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[Maxima-commits] Maxima, A Computer Algebra System branch master updated. ae3a5000b16705e9411daf0a6dae58bada9fb3ec
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From: Robert D. <rob...@us...> - 2011-12-09 05:58:22
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- Log -----------------------------------------------------------------
commit ae3a5000b16705e9411daf0a6dae58bada9fb3ec
Merge: 765e987 05e0868
Author: robert_dodier <rob...@us...>
Date: Thu Dec 8 22:55:29 2011 -0700
Resolve merge conflict in git pull.
commit 765e987503f4823507f9242d2fc6638aba70a594
Author: Mario Rodriguez <rio...@us...>
Date: Tue Dec 6 10:48:00 2011 +0100
Update translation
diff --git a/doc/info/es/Polynomials.es.texi b/doc/info/es/Polynomials.es.texi
index ab4dadf..20c51e8 100644
--- a/doc/info/es/Polynomials.es.texi
+++ b/doc/info/es/Polynomials.es.texi
@@ -1,4 +1,4 @@
-@c English version 2011-07-03
+@c English version 2011-08-04
@menu
* Introducci@'on a los polinomios::
* Funciones y variables para polinomios::
@@ -65,6 +65,18 @@ Si @code{berlefact} vale @code{false} entonces se utiliza el algoritmo de factor
@deffn {Funci@'on} bezout (@var{p1}, @var{p2}, @var{x})
Es una alternativa a la funci@'on @code{resultant}. Devuelve una matriz.
+@example
+(%i1) bezout(a*x+b, c*x^2+d, x);
+ [ b c - a d ]
+(%o1) [ ]
+ [ a b ]
+(%i2) determinant(%);
+ 2 2
+(%o2) a d + b c
+(%i3) resultant(a*x+b, c*x^2+d, x);
+ 2 2
+(%o3) a d + b c
+@end example
@end deffn
@deffn {Funci@'on} bothcoef (@var{expr}, @var{x})
@@ -214,9 +226,11 @@ funci@'on.
(%i1) coeff ([4*a, -3*a, 2*a], a);
(%o1) [4, - 3, 2]
(%i2) coeff (matrix ([a*x, b*x], [-c*x, -d*x]), x);
+@group
[ a b ]
(%o2) [ ]
[ - c - d ]
+@end group
(%i3) coeff (a*u - b*v = 7*u + 3*v, u);
(%o3) a = 7
@end example
@@ -917,17 +931,46 @@ equation may be given as first argument.
@defvr {Variable opcional} modulus
Valor por defecto: @code{false}
-Si @code{modulus} es un n@'umero positivo @var{p}, las operaciones con n@'umeros racionales (como los devueltos por @code{rat} y funciones asociadas) se realizan m@'odulo @var{p}, utilizando el llamado sistema de m@'odulo balanceado, en el que @code{@var{n} m@'odulo @var{p}} se define como un entero @var{k} de @code{[-(@var{p}-1)/2, ..., 0, ..., (@var{p}-1)/2]}
-si @var{p} es impar, o de @code{[-(@var{p}/2 - 1), ..., 0, ...., @var{p}/2]} si @var{p} es par, de tal manera que @code{@var{a} @var{p} + @var{k}} es igual a @var{n} para alg@'un entero @var{a}.
-@c NEED EXAMPLES OF "BALANCED MODULUS" HERE
+Si @code{modulus} es un n@'umero positivo @var{p}, las operaciones con n@'umeros racionales
+(como los devueltos por @code{rat} y funciones relacionadas) se realizan m@'odulo @var{p},
+utilizando el llamado sistema de m@'odulo balanceado, en el que @code{@var{n} m@'odulo @var{p}}
+se define como un entero @var{k} de @code{[-(@var{p}-1)/2, ..., 0, ..., (@var{p}-1)/2]}
+si @var{p} es impar, o de @code{[-(@var{p}/2 - 1), ..., 0, ...., @var{p}/2]} si @var{p} es par,
+de tal manera que @code{@var{a} @var{p} + @var{k}} es igual a @var{n} para alg@'un entero @var{a}.
-@c FALTA PARRAFO POCO LEGIBLE
+Normalmente a @code{modulus} se le asigna un n@'umero primo. Se acepta que a @code{modulus}
+se le asigne un entero positivo no primo, pero se obtendr@'a un mensaje de aviso. Maxima
+responder@'a con un mensaje de error cuando se le asigne a @code{modulus} cero o un n@'umero
+negativo.
-Normalmente a @code{modulus} se le asigna un n@'umero primo. Se acepta que a @code{modulus} se le asigne un entero positivo no primo, pero se obtendr@'a un mensaje de aviso. Maxima permitir@'a que a @code{modulus} se le asigne cero o un entero negativo, aunque no est@'e clara su utilidad.
+Ejemplos:
-@c NEED EXAMPLES HERE
+@example
+(%i1) modulus:7;
+(%o1) 7
+(%i2) polymod([0,1,2,3,4,5,6,7]);
+(%o2) [0, 1, 2, 3, - 3, - 2, - 1, 0]
+(%i3) modulus:false;
+(%o3) false
+(%i4) poly:x^6+x^2+1;
+ 6 2
+(%o4) x + x + 1
+(%i5) factor(poly);
+ 6 2
+(%o5) x + x + 1
+(%i6) modulus:13;
+(%o6) 13
+(%i7) factor(poly);
+ 2 4 2
+(%o7) (x + 6) (x - 6 x - 2)
+(%i8) polymod(%);
+ 6 2
+(%o8) x + x + 1
+@end example
@end defvr
+
+
@deffn {Funci@'on} num (@var{expr})
Devuelve el numerador de @var{expr} si se trata de una fracci@'on. Si @var{expr} no es una fracci@'on, se devuelve @var{expr}.
@@ -1155,7 +1198,8 @@ Valor por defecto: @code{true}
Si @code{ratdenomdivide} vale @code{true}, la funci@'on @code{ratexpand} expande una fracci@'on en la que el numerador es una suma en una suma de divisiones. En otro caso, @code{ratexpand} reduce una suma de divisiones a una @'unica fracci@'on, cuyo numerador es la suma de los denominadores de cada fracci@'on.
-Examples:
+Ejemplos:
+
@example
(%i1) expr: (x^2 + x + 1)/(y^2 + 7);
2
@@ -1172,11 +1216,13 @@ Examples:
y + 7 y + 7 y + 7
(%i4) ratdenomdivide: false$
(%i5) ratexpand (expr);
+@group
2
x + x + 1
(%o5) ----------
2
y + 7
+@end group
(%i6) expr2: a^2/(b^2 + 3) + b/(b^2 + 3);
2
b a
@@ -1610,20 +1656,68 @@ La funci@'on @code{remainder} devuelve el segundo elemento de la lista retornada
@end deffn
@deffn {Funci@'on} resultant (@var{p_1}, @var{p_2}, @var{x})
-@deffnx {Variable} resultant
-Calcula la resultante de los dos polinomios @var{p_1} y @var{p_2}, eliminando la variable @var{x}.
-La resultante es un determinante de los coeficientes de @var{x} en @var{p_1} y @var{p_2}, que es igual a cero si s@'olo si @var{p_1} y @var{p_2} tienen un factor com@'un no constante.
+Calcula la resultante de los dos polinomios @var{p_1} y @var{p_2}, eliminando
+la variable @var{x}. La resultante es un determinante de los coeficientes de @var{x}
+en @var{p_1} y @var{p_2}, que es igual a cero si s@'olo si @var{p_1} y @var{p_2}
+tienen un factor com@'un no constante.
