login  home  contents  what's new  discussion  bug reports help  links  subscribe  changes  refresh  edit
 Download: pdf dvi ps src tex log * category LALG LeftAlgebra domain EAB ExtAlgBasis domain ANTISYM AntiSymm domain DERHAM DeRhamComplex license

Correction to degree

)abbrev domain DERHAM DeRhamComplex
++ Author: Larry A. Lambe
++ Date    : 01/26/91.
++ Revised : 12/01/91.
++
++ based on code from '89 (AntiSymmetric)
++
++ Needs: LeftAlgebra, ExtAlgBasis, FreeMod(Ring,OrderedSet)
++
++ Description: The deRham complex of Euclidean space, that is, the
++ class of differential forms of arbitary degree over a coefficient ring.
++ See Flanders, Harley, Differential Forms, With Applications to the Physical
++ Sciences, New York, Academic Press, 1963.
DeRhamComplex(CoefRing,listIndVar:List Symbol): Export == Implement where CoefRing : Join(Ring, Comparable) ASY ==> AntiSymm(R,listIndVar) DIFRING ==> DifferentialRing LALG ==> LeftAlgebra FMR ==> FreeMod(R,EAB) I ==> Integer L ==> List EAB ==> ExtAlgBasis -- these are exponents of basis elements in order NNI ==> NonNegativeInteger O ==> OutputForm R ==> Expression(CoefRing)
Export == Join(LALG(R), RetractableTo(R)) with leadingCoefficient : % -> R ++ leadingCoefficient(df) returns the leading ++ coefficient of differential form df. leadingBasisTerm : % -> % ++ leadingBasisTerm(df) returns the leading ++ basis term of differential form df. reductum : % -> % ++ reductum(df), where df is a differential form, ++ returns df minus the leading ++ term of df if df has two or more terms, and ++ 0 otherwise. coefficient : (%,%) -> R ++ coefficient(df,u), where df is a differential form, ++ returns the coefficient of df containing the basis term u ++ if such a term exists, and 0 otherwise. generator : NNI -> % ++ generator(n) returns the nth basis term for a differential form. homogeneous? : % -> Boolean ++ homogeneous?(df) tests if all of the terms of ++ differential form df have the same degree. retractable? : % -> Boolean ++ retractable?(df) tests if differential form df is a 0-form, ++ i.e., if degree(df) = 0. degree : % -> NNI ++ degree(df) returns the homogeneous degree of differential form df. map : (R -> R, %) -> % ++ map(f,df) replaces each coefficient x of differential ++ form df by \spad{f(x)}. totalDifferential : R -> % ++ totalDifferential(x) returns the total differential ++ (gradient) form for element x. exteriorDifferential : % -> % ++ exteriorDifferential(df) returns the exterior ++ derivative (gradient, curl, divergence, ...) of ++ the differential form df.
Implement == ASY add Rep := ASY
dim := #listIndVar
totalDifferential(f) == divs:=[differentiate(f,listIndVar.i)*generator(i)$ASY for i in 1..dim] reduce("+",divs) termDiff : (R, %) -> % termDiff(r,e) == totalDifferential(r) * e exteriorDifferential(x) == x = 0 => 0 termDiff(leadingCoefficient(x)$Rep,leadingBasisTerm x) + exteriorDifferential(reductum x)
lv := [concat("d",string(liv))$String::Symbol for liv in listIndVar] displayList:EAB -> O displayList(x):O == le: L I := exponents(x)$EAB -- reduce(_*,[(lv.i)::O for i in 1..dim | le.i = 1])$L(O) -- reduce(_*,[(lv.i)::O for i in 1..dim | one?(le.i)])$L(O) reduce(_*,[(lv.i)::O for i in 1..dim | ((le.i) = 1)])$L(O) makeTerm:(R,EAB) -> O makeTerm(r,x) == -- we know that r ~= 0 x = Nul(dim)$EAB => r::O -- one? r => displayList(x) (r = 1) => displayList(x) -- r = 1 => displayList(x) r::O * displayList(x)
