F F))): Not documented!!!!
--->-->Surface((lb (F %))): Not documented!!!!
--->-->Surface((ub (F %))): Not documented!!!!
--->-->Surface((topology ((OutputForm) %))): Not documented!!!!
--->-->Surface((measure (F %))): Not documented!!!!
--->-->Surface((* (% F %))): Not documented!!!!
--->-->Surface((+ (% F %))): Not documented!!!!
--------constructor---------
--->-->Surface((coerce (% (Matrix F)))): Not documented!!!!
--->-->Surface((coerce ((List (Interval F)) %))): Not documented!!!!
--->-->Surface((coerce (% (List (Interval F))))): Not documented!!!!
--->-->Surface((coerce ((OutputForm) %))): Not documented!!!!
--->-->Surface((measure (F %))): Not documented!!!!
--------constructor---------
--->-->Surface((coerce ((OutputForm) %))): Not documented!!!!
--->-->Surface((coerce (% (Mapping (RealSpace n) (RealSpace k))))): Not documented!!!!
--->-->Surface((dim ((PositiveInteger) %))): Not documented!!!!
--->-->Surface((codim ((PositiveInteger) %))): Not documented!!!!
--->-->Surface((eval ((RealSpace n) % (List (Expression (Integer)))))): Not documented!!!!
--->-->Surface((eval ((RealSpace n) % (List (Symbol))))): Not documented!!!!
--->-->Surface((JacobianMatrix ((Matrix (Expression (Integer))) % (List (Symbol))))): Not documented!!!!
--->-->Surface((TangentSpace ((List (RealSpace n)) % (List (Symbol))))): Not documented!!!!
--->-->Surface((CoTangentSpace ((List (RealSpace n)) % (List (Symbol))))): Not documented!!!!
--->-->Surface((normal ((RealSpace n) % (List (Symbol))))): Not documented!!!!
; compiling file "/var/aw/var/LatexWiki/SURF.NRLIB/SURF.lsp" (written 23 NOV 2014 02:37:55 AM):
; /var/aw/var/LatexWiki/SURF.NRLIB/SURF.fasl written
; compilation finished in 0:00:00.049
------------------------------------------------------------------------
Surface is now explicitly exposed in frame initial
Surface will be automatically loaded when needed from
/var/aw/var/LatexWiki/SURF.NRLIB/SURF

fricas
R ==> EXPR INT
Type: Void
fricas
R2 ==> RealSpace(2)
Type: Void
fricas
R3 ==> RealSpace(3)
Type: Void
fricas
f: R2 -> R3
Type: Void
fricas
f(X) == [(1+X.2/2*cos(X.1/2))*cos(X.1),(1+X.2/2*cos(X.1/2))*sin(X.1),X.2/2*sin(X.1/2)::R]::R3
Type: Void
fricas
f
 (1)
Type: FunctionCalled?(f)
fricas
typeOf f
 (2)
Type: Type

Example MÃ¶bius strip as 2-surface

fricas
f([u,v::Symbol]::R2)
fricas
Compiling function f with type RealSpace(2) -> RealSpace(3)
 (3)
Type: RealSpace?(3)
fricas
-- proxy error without tex off !!
fricas
)set output tex off