+
+Si @var{p_1} o @var{p_2} pueden ser factorizados, puede ser necesario llamar a
+@code{factor} antes que invocar a @code{resultant}.
-Si @var{p_1} o @var{p_2} pueden ser factorizados, puede ser necesario llamar a @code{factor} antes que invocar a @code{resultant}.
+La variable opcional @code{resultant} controla qu@'e algoritmo ser@'a utilizado para calcular la resultante.
+V@'eanse @code{option_resultant} y @code{resultant}.
-La variable @code{resultant} controla qu@'e algoritmo ser@'a utilizado para calcular la resultante.
-@c FALTA COMPLETAR PARRAFO
+La funci@'on @code{bezout} toma los mismos argumentos que @code{resultant} y devuelve una matriz.
+El determinante del valor retornado es la resultante buscada.
-La funci@'on @code{bezout} toma los mismos argumentos que @code{resultant} y devuelve una matriz. El determinante del valor retornado es la resultante buscada.
+Ejemplos:
-@c NEED AN EXAMPLE HERE
+@example
+(%i1) resultant(2*x^2+3*x+1, 2*x^2+x+1, x);
+(%o1) 8
+(%i2) resultant(x+1, x+1, x);
+(%o2) 0
+(%i3) resultant((x+1)*x, (x+1), x);
+(%o3) 0
+(%i4) resultant(a*x^2+b*x+1, c*x + 2, x);
+ 2
+(%o4) c - 2 b c + 4 a
+
+(%i5) bezout(a*x^2+b*x+1, c*x+2, x);
+@group
+ [ 2 a 2 b - c ]
+(%o5) [ ]
+ [ c 2 ]
+@end group
+(%i6) determinant(%);
+(%o6) 4 a - (2 b - c) c
+@end example
@end deffn
+
+
+@defvr {Variable opcional} resultant
+Valor por defecto: @code{subres}
+
+La variable opcional @code{resultant} controla qu@'e algoritmo ser@'a utilizado para
+calcular la resultante con la funci@'on @code{resultant}. Los valores posibles son:
+
+@table @code
+@item subres
+para el algoritmo PRS (@i{polynomial remainder sequence}) subresultante,
+@item mod
+para el algoritmo resultante modular y
+@item red
+para el algoritmo PRS (@i{polynomial remainder sequence}) reducido.
+@end table
+
+En la mayor parte de problemas, el valor por defecto, @code{subres}, es el
+m@'as apropiado. Pero en el caso de problemas bivariantes o univariantes
+de grado alto, puede ser mejor utilizar @code{mod}.
+@end defvr
+
+
+
@defvr {Variable opcional} savefactors
Valor por defecto: @code{false}
commit 14d31d4df832a7c3cd8478bc2d04a0f3a24bc9d8
Author: Mario Rodriguez <rio...@us...>
Date: Tue Dec 6 08:38:28 2011 +0100
Update translation
diff --git a/doc/info/es/MathFunctions.es.texi b/doc/info/es/MathFunctions.es.texi
index 8b8932b..adb2fe0 100644
--- a/doc/info/es/MathFunctions.es.texi
+++ b/doc/info/es/MathFunctions.es.texi
@@ -1,4 +1,4 @@
-@c English version 2011-06-15
+@c English version 2011-12-01
@menu
* Funciones para los n@'umeros::
* Funciones para los n@'umeros complejos::
@@ -15,18 +15,103 @@
@section Funciones para los n@'umeros
-@deffn {Funci@'on} abs (@var{expr})
-Devuelve el valor absoluto de @var{expr}. Si la expresi@'on es compleja, retorna
-el m@'odulo de @var{expr}.
+@deffn {Funci@'on} abs (@var{z})
+La funci@'on @code{abs} representa el valor absoluto y se puede aplicar tanto
+a argumentos num@'ericos como simb@'olicos. Si el argumento @var{z} es un
+n@'umero real o complejo, @code{abs} devuelve el valor absoluto de @var{z}.
+Si es posible, las expresiones simb@'olicas que utilizan la funci@'on del
+valor absoluto tambi@'en se simplifican.
-@code{abs} se distribuye sobre listas, matrices y ecuaciones.
-V@'ease tambi@'en @code{distribute_over}.
+Maxima puede derivar, integrar y calcular l@'{@dotless{i}}mites de expresiones
+que contengan a @code{abs}. El paquete @code{abs_integrate} extiende las
+capacidades de Maxima para calcular integrales que contengan llamadas a
+@code{abs}. V@'ease @code{(%i12)} en el ejemplo de m@'as abajo.
+
+Cuando se aplica a una lista o matriz, @code{abs} se distribuye autom@'aticamente
+sobre sus elementos. De forma similar, tambi@'en se distribuye sobre los dos
+miembros de una igualdad. Para cambiar este comportamiento por defecto,
+v@'ease la variable @code{distribute_over}.
+
+Ejemplos:
+
+C@'alculo del valor absoluto de n@'umeros reales y complejos, incluyendo
+constantes num@'ericas e infinitos. El primer ejemplo muestra c@'omo
+@code{abs} se distribuye sobre los elementos de una lista.
+
+@example
+(%i1) abs([-4, 0, 1, 1+%i]);
+(%o1) [4, 0, 1, sqrt(2)]
+
+(%i2) abs((1+%i)*(1-%i));
+(%o2) 2
+(%i3) abs(%e+%i);
+ 2
+(%o3) sqrt(%e + 1)
+(%i4) abs([inf, infinity, minf]);
+(%o4) [inf, inf, inf]
+@end example
+
+Simplificaci@'on de expresiones que contienen @code{abs}:
+
+@example
+(%i5) abs(x^2);
+ 2
+(%o5) x
+(%i6) abs(x^3);
+ 2
+(%o6) x abs(x)
+
+(%i7) abs(abs(x));
+(%o7) abs(x)
+(%i8) abs(conjugate(x));
+(%o8) abs(x)
+@end example
+
+Integrando y derivando con la funci@'on @code{abs}. N@'otese que se pueden
+calcular m@'as integrales que involucren la funci@'on @code{abs} si se
+carga el paquete @code{abs_integrate}. El @'ultimo ejemplo muestra la
+transformada de Laplace de @code{abs}. V@'ease @code{laplace}.
+
+@example
+(%i9) diff(x*abs(x),x),expand;
+(%o9) 2 abs(x)
+
+(%i10) integrate(abs(x),x);
+ x abs(x)
+(%o10) --------
+ 2
+
+(%i11) integrate(x*abs(x),x);
+ /
+ [
+(%o11) I x abs(x) dx
+ ]
+ /
+
+(%i12) load(abs_integrate)$
+(%i13) integrate(x*abs(x),x);
+ 2 3
+ x abs(x) x signum(x)
+(%o13) --------- - ------------
+ 2 6
+
+(%i14) integrate(abs(x),x,-2,%pi);
+ 2
+ %pi
+(%o14) ---- + 2
+ 2
+
+(%i15) laplace(abs(x),x,s);
+ 1
+(%o15) --
+ 2
+ s
+@end example
@end deffn
@deffn {Funci@'on} ceiling (@var{x})
-
Si @var{x} es un n@'umero real, devuelve el menor entero mayor o igual que @var{x}.
Si @var{x} es una expresi@'on constante (por ejemplo, @code{10 * %pi}),
@@ -227,8 +312,70 @@ matrices o ecuaciones. V@'ease @code{distribute_over}.
@deffn {Funci@'on} cabs (@var{expr})
-Devuelve el valor absoluto complejo (m@'odulo complejo) de @var{expr}.
