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[Maxima-commits] CVS: maxima/doc/info Ctensor.texi,1.28,1.29
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From: Vadim V. Z. <vv...@us...> - 2005-11-27 21:05:42
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Update of /cvsroot/maxima/maxima/doc/info In directory sc8-pr-cvs1.sourceforge.net:/tmp/cvs-serv23159 Modified Files: Ctensor.texi Log Message: Valious small inprovments. Index: Ctensor.texi =================================================================== RCS file: /cvsroot/maxima/maxima/doc/info/Ctensor.texi,v retrieving revision 1.28 retrieving revision 1.29 diff -u -d -r1.28 -r1.29 --- Ctensor.texi 16 Jun 2005 17:05:28 -0000 1.28 +++ Ctensor.texi 27 Nov 2005 21:05:34 -0000 1.29 @@ -1,6 +1,6 @@ @menu -* Introduction to ctensor:: -* Definitions for ctensor:: +* Introduction to ctensor:: +* Definitions for ctensor:: @end menu @node Introduction to ctensor, Definitions for ctensor, ctensor, ctensor @@ -14,15 +14,16 @@ or @code{[x,y,z,t]} respectively. These names may be changed by assigning a new list of coordinates to the variable @code{ct_coords} (described below) and the user is queried about -this. -** Care must be taken to avoid the coordinate names conflicting -with other object definitions **. +this. Care must be taken to avoid the coordinate names conflicting +with other object definitions. Next, the user enters the metric either directly or from a file by -specifying its ordinal position. As an example of a file of common -metrics, see @code{share/tensor/metrics.mac}. The metric is stored in the matrix -LG. Finally, the metric inverse is computed and stored in the matrix -UG. One has the option of carrying out all calculations in a power +specifying its ordinal position. +@c NO SUCH FILE ! +@c As an example of a file of common metrics, see @code{share/tensor/metrics.mac}. +The metric is stored in the matrix +@code{lg}. Finally, the metric inverse is computed and stored in the matrix +@code{ug}. One has the option of carrying out all calculations in a power series. A sample protocol is begun below for the static, spherically symmetric @@ -33,9 +34,9 @@ @example (%i1) load(ctensor); -(%o1) /usr/local/lib/maxima/share/tensor/ctensor.mac +(%o1) /share/tensor/ctensor.mac (%i2) csetup(); -Enter the dimension of the coordinate system: +Enter the dimension of the coordinate system: 4; Do you wish to change the coordinate names? n; @@ -60,9 +61,9 @@ -d; Matrix entered. -Enter functional dependencies with the DEPENDS function or 'N' if none +Enter functional dependencies with the DEPENDS function or 'N' if none depends([a,d],x); -Do you wish to see the metric? +Do you wish to see the metric? y; [ a 0 0 0 ] [ ] @@ -137,14 +138,14 @@ further calculations. If @code{cframe_flag} is false, the function computes the inverse metric -ug from the (user-defined) matrix @code{lg}. The metric determinant is +@code{ug} from the (user-defined) matrix @code{lg}. The metric determinant is also computed and stored in the variable @code{gdet}. Furthermore, the package determines if the metric is diagonal and sets the value of @code{diagmetric} accordingly. If the optional argument @var{dis} -is present and not equal to false, the user is prompted to see +is present and not equal to @code{false}, the user is prompted to see the metric inverse. -If @code{cframe_flag} is true, the function expects that the values of +If @code{cframe_flag} is @code{true}, the function expects that the values of @code{fri} (the inverse frame matrix) and @code{lfg} (the frame metric) are defined. From these, the frame matrix @code{fr} and the inverse frame metric @code{ufg} are computed. @@ -162,27 +163,27 @@ -------------------------------------------------------------------------- cartesian2d 2 [x,y] Cartesian 2D coordinate system polar 2 [r,phi] Polar coordinate system - elliptic 2 [u,v] - confocalelliptic 2 [u,v] - bipolar 2 [u,v] - parabolic 2 [u,v] + elliptic 2 [u,v] Elliptic coordinate system + confocalelliptic 2 [u,v] Confocal elliptic coordinates + bipolar 2 [u,v] Bipolar coordinate system + parabolic 2 [u,v] Parabolic coordinate system cartesian3d 3 [x,y,z] Cartesian 3D coordinate system - polarcylindrical 3 [r,theta,z] - ellipticcylindrical 3 [u,v,z] Elliptic 2D with cylindrical Z - confocalellipsoidal 3 [u,v,w] - bipolarcylindrical 3 [u,v,z] Bipolar 2D with cylintrical Z - paraboliccylindrical 3 [u,v,z] Parabolic 2D with cylindrical Z - paraboloidal 3 [u,v,phi] - conical 3 [u,v,w] - toroidal 3 [u,v,phi] + polarcylindrical 3 [r,theta,z] Polar 2D with cylindrical z + ellipticcylindrical 