fricas
)set output algebra on
S:=f::Surface(2,3)
(9)  2-surface in R^3
Type: Surface(2,3)
fricas
dim S
(10)  2
Type: PositiveInteger?
fricas
eval(S,[u,v])
u                    u                   u
(v cos(-) + 2)cos(u) (v cos(-) + 2)sin(u) v sin(-)
2                    2                   2
(11)  [--------------------,--------------------,--------]
2                    2              2
Type: RealSpace?(3)
fricas
ts:=TangentSpace(S,[u,v])
(12)
[
u                           u
(- 2v cos(-) - 4)sin(u) - v cos(u)sin(-)
2                           2
[----------------------------------------,
4
u                  u                    u
- v sin(-)sin(u) + (2v cos(-) + 4)cos(u)  v cos(-)
2                  2                    2
----------------------------------------, --------]
4                         4
,
u            u            u
cos(-)cos(u) cos(-)sin(u) sin(-)
2            2            2
[------------,------------,------]]
2            2         2
Type: List(RealSpace?(3))
fricas
cs:=CoTangentSpace(S,[u,v])
(13)
[
u 2         u 2                    u                u
(v sin(-)  + v cos(-) )sin(u) + (- 2v cos(-) - 4)cos(u)sin(-)
2           2                      2                2
[-------------------------------------------------------------,
u 2        u        2           u 2        u        2
(2v cos(-)  + 4cos(-))sin(u)  + (2v cos(-)  + 4cos(-))cos(u)
2          2                    2          2
u          u                      u 2         u 2
(- 2v cos(-) - 4)sin(-)sin(u) - v cos(u)sin(-)  - v cos(-) cos(u)
2          2                      2           2
-----------------------------------------------------------------, 1]
u 2        u        2           u 2        u        2
(2v cos(-)  + 4cos(-))sin(u)  + (2v cos(-)  + 4cos(-))cos(u)
2          2                    2          2
]
Type: List(RealSpace?(3))
fricas
ns:=normal(S,[u,v])
(14)
[
u 2         u 2                    u                u
(v sin(-)  + v cos(-) )sin(u) + (- 2v cos(-) - 4)cos(u)sin(-)
2           2                      2                2
/
u 2        u        2           u 2        u        2
((2v cos(-)  + 4cos(-))sin(u)  + (2v cos(-)  + 4cos(-))cos(u) )
2          2                    2          2
*
ROOT
2    u 4           u 3         u 2       2    2    u 4
(4v cos(-)  + 16v cos(-)  + 16cos(-) )sin(u)  + v sin(-)
2             2           2                   2
+
2    u 2           u           u 2
(6v cos(-)  + 16v cos(-) + 16)sin(-)
2             2           2
+
2    u 4           u 3         u 2       2    2    u 4
(4v cos(-)  + 16v cos(-)  + 16cos(-) )cos(u)  + v cos(-)
2             2           2                   2
/
2    u 4           u 3         u 2       2
(4v cos(-)  + 16v cos(-)  + 16cos(-) )sin(u)
2             2           2
+
2    u 4           u 3         u 2       2
(4v cos(-)  + 16v cos(-)  + 16cos(-) )cos(u)
2             2           2
,
u          u                      u 2         u 2
(- 2v cos(-) - 4)sin(-)sin(u) - v cos(u)sin(-)  - v cos(-) cos(u)
2          2                      2           2
/
u 2        u        2           u 2        u        2
((2v cos(-)  + 4cos(-))sin(u)  + (2v cos(-)  + 4cos(-))cos(u) )
2          2                    2          2
*
ROOT
2    u 4           u 3         u 2       2    2    u 4
(4v cos(-)  + 16v cos(-)  + 16cos(-) )sin(u)  + v sin(-)
2             2           2                   2
+
2    u 2           u           u 2
(6v cos(-)  + 16v cos(-) + 16)sin(-)
2             2           2
+
2    u 4           u 3         u 2       2    2    u 4
(4v cos(-)  + 16v cos(-)  + 16cos(-) )cos(u)  + v cos(-)
2             2           2                   2
/
2    u 4           u 3         u 2       2
(4v cos(-)  + 16v cos(-)  + 16cos(-) )sin(u)
2             2           2
+
2    u 4           u 3         u 2       2
(4v cos(-)  + 16v cos(-)  + 16cos(-) )cos(u)
2             2           2
,
1
/
ROOT
2    u 4           u 3         u 2       2    2    u 4
(4v cos(-)  + 16v cos(-)  + 16cos(-) )sin(u)  + v sin(-)
2             2           2                   2
+
2    u 2           u           u 2
(6v cos(-)  + 16v cos(-) + 16)sin(-)
2             2           2
+
2    u 4           u 3         u 2       2    2    u 4
(4v cos(-)  + 16v cos(-)  + 16cos(-) )cos(u)  + v cos(-)
2             2           2                   2
/
2    u 4           u 3         u 2       2
(4v cos(-)  + 16v cos(-)  + 16cos(-) )sin(u)
2             2           2
+
2    u 4           u 3         u 2       2
(4v cos(-)  + 16v cos(-)  + 16cos(-) )cos(u)
2             2           2
]
Type: RealSpace?(3)

Cross product in R^n

fricas
X:=[a,b,c,d::R]::RealSpace(4)
(15)  [a,b,c,d]
Type: RealSpace?(4)
fricas
Y:=[p,q,r,s::R]::RealSpace(4)
(16)  [p,q,r,s]
Type: RealSpace?(4)
fricas
Z:=[t,u,v,w::R]::RealSpace(4)
(17)  [t,u,v,w]
Type: RealSpace?(4)
fricas
C:=cross [X,Y,Z]
(18)
[(b r - c q)w + (- b s + d q)v + (c s - d r)u,
(- a r + c p)w + (a s - d p)v + (- c s + d r)t,
(a q - b p)w + (- a s + d p)u + (b s - d q)t,
(- a q + b p)v + (a r - c p)u + (- b r + c q)t]
Type: RealSpace?(4)
fricas
[dot(C,v) for v in [X,Y,Z]]
(19)  [0,0,0]
Type: List(Expression(Integer))

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