+Calcula el valor absoluto de una expresi@'on que representa a un n@'umero
+complejo. Al contrario que @code{abs}, la funci@'on @code{cabs} siempre
+descompone su argumento en sus partes real e imaginaria. Si @code{x} e
+@code{y} representan variables o expresiones reales, la funci@'on @code{cabs}
+calcula el valor absoluto de @code{x + %i*y} como
+@example
+ 2 2
+ sqrt(y + x )
+@end example
+La funci@'on @code{cabs} puede utilizar propiedades como la simetr@'{@dotless{i}}a
+de funciones complejas para calcular el valor absoluto de una expresi@'on.
+
+@code{cabs} no es una funci@'on apropiada para c@'alculos simb@'olicos; en tales
+casos, que incluyen la integraci@'on, diferenciaci@'on y l@'{@dotless{i}}mites que
+contienen valores absolutos, es mejor utilizar @code{abs}.
+
+El resultado devuelto por @code{cabs} puede incluir la funci@'on de valor absoluto,
+@code{abs}, y el arco tangente, @code{atan2}.
+
+Cuando se aplica a una lista o matriz, @code{cabs} autom@'aticamente se distribuye
+sobre sus elementos. Tambi@'en se distribuye sobre los dos miembros de una
+igualdad.
+
+Para otras formas de operar con n@'umeros complejos, v@'eanse las funciones
+@code{rectform}, @code{realpart}, @code{imagpart}, @code{carg}, @code{conjugate}
+y @code{polarform}.
+
+Ejemplos:
+
+Ejemplos con @code{sqrt} and @code{sin}:
+
+@example
+(%i1) cabs(sqrt(1+%i*x));
+ 2 1/4
+(%o1) (x + 1)
+(%i2) cabs(sin(x+%i*y));
+ 2 2 2 2
+(%o2) sqrt(cos (x) sinh (y) + sin (x) cosh (y))
+@end example
+
+La simetr@'{@dotless{i}}a especular de la funci@'on de error @code{erf} se utiliza
+para calcular el valor absoluto del argumento complejo:
+
+@example
+(%i3) cabs(erf(x+%i*y));
+ 2
+ (erf(%i y + x) - erf(%i y - x))
+(%o3) sqrt(--------------------------------
+ 4
+ 2
+ (erf(%i y + x) + erf(%i y - x))
+ - --------------------------------)
+ 4
+@end example
+
+Dado que Maxima reconoce algunas identidades complejas de las funciones de
+Bessel, le permite calcular su valor absoluto cuando tiene argumentos
+complejos. Un ejemplo para @code{bessel_j}:
+
+@example
+(%i4) cabs(bessel_j(1,%i));
+(%o4) abs(bessel_j(1, %i))
+@end example
@end deffn
@@ -793,13 +940,6 @@ Si toma el valor @code{false}, se desactivar@'an todas estas simplificaciones.
@item @code{logsimp}
si vale @code{false}, entonces no se transforma @code{%e} a potencias que contengan logaritmos.
-@item @code{lognumer}
-si vale @code{true}, entonces los argumentos de @code{log} que sean n@'umeros
-decimales negativos en coma flotante se convertir@'an siempre a su valor
-absoluto antes de aplicar @code{log}. Si @code{numer} vale tambi@'en @code{true},
-entonces los argumentos enteros negativos de @code{log} tambi@'en se convertir@'an
-en su valor absoluto.
-
@item @code{lognegint}
si vale @code{true} se aplica la regla @code{log(-n)} -> @code{log(n)+%i*%pi},
siendo @code{n} un entero positivo.
@@ -872,7 +1012,7 @@ Si se hace @code{declare(n,integer);} entonces @code{logcontract(2*a*n*log(x));}
@defvr {Variable opcional} logexpand
-Valor por defecto: @code{false}
+Valor por defecto: @code{true}
Si @code{logexpand} vale @code{true} hace que @code{log(a^b)} se convierta
en @code{b*log(a)}. Si toma el valor @code{all}, @code{log(a*b)} tambi@'en se
@@ -880,7 +1020,7 @@ reducir@'a a @code{log(a)+log(b)}. Si toma el valor @code{super}, entonces
@code{log(a/b)} tambi@'en se reducir@'a a @code{log(a)-log(b)}, siendo
@code{a/b} racional con @code{a#1}, (la expresi@'on @code{log(1/b)}, para
@code{b} entero, se simplifica siempre). Si toma el valor @code{false},
-que es el valor por defecto, se desactivar@'an todas estas simplificaciones.
+se desactivar@'an todas estas simplificaciones.
@end defvr
@@ -892,13 +1032,6 @@ Si @code{lognegint} vale @code{true} se aplica la regla @code{log(-n)} -> @code{
@end defvr
-@defvr {Variable opcional} lognumer
-Valor por defecto: @code{false}
-
-Si @code{lognumer} vale @code{true}, entonces los argumentos de @code{log} que sean n@'umeros decimales negativos en coma flotante se convertir@'an siempre a su valor absoluto antes de aplicar @code{log}. Si @code{numer} vale tambi@'en @code{true}, entonces los argumentos enteros negativos de @code{log} tambi@'en se convertir@'an en su valor absoluto.
-
-@end defvr
-
@defvr {Variable opcional} logsimp
@@ -1480,7 +1613,12 @@ La instrucci@'on @code{demo ("trgsmp.dem")} muestra algunos ejemplos de @code{tr
@end deffn
@deffn {Funci@'on} trigrat (@var{expr})
-Devuelve una forma can@'onica simplificada cuasi-lineal de una expresi@'on trigonom@'etrica; @var{expr} es una fracci@'on racional que contiene @code{sin}, @code{cos} o @code{tan}, cuyos argumentos son formas lineales respecto de ciertas variables (o kernels) y @code{%pi/@var{n}} (@var{n} entero) con coeficientes enteros. El resultado es una fracci@'on simplificada con el numerador y denominador lineales respecto de @code{sin} y @code{cos}.
+Devuelve una forma can@'onica simplificada cuasi-lineal de una expresi@'on trigonom@'etrica;
+@var{expr} es una fracci@'on racional que contiene @code{sin}, @code{cos} o @code{tan},
+cuyos argumentos son formas lineales respecto de ciertas variables (o kernels) y @code{%pi/@var{n}}
+(@var{n} entero) con coeficientes enteros. El resultado es una fracci@'on simplificada con
+el numerador y denominador lineales respecto de @code{sin} y @code{cos}. As@'{@dotless{i}},
+ @code{trigrat} devuelve una expresi@'on lineal siempre que sea posible.
@c ===beg===
@c trigrat(sin(3*a)/sin(a+%pi/3));
commit b804115e791a3f5bfc089f4ee7d11204397b0896
Author: Mario Rodriguez <rio...@us...>
Date: Tue Dec 6 07:22:50 2011 +0100
Change version
diff --git a/doc/info/es/maxima.texi b/doc/info/es/maxima.texi
index 017409b..fa1ff68 100644
--- a/doc/info/es/maxima.texi
+++ b/doc/info/es/maxima.texi
@@ -17,7 +17,7 @@
@documentencoding ISO-8859-1
@synindex vr fn
@settitle Manual de Maxima
-Ver. 5.25
+Ver. 5.26
@c %**end of header
@setchapternewpage odd
@ifinfo
commit 238c947faef88548f67f68c5f1f534a48293173c
Author: crategus <cra...@us...>
Date: Mon Dec 5 00:11:31 2011 +0100
Work on the translation
diff --git a/doc/info/de/Itensor.de.texi b/doc/info/de/Itensor.de.texi
index d6f6d52..551a5f3 100755
--- a/doc/info/de/Itensor.de.texi
+++ b/doc/info/de/Itensor.de.texi
@@ -5,7 +5,7 @@
@c Original : Itensor.texi revision 15.06.2011
@c Translation : Dr. Dieter Kaiser
@c Date : 20.11.2010
-@c Revision : 24.09.2011
+@c Revision : 04.12.2011
@c
@c This file is part of Maxima -- GPL CAS based on DOE-MACSYMA
@c -----------------------------------------------------------------------------
@@ -22,85 +22,41 @@
@section Tensorpakete in Maxima
@c -----------------------------------------------------------------------------
-@c Maxima implements symbolic tensor manipulation of two distinct types:
-@c component tensor manipulation (@code{ctensor} package) and indicial tensor
-@c manipulation (@code{itensor} package).