terms : % -> List Record(k: EAB, c: R) terms(a) == -- it is the case that there are at least two terms in a a pretend List Record(k: EAB, c: R)
coerce(a):O == a = 0$Rep => 0$I::O ta := terms a -- reductum(a) = 0$Rep => makeTerm(leadingCoefficient a, a.first.k) null ta.rest => makeTerm(ta.first.c, ta.first.k) reduce(_+,[makeTerm(t.c,t.k) for t in ta])$L(O)
Compiling FriCAS source code from file
using old system compiler.
DERHAM abbreviates domain DeRhamComplex
------------------------------------------------------------------------
initializing NRLIB DERHAM for DeRhamComplex
compiling into NRLIB DERHAM
****** Domain: CoefRing already in scope
compiling exported totalDifferential : Expression CoefRing -> $Time: 0.05 SEC. compiling local termDiff : (Expression CoefRing,$) -> $Time: 0.01 SEC. compiling exported exteriorDifferential :$ -> $Time: 0 SEC. compiling local displayList : ExtAlgBasis -> OutputForm Time: 0.04 SEC. compiling local makeTerm : (Expression CoefRing,ExtAlgBasis) -> OutputForm Time: 0.01 SEC. compiling local terms :$ -> List Record(k: ExtAlgBasis,c: Expression CoefRing) DERHAM;terms is replaced by a Time: 0 SEC.
compiling exported coerce : $-> OutputForm Time: 0.01 SEC. (time taken in buildFunctor: 0) ;;; *** |DeRhamComplex| REDEFINED ;;; *** |DeRhamComplex| REDEFINED Time: 0 SEC. Warnings: [1] coerce: c has no value [2] coerce: k has no value Cumulative Statistics for Constructor DeRhamComplex Time: 0.12 seconds --------------non extending category---------------------- .. DeRhamComplex(#1,#2) of cat (|Join| (|LeftAlgebra| (|Expression| |#1|)) (|RetractableTo| (|Expression| |#1|)) (CATEGORY |domain| (SIGNATURE |leadingCoefficient| ((|Expression| |#1|)$)) (SIGNATURE |leadingBasisTerm| ()) (SIGNATURE |reductum| ()) (SIGNATURE |coefficient| ((|Expression| |#1|) )) (SIGNATURE |generator| ($(|NonNegativeInteger|))) (SIGNATURE |homogeneous?| ((|Boolean|)$)) (SIGNATURE |retractable?| ((|Boolean|) $)) (SIGNATURE |degree| ((|NonNegativeInteger|)$)) (SIGNATURE |map| ($(|Mapping| (|Expression| |#1|) (|Expression| |#1|))$)) (SIGNATURE |totalDifferential| ($(|Expression| |#1|))) (SIGNATURE |exteriorDifferential| ($ \$)))) has no exp : List(Integer) -> % finalizing NRLIB DERHAM Processing DeRhamComplex for Browser database: --------constructor--------- --------(leadingCoefficient ((Expression CoefRing) %))--------- --------(leadingBasisTerm (% %))--------- --------(reductum (% %))--------- --------(coefficient ((Expression CoefRing) % %))--------- --------(generator (% (NonNegativeInteger)))--------- --------(homogeneous? ((Boolean) %))--------- --------(retractable? ((Boolean) %))--------- --------(degree ((NonNegativeInteger) %))--------- --------(map (% (Mapping (Expression CoefRing) (Expression CoefRing)) %))--------- --------(totalDifferential (% (Expression CoefRing)))--------- --------(exteriorDifferential (% %))--------- ; compiling file "/var/aw/var/LatexWiki/DERHAM.NRLIB/DERHAM.lsp" (written 09 OCT 2014 09:54:39 PM):
; /var/aw/var/LatexWiki/DERHAM.NRLIB/DERHAM.fasl written ; compilation finished in 0:00:00.053 ------------------------------------------------------------------------ DeRhamComplex is now explicitly exposed in frame initial DeRhamComplex will be automatically loaded when needed from /var/aw/var/LatexWiki/DERHAM.NRLIB/DERHAM

 Subject:   Be Bold !! ( 13 subscribers )