-
-In Maxima sind zwei verschiedene Methoden implementiert, um mit Tensoren zu
-rechnen. Das Paket @code{ctensor} implementiert das Rechnen in der
+Maxima hat drei verschiedene Pakete, um mit Tensoren zu rechnen. Das Paket
+@code{ctensor} implementiert das Rechnen mit Tensoren in der
Koordinatendarstellung und das Paket @code{itensor} das Rechnen in einer
-Indexnotation.
-
-@c Nota bene: Please see the note on 'new tensor notation' below.
-
-@c Component tensor manipulation means that geometrical tensor
-@c objects are represented as arrays or matrices. Tensor operations such
-@c as contraction or covariant differentiation are carried out by
-@c actually summing over repeated (dummy) indices with @code{do} statements.
-@c That is, one explicitly performs operations on the appropriate tensor
-@c components stored in an array or matrix.
-
-Beim Rechnen in einer Koordinatendarstellung werden Tensoren als Arrays oder
-Matrizen dargestellt. Operationen mit Tensoren wie die Tensorverj@"ungung oder
-die kovariante Ableitung werden ausgef@"uhrt als Operationen mit den
-Komponenten des Tensors, die in einem Array oder einer Matrix gespeichert sind.
-
-@c Indicial tensor manipulation is implemented by representing
-@c tensors as functions of their covariant, contravariant and derivative
-@c indices. Tensor operations such as contraction or covariant
-@c differentiation are performed by manipulating the indices themselves
-@c rather than the components to which they correspond.
-
-Beim Rechnen in der Indexnotation werden Tensoren als Funktionen ihrer
-kovarianten und kontravarianten Indizes sowie den Ableitungen nach den
-Komponenten dargestellt. Operationen wie die Tensorverj@"ungung oder die
-kovariante Ableitung werden ausgef@"uhrt, in dem die Indizes manipuliert werden.
-
-@c These two approaches to the treatment of differential, algebraic and
-@c analytic processes in the context of Riemannian geometry have various
-@c advantages and disadvantages which reveal themselves only through the
-@c particular nature and difficulty of the user's problem. However, one
-@c should keep in mind the following characteristics of the two
-@c implementations:
-
-Diese beiden Ans@"atze der Behandlung von mathematischen Problemen im
-Zusammenhang mit der Riemannschen Geometrie haben verschiedene Vor- und
-Nachteile, die sich erst anhand des zu behandelnden Problems und dessen
-Schwierigkeitsgrad zeigen. Folgenden Eigenschaften der beiden
-Implementierungen sollten beachtet werden:
-
-@c The representation of tensors and tensor operations explicitly in
-@c terms of their components makes @code{ctensor} easy to use. Specification of
-@c the metric and the computation of the induced tensors and invariants
-@c is straightforward. Although all of Maxima's powerful simplification
-@c capacity is at hand, a complex metric with intricate functional and
-@c coordinate dependencies can easily lead to expressions whose size is
-@c excessive and whose structure is hidden. In addition, many calculations
-@c involve intermediate expressions which swell causing programs to
-@c terminate before completion. Through experience, a user can avoid
-@c many of these difficulties.
+Indexnotation. Das Paket @code{atensor} erlaubt die algebraische Manipulation
+von Tensoren in verschiedenen Algebren.
+
+Beim Rechnen in einer Koordinatendarstellung mit dem Paket @code{ctensor} werden
+Tensoren als Arrays oder Matrizen dargestellt. Operationen mit Tensoren wie die
+Tensorverj@"ungung oder die kovariante Ableitung werden ausgef@"uhrt als
+Operationen mit den Komponenten des Tensors, die in einem Array oder einer
+Matrix gespeichert sind.
+
+Beim Rechnen in der Indexnotation mit dem Paket @code{itensor} werden Tensoren
+als Funktionen ihrer kovarianten und kontravarianten Indizes sowie den
+Ableitungen nach den Komponenten dargestellt. Operationen wie die
+Tensorverj@"ungung oder die kovariante Ableitung werden ausgef@"uhrt, in dem die
+Indizes manipuliert werden.
+
+Die beiden genannten Pakete @code{itensor} und @code{ctensor} f@"ur die
+Behandlung von mathematischen Problemen im Zusammenhang mit der Riemannschen
+Geometrie haben verschiedene Vor- und Nachteile, die sich erst anhand des zu
+behandelnden Problems und dessen Schwierigkeitsgrad zeigen. Folgenden
+Eigenschaften der beiden Implementierungen sollten beachtet werden:
Die Darstellung von Tensoren und Tensoroperationen in einer expliziten
-Koordinatendarstellung macht die Nutzung des Paketes @code{ctensor} einfach.
-Die Spezifikation der Metrik und die Ableitung von Tensoren sowie von
-Invarianten ist unkompliziert. Trotz Maximas Methoden f@"ur die Vereinfachung
-von Ausdr@"ucken kann jedoch eine komplexe Metrik mit komplizierten funktionalen
+Koordinatendarstellung vereinfacht die Nutzung des Paketes @code{ctensor}. Die
+Spezifikation der Metrik und die Ableitung von Tensoren sowie von Invarianten
+ist unkompliziert. Trotz Maximas Methoden f@"ur die Vereinfachung von
+Ausdr@"ucken kann jedoch eine komplexe Metrik mit komplizierten funktionalen
Abh@"angigkeiten der Koordinaten leicht zu sehr gro@ss{}en Ausdr@"ucken
f@"uhren, die die Struktur eines Ergebnisses verbergen. Weiterhin k@"onnen
Rechnungen zu sehr gro@ss{}en Zwischenergebnisse f@"uhren, die zu einem
Programmabbruch f@"uhren, bevor die Rechnung beendet werden kann. Jedoch kann
der Nutzer mit einiger Erfahrung viele dieser Probleme vermeiden.
-@c Because of the special way in which tensors and tensor operations
-@c are represented in terms of symbolic operations on their indices,
-@c expressions which in the component representation would be unmanageable can
-@c sometimes be greatly simplified by using the special routines for symmetrical
-@c objects in @code{itensor}. In this way the structure of a large expression
-@c may be more transparent. On the other hand, because of the the special
-@c indicial representation in @code{itensor}, in some cases the
-@c user may find difficulty with the specification of the metric, function
-@c definition, and the evaluation of differentiated "indexed" objects.
-
Aufgrund der besonderen Weise, wie Tensoren und Tensoroperationen als
symbolische Operationen ihrer Indizes dargestellt werden, k@"onnen Ausdr@"ucke,
die in einer Koordinatendarstellung sehr unhandlich sind, mit Hilfe spezieller
@@ -110,22 +66,34 @@ transparenter sein. Auf der anderen Seite kann die Spezifikation einer Metrik,
die Definition von Funktionen und die Auswertung von abgeleiteten indizierten
Objekten f@"ur den Nutzer schwierig sein.
-The @code{itensor} package can carry out differentiation with respect to an
-indexed variable, which allows one to use the package when dealing with
-Lagrangian and Hamiltonian formalisms. As it is possible to differentiate a
-field Lagrangian with respect to an (indexed) field variable, one can use
-Maxima to derive the corresponding Euler-Lagrange equations in indicial form.
-These equations can be translated into component tensor (@code{ctensor})
-programs using the @code{ic_convert} function, allowing us to solve the field
-equations in a particular coordinate representation, or to recast the equations
-of motion in Hamiltonian form. See @code{einhil.dem} and @code{bradic.dem} for
-two comprehensive examples. The first, @code{einhil.dem}, uses the
-Einstein-Hilbert action to derive the Einstein field tensor in the homogeneous
-and isotropic case (Friedmann equations) and the spherically symmetric, static
-case (Schwarzschild solution.) The second, @code{bradic.dem}, demonstrates how
-to compute the Friedmann equations from the action of Brans-Dicke gravity
-theory, and also derives the Hamiltonian associated with the theory's scalar
-field.
+@c The @code{itensor} package can carry out differentiation with respect to an
+@c indexed variable, which allows one to use the package when dealing with
+@c Lagrangian and Hamiltonian formalisms. As it is possible to differentiate a
+@c field Lagrangian with respect to an (indexed) field variable, one can use
+@c Maxima to derive the corresponding Euler-Lagrange equations in indicial form.
+@c These equations can be translated into component tensor (@code{ctensor})
+@c programs using the @code{ic_convert} function, allowing us to solve the field
+@c equations in a particular coordinate representation, or to recast the
+@c equations of motion in Hamiltonian form. See @code{einhil.dem} and
+@c @code{bradic.dem} for two comprehensive examples. The first,
+@c @code{einhil.dem}, uses the Einstein-Hilbert action to derive the Einstein
+@c field tensor in the homogeneous and isotropic case (Friedmann equations) and
+@c the spherically symmetric, static case (Schwarzschild solution.) The second,
+@c @code{bradic.dem}, demonstrates how to compute the Friedmann equations from
+@c the action of Brans-Dicke gravity theory, and also derives the Hamiltonian
+@c associated with the theory's scalar field.
+
+Mit dem Paket @code{itensor} k@"onnen Ableitungen nach einer indizierten
+Variablen ausgef@"uhrt werden, wodurch es m@"oglich ist, @code{itensor} auch
+f@"ur Probleme im Zusammenhang mit dem Lagrange- oder Hamiltonian-Formalismus
+einzusetzen. Da es m@"oglich ist, die Lagrangeschen Feldgleichungen nach einer
+indizierten Variablen abzuleiten, k@"onnen zum Beispiel die
+Euler-Lagrange-Gleichungen in einer Indexnotation aufgestellt werden. Werden
+die Gleichungen mit der Funktion @mref{ic_convert} in eine
+Komponentendarstellung f@"ur das Paket @code{ctensor} transformiert, k@"onnen
+die Feldgleichungen in einer bestimmten Koordinatendarstellung gel@"ost werden.
+Siehe dazu die ausf@"uhrlichen Beispiele in @code{einhil.dem} und
+@code{bradic.dem}.
@c -----------------------------------------------------------------------------
@page
@@ -144,116 +112,218 @@ field.
@c -----------------------------------------------------------------------------
@c -----------------------------------------------------------------------------
-@subsubsection New tensor notation
+@c @subsubsection New tensor notation
@c -----------------------------------------------------------------------------
-Earlier versions of the @code{itensor} package in Maxima used a notation that sometimes
-led to incorrect index ordering. Consider the following, for instance:
+@c Earlier versions of the @code{itensor} package in Maxima used a notation that
+@c sometimes led to incorrect index ordering. Consider the following, for
+@c instance:
+
+@c @example
+@c (%i2) imetric(g);
+@c (%o2) done
+@c (%i3) ishow(g([],[j,k])*g([],[i,l])*a([i,j],[]))$
+@c i l j k
+@c (%t3) g g a
+@c i j
+@c (%i4) ishow(contract(%))$
+@c k l
+@c (%t4) a
+@c @end example
+
+@c This result is incorrect unless @code{a} happens to be a symmetric tensor.
+@c The reason why this happens is that although @code{itensor} correctly
+@c maintains the order within the set of covariant and contravariant indices,
+@c once an index is raised or lowered, its position relative to the other set of
+@c indices is lost.
+
+@c To avoid this problem, a new notation has been developed that remains fully
+@c compatible with the existing notation and can be used interchangeably. In
+@c this notation, contravariant indices are inserted in the appropriate
+@c positions in the covariant index list, but with a minus sign prepended.
+@c Functions like @code{contract} and @code{ishow} are now aware of this
+@c new index notation and can process tensors appropriately.
+
+@c In this new notation, the previous example yields a correct result:
+
+@c @example
+@c (%i5) ishow(g([-j,-k],[])*g([-i,-l],[])*a([i,j],[]))$
+@c i l j k
+@c (%t5) g a g
+@c i j
+@c (%i6) ishow(contract(%))$
+@c l k
+@c (%t6) a
+@c @end example
+
+@c Presently, the only code that makes use of this notation is the @code{lc2kdt}
+@c function. Through this notation, it achieves consistent results as it
+@c applies the metric tensor to resolve Levi-Civita symbols without resorting
+@c to numeric indices.
+
+@c Since this code is brand new, it probably contains bugs. While it has been
+@c tested to make sure that it doesn't break anything using the "old" tensor
+@c notation, there is a considerable chance that "new" tensors will fail to
+@c interoperate with certain functions or features. These bugs will be fixed
+@c as they are encountered @dots{} until then, caveat emptor!
+
+@c -----------------------------------------------------------------------------
+@c @subsubsection Indicial tensor manipulation
+@c -----------------------------------------------------------------------------
+
+@c The indicial tensor manipulation package may be loaded by
+@c @code{load(itensor)}. Demos are also available: try @code{demo(tensor)}.
+
+Das Paket @code{itensor} f@"ur das Rechnen mit Tensoren in der Indexnotation
+wird mit dem Kommando @code{load(itensor)} geladen. Mit dem Kommando
+@code{demo(tensor)} wird eine Liste mit verschiedenen Beispielen angezeigt.
+
+@c In @code{itensor} a tensor is represented as an "indexed object" . This is a
+@c function of 3 groups of indices which represent the covariant,
+@c contravariant and derivative indices. The covariant indices are
+@c specified by a list as the first argument to the indexed object, and
+@c the contravariant indices by a list as the second argument. If the
+@c indexed object lacks either of these groups of indices then the empty list
+@c @code{[]} is given as the corresponding argument. Thus, @code{g([a,b],[c])}
+@c represents an indexed object called @code{g} which has two covariant indices
+@c @code{(a,b)}, one contravariant index (@code{c}) and no derivative indices.
+
+Im Paket @code{itensor} werden Tensoren als indiziertes Objekte dargestellt.
+Ein indiziertes Objekt ist eine Funktion mit drei Gruppen an Indizes, die die
+kovarianten, kontravarianten und Ableitungsindizes eines Tensors darstellen.
+Das erste Argument der Funktion ist eine Liste der kovarianten Indizes und das
+zweite Argument die Liste der kontravarianten Indizes. Hat der Tensor keine
+entsprechenden Komponenten, dann wird eine leere Liste als Argument angegeben.
+Zum Beispiel repr@"asentiert @code{g([a,b], [c]} einen Tensor @code{g}, der zwei
+kovariante Indizes @code{[a,b]}, einen kontravarianten Index @code{[c]} und
+keinen Ableitungsindex hat. Mit der Funktion @mref{ishow} werden Tensoren
+in einer besonderen Schreibweise ausgegeben.
+
+Beispiele:
@example
-(%i2) imetric(g);
-(%o2) done
-(%i3) ishow(g([],[j,k])*g([],[i,l])*a([i,j],[]))$
- i l j k
-(%t3) g g a
- i j
-(%i4) ishow(contract(%))$
- k l
-(%t4) a
-@end example
-
-This result is incorrect unless @code{a} happens to be a symmetric tensor.
-The reason why this happens is that although @code{itensor} correctly maintains
-the order within the set of covariant and contravariant indices, once an
-index is raised or lowered, its position relative to the other set of
-indices is lost.
-
-To avoid this problem, a new notation has been developed that remains fully
-compatible with the existing notation and can be used interchangeably. In
-this notation, contravariant indices are inserted in the appropriate
-positions in the covariant index list, but with a minus sign prepended.
-Functions like @code{contract} and @code{ishow} are now aware of this
-new index notation and can process tensors appropriately.
-
-In this new notation, the previous example yields a correct result:
-
-@example
-(%i5) ishow(g([-j,-k],[])*g([-i,-l],[])*a([i,j],[]))$
- i l j k
-(%t5) g a g
- i j
-(%i6) ishow(contract(%))$
- l k
-(%t6) a
-@end example
+(%i1) load(itensor)$
-Presently, the only code that makes use of this notation is the @code{lc2kdt}
-function. Through this notation, it achieves consistent results as it
-applies the metric tensor to resolve Levi-Civita symbols without resorting
-to numeric indices.
+(%i2) g([a,b], [c]);
+(%o2) g([a, b], [c])
+
+(%i3) ishow(g([a,b], [c]))$
+ c
+(%t3) g
+ a b
+@end example
+
+@c The derivative indices, if they are present, are appended as
+@c additional arguments to the symbolic function representing the tensor.
+@c They can be explicitly specified by the user or be created in the
+@c process of differentiation with respect to some coordinate variable.
+@c Since ordinary differentiation is commutative, the derivative indices
+@c are sorted alphanumerically, unless @code{iframe_flag} is set to @code{true},
+@c indicating that a frame metric is being used. This canonical ordering makes
+@c it possible for Maxima to recognize that, for example, @code{t([a],[b],i,j)}
+@c is the same as @code{t([a],[b],j,i)}. Differentiation of an indexed object
+@c with respect to some coordinate whose index does not appear as an argument
+@c to the indexed object would normally yield zero. This is because
+@c Maxima would not know that the tensor represented by the indexed object
+@c might depend implicitly on the corresponding coordinate. By modifying the
+@c existing Maxima function @code{diff} in @code{itensor}, Maxima now assumes
+@c that all indexed objects depend on any variable of differentiation unless
+@c otherwise stated. This makes it possible for the summation convention to be
+@c extended to derivative indices. It should be noted that @code{itensor} does
+@c not possess the capabilities of raising derivative indices, and so they are
+@c always treated as covariant.
+
+Ableitungsindizes werden als weitere Argumente der Funktion hinzugef@"ugt, die
+den Tensor repr@"asentiert. Ableitungsindizes k@"onnen vom Nutzer angegeben
+oder bei der Ableitung von Tensoren von Maxima hinzugef@"ugt werden. Im
+Allgemeinen ist die Differentiation kommutativ, so dass die Reihenfolge der
+Ableitungsindizes keine Rolle spielt. Daher werden die Indizes von Maxima
+bei der Vereinfachung mit Funktionen wie @mref{rename} alphabetisch sortiert.
+Dies ist jedoch nicht der Fall, wenn bewegte Bezugssysteme genutzt werden, was
+mit der Optionsvariablen @mref{iframe_flag} angezeigt wird, die in diesem Fall
+den Wert @code{true} erh@"alt. Es ist zu beachten, dass mit dem Paket
+@code{itensor} Ableitungsindizes nicht angehoben werden k@"onnen und
+nur als kovariante Indizes auftreten.
-Since this code is brand new, it probably contains bugs. While it has been
-tested to make sure that it doesn't break anything using the "old" tensor
-notation, there is a considerable chance that "new" tensors will fail to
-interoperate with certain functions or features. These bugs will be fixed
-as they are encountered @dots{} until then, caveat emptor!
+Beispiele:
-@c -----------------------------------------------------------------------------
-@subsubsection Indicial tensor manipulation
-@c -----------------------------------------------------------------------------
+@example
+(%i1) load(itensor)$
-The indicial tensor manipulation package may be loaded by
-@code{load(itensor)}. Demos are also available: try @code{demo(tensor)}.
-
-In @code{itensor} a tensor is represented as an "indexed object" . This is a
-function of 3 groups of indices which represent the covariant,
-contravariant and derivative indices. The covariant indices are
-specified by a list as the first argument to the indexed object, and
-the contravariant indices by a list as the second argument. If the
-indexed object lacks either of these groups of indices then the empty
-list @code{[]} is given as the corresponding argument. Thus, @code{g([a,b],[c])}
-represents an indexed object called @code{g} which has two covariant indices
-@code{(a,b)}, one contravariant index (@code{c}) and no derivative indices.
-
-The derivative indices, if they are present, are appended as
-additional arguments to the symbolic function representing the tensor.
-They can be explicitly specified by the user or be created in the
-process of differentiation with respect to some coordinate variable.
-Since ordinary differentiation is commutative, the derivative indices
-are sorted alphanumerically, unless @code{iframe_flag} is set to @code{true},
-indicating that a frame metric is being used. This canonical ordering makes it
-possible for Maxima to recognize that, for example, @code{t([a],[b],i,j)} is
-the same as @code{t([a],[b],j,i)}. Differentiation of an indexed object with
-respect to some coordinate whose index does not appear as an argument
-to the indexed object would normally yield zero. This is because
-Maxima would not know that the tensor represented by the indexed
-object might depend implicitly on the corresponding coordinate. By
-modifying the existing Maxima function @code{diff} in @code{itensor}, Maxima now
-assumes that all indexed objects depend on any variable of
-differentiation unless otherwise stated. This makes it possible for
-the summation convention to be extended to derivative indices. It
-should be noted that @code{itensor} does not possess the capabilities of
-raising derivative indices, and so they are always treated as
-covariant.
-
-The following functions are available in the tensor package for
-manipulating indexed objects. At present, with respect to the
-simplification routines, it is assumed that indexed objects do not
-by default possess symmetry properties. This can be overridden by
-setting the variable @code{allsym[false]} to @code{true}, which will
-result in treating all indexed objects completely symmetric in their
-lists of covariant indices and symmetric in their lists of
-contravariant indices.
-
-The @code{itensor} package generally treats tensors as opaque objects.
-Tensorial equations are manipulated based on algebraic rules, specifically
-symmetry and contraction rules. In addition, the @code{itensor} package
-understands covariant differentiation, curvature, and torsion. Calculations
-can be performed relative to a metric of moving frame, depending on the setting
-of the @code{iframe_flag} variable.
-
-A sample session below demonstrates how to load the @code{itensor} package,
-specify the name of the metric, and perform some simple calculations.
+(%i2) ishow(t([a,b],[c],j,i))$
+ c
+(%t2) t
+ a b,j i
+(%i3) ishow(rename(%))$
+ c
+(%t3) t
+ a b,i j
+(%i4) ishow(t([a,b],[c],j,i) - t([a,b],[c],i,j))$
+ c c
+(%t4) t - t
+ a b,j i a b,i j
+(%i5) ishow(rename(%))$
+(%t5) 0
+(%i6) iframe_flag:true;
+(%o6) true
+(%i7) ishow(t([a,b],[c],j,i) - t([a,b],[c],i,j))$
+ c c
+(%t7) t - t
+ a b,j i a b,i j
+(%i8) ishow(rename(%))$
+ c c
+(%t8) t - t
+ a b,j i a b,i j
+@end example
+
+Das folgende Beispiel zeigt einen Ausdruck mit verschiedenen Ableitungen eines
+Tensors @code{g}. Ist @code{g} der metrische Tensor, dann entspricht das
+Ergebnis der Definition des Christoffel-Symbols der ersten Art.
+
+@example
+(%i1) load(itensor)$
+
+(%i2) ishow(1/2*(idiff(g([i,k],[]),j) + idiff(g([j,k],[]),i)
+ - idiff(g([i,j],[]),k)))$
+ g + g - g
+ j k,i i k,j i j,k
+(%t2) ------------------------
+ 2
+@end example
+
+@c The following functions are available in the tensor package for
+@c manipulating indexed objects. At present, with respect to the
+@c simplification routines, it is assumed that indexed objects do not
+@c by default possess symmetry properties. This can be overridden by
+@c setting the variable @code{allsym[false]} to @code{true}, which will
+@c result in treating all indexed objects completely symmetric in their
+@c lists of covariant indices and symmetric in their lists of
+@c contravariant indices.
+
+Tensoren werden standardm@"a@ss{}ig nicht als symmetrisch angenommen. Erh@"alt
+die Optionsvariable @mref{allsym} den Wert @code{true}, dann werden alle
+Tensoren als symmetrisch in den kovarianten und kontravarianten Indizes
+angenommen.
+
+@c The @code{itensor} package generally treats tensors as opaque objects.
+@c Tensorial equations are manipulated based on algebraic rules, specifically
+@c symmetry and contraction rules. In addition, the @code{itensor} package
+@c understands covariant differentiation, curvature, and torsion. Calculations
+@c can be performed relative to a metric of moving frame, depending on the
+@c setting of the @code{iframe_flag} variable.
+
+Das Paket @code{itensor} behandelt Tensoren im Allgemeinen als opake Objekte.
+Auf Tensorgleichungen werden algebraischen Regeln insbesondere Symmetrieregeln
+und Regeln f@"ur die Tensorverj@"ungung angewendet. Weiterhin kennt
+@code{itensor} die kovariante Ableitung, Kr@"ummung und die Torsion. Rechnungen
+k@"onnen in bewegten Bezugssystemen ausgef@"uhrt werden.
+
+@c A sample session below demonstrates how to load the @code{itensor} package,
+@c specify the name of the metric, and perform some simple calculations.
+
+Beispiele:
+
+Die folgenden Beispiele zeigen einige Anwendungen des Paketes @code{itensor}.
@example
(%i1) load(itensor);
@@ -306,6 +376,7 @@ specify the name of the metric, and perform some simple calculations.
(%t15) v - v ifc2
i,j %6 i j
(%i16) ishow(radcan(ev(%,ifc2,ifc1)))$
+@group
%6 %7 %6 %7
(%t16) - (ifg v ifb + ifg v ifb - 2 v
%6 j %7 i %6 i j %7 i,j
@@ -313,6 +384,7 @@ specify the name of the metric, and perform some simple calculations.
%6 %7
- ifg v ifb )/2
%6 %7 i j
+@end group
(%i17) ishow(canform(s([i,j],[])-s([j,i])))$
(%t17) s - s
i j j i
@@ -352,7 +424,7 @@ specify the name of the metric, and perform some simple calculations.
@subsubsection Behandlung indizierter Gr@"o@ss{}en
@c -----------------------------------------------------------------------------
-@c --- 04.09.2011 DK -----------------------------------------------------------
+@c --- 04.12.2011 DK -----------------------------------------------------------
@anchor{canten}
@deffn {Funktion} canten (@var{expr})
@@ -363,11 +435,12 @@ specify the name of the metric, and perform some simple calculations.
@c required simplification.
Ist vergleichbar mit der Funktion @mref{rename} und vereinfacht den Ausdruck
-@var{expr} indem Dummy-Indizes umbenannt und permutiert werden. Die Funktion
-@code{rename} kann nur Ausdr@"ucke mit Summen von Tensorprodukten vereinfachen,
-in denen keine Ableitungen nach Tensorkomponenten auftreten. Die Funktion
-@code{canten} hat dieselben Beschr@"ankungen und sollte nur verwendet werden,
-wenn die Funktion @mref{canform} nicht die gew@"unschte Vereinfachung erzielt.
+@var{expr} indem gebundene Indizes umbenannt und permutiert werden. Wie die
+Funktion @code{rename} kann @code{canten} nur Ausdr@"ucke mit Summen von
+Tensorprodukten vereinfachen, in denen keine Ableitungen nach Tensorkomponenten
+auftreten. Daher sollte @code{canten} nur verwendet werden, wenn sich mit
+der Funktion @mref{canform} nicht die gew@"unschte Vereinfachung eines
+Ausdrucks erzielen l@"asst.
@c The @code{canten} function returns a mathematically correct result only
@c if its argument is an expression that is fully symmetric in its indices.
@@ -378,6 +451,10 @@ Das Ergebnis der Funktion @code{canten} ist mathematisch nur korrekt, wenn
die Tensoren symmetrisch in ihren Indizes sind. Hat die Optionsvariable
@mref{allsym} @emph{nicht} den Wert @code{true}, bricht @code{canten} mit einer
Fehlermeldung ab.
+
+Siehe auch die Funktion @mrefcomma{concan} mit der Ausdr@"ucke mit Tensoren
+ebenfalls vereinfacht werden k@"onnen, wobei @code{concan} zus@"atzlich
+Tensorverj@"ungungen ausf@"uhrt.
@end deffn
@c --- 22.08.2011 DK -----------------------------------------------------------
@@ -390,10 +467,10 @@ Fehlermeldung ab.
@c objects called @var{name} with @var{m} covariant and @var{n} contravariant
@c indices will be renamed to @var{new}.
-@"Andert den Namen aller indizierten Objekte im Ausdruck @var{expr} von
+@"Andert den Namen aller Tensoren im Ausdruck @var{expr} von
@var{old} nach @var{new}. Das Argument @var{old} kann ein Symbol oder eine
Liste der Form @code{[@var{name}, @var{m}, @var{n}]} sein. Im letzteren Fall
-werden nur die Tensoren zu @var{new} umbenannt, die den Namen @var{name} haben
+werden nur die Tensoren zu @var{new} umbenannt, die den Namen @var{name}
sowie @var{m} kovariante und @var{n} kontravariante Indizes haben.
Beispiel:
@@ -549,7 +626,7 @@ Funktion gegeben.
@c Similar to @code{canten} but also performs index contraction.
Ist vergleichbar mit der Funktion @mrefdot{canten} Im Unterschied zu
-@code{canten} wird zus@"atzlich die Tensorverj@"ungung ausgef@"uhrt.
+@code{canten} werden zus@"atzlich Tensorverj@"ungungen ausgef@"uhrt.
@end deffn
@c --- 03.09.2011 DK -----------------------------------------------------------
@@ -1025,19 +1102,67 @@ To reenable the Lisp debugger set *debugger-hook* to nil.
@end example
@end deffn
-@c --- 04.09.2011 DK -----------------------------------------------------------
+@c --- 02.12.2011 DK -----------------------------------------------------------
@anchor{levi_civita}
@deffn {Funktion} levi_civita (@var{L})
-@c is the permutation (or Levi-Civita) tensor which yields 1 if
-@c the list @var{L} consists of an even permutation of integers, -1 if it
-@c consists of an odd permutation, and 0 if some indices in @var{L} are
-@c repeated.
-
Ist der Levi-Civita-Tensor, der auch Permutationstensor genannt wird. Der
Tensor hat den Wert @code{1}, wenn die Liste @var{L} eine gerade Permutation
-ganzer Zahlen ist, @code{-1} f@"ur eine ungerade Permutation und ansonsten
-@code{0}.
+ganzer Zahlen ist, den Wert @code{-1} f@"ur eine ungerade Permutation und
+ansonsten den Wert @code{0}.
+
+Beispiel:
+
+F@"ur eine Kreisbewegung ist die Bahngeschwindigkeit @code{v} das Kreuzprodukt
+aus Winkelgeschwindigkeit @code{w} und Ortsvektor @code{r}. Wir haben also
+@code{v = w x r}. Hier wird eine tensorielle Schreibweise des Kreuzproduktes
+mit dem Levi-Civita-Tensor eingef@"uhrt. Der Ausdruck wird sodann f@"ur die
+erste Komponente zu der bekannten Definition des Kreuzproduktes vereinfacht.
+
+@example
+(%i1) load(itensor)$
+
+(%i2) ishow(v([],[a])=
+ 'levi_civita([],[a,b,c])*w([b],[])*r([c],[]))$
+@group
+ a a b c
+(%t2) v = levi_civita w r
+ b c
+@end group
+(%i3) ishow(subst([a=1],%))$
+ 1 1 b c
+(%t3) v = levi_civita w r
+ b c
+(%i4) ishow(ev(%, levi_civita))$
+ 1 1 b c
+(%t4) v = kdelta w r
+ 1 2 3 b c
+(%i5) ishow(expand(ev(%, kdelta)))$
+ 1 b c c b
+(%t5) v = kdelta kdelta w r - kdelta kdelta w r
+ 2 3 b c 2 3 b c
+(%i6) ishow(contract(%))$
+ 1
+(%t6) v = w r - r w
+ 2 3 2 3
+@end example
+
+In diesem Beispiel wird das Spatprodukt von drei Vektoren @code{a}, @code{b} und
+@code{b} mit dem Levi-Civita-Tensor definiert und dann vereinfacht.
+
+@example
+(%i1) load(itensor)$
+
+(%i2) ishow(levi_civita([],[i,j,k])*a([i],[])*b([j],[])*c([k],[]))$
+ i j k
+(%t2) kdelta a b c
+ 1 2 3 i j k
+(%i3) ishow(contract(expand(ev(%,kdelta))))$
+(%t3) a b c - b a c - a c b + c a b + b c a
+ 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
+ - c b a
+ 1 2 3
+@end example
@end deffn
@c --- 22.08.2011 DK -----------------------------------------------------------
@@ -1143,12 +1268,14 @@ Beispiele:
*ichr2([%7,r],[%2])$
(%i4) expr: ishow(%)$
+@group
%4 %5 %6 %7 %3 u %1 %2
(%t4) g g ichr2 ichr2 ichr2 ichr2
%1 %4 %2 %3 %5 %6 %7 r
%4 %5 %6 %7 u %1 %3 %2
- g g ichr2 ichr2 ichr2 ichr2
%1 %2 %3 %5 %4 %6 %7 r
+@end group
(%i5) flipflag: true;
(%o5) true
(%i6) ishow(rename(expr))$
@@ -1193,12 +1320,11 @@ eines Tensors mit einer h@"oheren Stufe als 2 zeigen.
Beispiel:
@example
-(%i1) load(ctensor);
-(%o1) /share/tensor/ctensor.mac
-(%i2) load(itensor);
-(%o2) /share/tensor/itensor.lisp
+(%i1) load(ctensor)$
+(%i2) load(itensor)$
(%i3) lg:matrix([sqrt(r/(r-2*m)),0,0,0],[0,r,0,0],
[0,0,sin(theta)*r,0],[0,0,0,sqrt((r-2*m)/r)]);
+@group
[ r ]
[ sqrt(-------) 0 0 0 ]
[ r - 2 m ]
@@ -1210,6 +1336,7 @@ Beispiel:
[ r - 2 m ]
[ 0 0 0 sqrt(-------) ]
[ r ]
+@end group
(%i4) components(g([i,j],[]),lg);
(%o4) done
(%i5) showcomps(g([i,j],[]));
@@ -1460,7 +1587,7 @@ Beispiele:
@c is set to @code{true}.
@code{idiff} f@"uhrt Ableitungen nach den Koordinaten einer tensoriellen
-Gr@"o@ss{}e aus. Im Unterschied dazu f@"uhrt die Funktion @mref{diff}
+Gr@"o@ss{}e aus. Im Unterschied dazu f@"uhrt die Funktion @mref{diff}@w{}
Ableitungen nach den unabh@"angigen Variablen aus. Eine tensorielle Gr@"o@ss{}e
erh@"alt zus@"atzlich den Index @var{v_1}, der die Ableitung bezeichnet.
Mehrfache Indizes f@"ur Ableitungen werden sortiert, au@ss{}er wenn die
@@ -1690,6 +1817,7 @@ Kronecker-Delta hat.
enth@"alt, die mit der Funktion @code{coord} die Eigenschaft der kovarianten
Ableitung erhalten haben.
+@need 800
Beispiel:
@example
@@ -1779,17 +1907,21 @@ enthalten. Im besonderen erkennt @code{simpmetderiv} die folgenden
Identit@"aten:
@example
+@group
ab ab ab a
g g + g g = (g g ) = (kdelta ) = 0
,d bc bc,d bc ,d c ,d
+@end group
@end example
-daher
+daher ist
@example
+@group
ab ab
g g = - g g
,d bc bc,d
+@end group
@end example
und
@@ -1881,6 +2013,16 @@ Setzt alle tensoriellen Gr@"o@ss{}en, die genau einen Ableitungsindex haben,
auf den Wert Null.
@end deffn
+@c --- 04.12.2011 DK -----------------------------------------------------------
+@anchor{vect_coords}
+@defvr {Optionsvariable} vect_coords
+Standardwert: @code{false}
+
+Tensoren k@"onnen durch Angabe von ganzen Zahlen nach den einzelnen Komponenten
+abgeleitet werden. In diesem Fall bezeichnen die ganzen Zahlen der Reihe nach
+die Indizes, die in der Optionsvariablen @code{vect_coords} abgelegt sind.
+@end defvr
+
@c -----------------------------------------------------------------------------
@need 1200
@node Tensoren in gekr@"ummten R@"aumen, Begleitende Vielbeine, Tensoranalysis, Funktionen und Variablen f@"ur itensor
@@ -2176,6 +2318,35 @@ the @code{iframe_bracket_form} flag is set to @code{false}:
@end example
@c -----------------------------------------------------------------------------
+@anchor{iframe_flag}
+@defvr {Optionsvariable} iframe_flag
+Standardwert: @code{false}
+
+To use frames, you must first set @code{iframe_flag} to @code{true}. This
+causes the Christoffel-symbols, @code{ichr1} and @code{ichr2}, to be replaced
+by the more general frame connection coefficients @code{icc1} and @code{icc2}
+in calculations. Speficially, the behavior of @code{covdiff} and
+@code{icurvature} is changed.
+
+The frame is defined by two tensors: the inverse frame field (@code{ifri},
+the dual basis tetrad),
+and the frame metric @code{ifg}. The frame metric is the identity matrix for
+orthonormal frames, or the Lorentz metric for orthonormal frames in Minkowski
+spacetime. The inverse frame field defines the frame base (unit vectors).
+Contraction properties are defined...
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