3 [u,v,z] Elliptic 2D with cylindrical z + confocalellipsoidal 3 [u,v,w] Confocal ellipsoidal + bipolarcylindrical 3 [u,v,z] Bipolar 2D with cylintrical z + paraboliccylindrical 3 [u,v,z] Parabolic 2D with cylindrical z + paraboloidal 3 [u,v,phi] Paraboloidal coordinates + conical 3 [u,v,w] Conical coordinates + toroidal 3 [u,v,phi] Thoroidal coordinates spherical 3 [r,theta,phi] Spherical coordinate system - oblatespheroidal 3 [u,v,phi] + oblatespheroidal 3 [u,v,phi] Oblate spheroidal coordinates oblatespheroidalsqrt 3 [u,v,phi] - prolatespheroidal 3 [u,v,phi] + prolatespheroidal 3 [u,v,phi] Prolate speroidal coordinates prolatespheroidalsqrt 3 [u,v,phi] - ellipsoidal 3 [r,theta,phi] + ellipsoidal 3 [r,theta,phi] Ellipsoidal coordinates cartesian4d 4 [x,y,z,t] Cartesian 4D coordinate system - spherical4d 4 [r,theta,eta,phi] + spherical4d 4 [r,theta,eta,phi] Spherical 4D coordinate system exteriorschwarzschild 4 [t,r,theta,phi] Schwarzschild metric interiorschwarzschild 4 [t,z,u,v] Interior Schwarzschild metric kerr_newman 4 [t,r,theta,phi] Charged axially symmetric metric @@ -279,7 +280,7 @@ lg -- ug \ \ - lcs -- mcs -- ric -- uric + lcs -- mcs -- ric -- uric \ \ \ \ tracer - ein -- lein \ @@ -320,6 +321,7 @@ displayed. If the argument is @code{false} then the display of the elements will not occur. The array elements @code{mcs[i,j,k]} are defined in such a manner that the final index is contravariant. + @end deffn @deffn {Function} ricci (@var{dis}) @@ -327,6 +329,7 @@ package. @code{ricci} computes the covariant (symmetric) components @code{ric[i,j]} of the Ricci tensor. If the argument @var{dis} is @code{true}, then the non-zero components are displayed. + @end deffn @deffn {Function} uricci (@var{dis}) @@ -334,18 +337,19 @@ covariant components @code{ric[i,j]} of the Ricci tensor. Then the mixed Ricci tensor is computed using the contravariant metric tensor. If the value of the argument @var{dis} -is @code{true}, then these mixed components, @code{uric[i,j]} (the index i is -covariant and the index j is contravariant), will be displayed +is @code{true}, then these mixed components, @code{uric[i,j]} (the +index @code{i} is covariant and the index @code{j} is contravariant), will be displayed directly. Otherwise, @code{ricci(false)} will simply compute the entries of the array @code{uric[i,j]} without displaying the results. @end deffn -@deffn {Function} scurvature () -returns the scalar curvature (obtained by contracting +@deffn {Function} scurvature () +Returns the scalar curvature (obtained by contracting the Ricci tensor) of the Riemannian manifold with the given metric. @end deffn + @deffn {Function} einstein (@var{dis}) A function in the @code{ctensor} (component tensor) package. @code{einstein} computes the mixed Einstein tensor @@ -358,6 +362,7 @@ also be factored. @end deffn + @deffn {Function} leinstein (@var{dis}) Covariant Einstein-tensor. @code{leinstein} stores the values of the covariant Einstein tensor in the array @code{lein}. The covariant Einstein-tensor is computed from the mixed Einstein tensor @code{ein} by multiplying it with the metric tensor. If the argument @var{dis} is @code{true}, then the non-zero values of the covariant Einstein tensor are displayed. @@ -375,7 +380,7 @@ ijk ij,k ik,j mk ij mj ik @end example -This notation is consistent with the notation used by the ITENSOR +This notation is consistent with the notation used by the @code{itensor} package and its @code{icurvature} function. If the optional argument @var{dis} is @code{true}, the non-zero components @code{riem[i,j,k,l]} will be displayed. @@ -518,22 +523,22 @@ (%i10) einstein(false); (%o10) done (%i11) ntermst(ein); -[[1, 1], 62] -[[1, 2], 0] -[[1, 3], 0] -[[1, 4], 0] -[[2, 1], 0] -[[2, 2], 24] -[[2, 3], 0] -[[2, 4], 0] -[[3, 1], 0] -[[3, 2], 0] -[[3, 3], 46] -[[3, 4], 0] -[[4, 1], 0] -[[4, 2], 0] -[[4, 3], 0] -[[4, 4], 46] +[[1, 1], 62] +[[1, 2], 0] +[[1, 3], 0] +[[1, 4], 0] +[[2, 1], 0] +[[2, 2], 24] +[[2, 3], 0] +[[2, 4], 0] +[[3, 1], 0] +[[3, 2], 0] +[[3, 3], 46] +[[3, 4], 0] +[[4, 1], 0] +[[4, 2], 0] +[[4, 3], 0] +[[4, 4], 46] (%o12) done @end example @@ -558,22 +563,22 @@ (%i20) einstein(false); (%o20) done (%i21) ntermst(ein); -[[1, 1], 6] -[[1, 2], 0] -[[1, 3], 0] -[[1, 4], 0] -[[2, 1], 0] -[[2, 2], 13] -[[2, 3], 2] -[[2, 4], 0] -[[3, 1], 0] -[[3, 2], 2] -[[3, 3], 9] -[[3, 4], 0] -[[4, 1], 0] -[[4, 2], 0] -[[4, 3], 0] -[[4, 4], 9] +[[1, 1], 6] +[[1, 2], 0] +[[1, 3], 0] +[[1, 4], 0] +[[2, 1], 0] +[[2, 2], 13] +[[2, 3], 2] +[[2, 4], 0] +[[3, 1], 0] +[[3, 2], 2] +[[3, 3], 9] +[[3, 4], 0] +[[4, 1], 0] +[[4, 2], 0] +[[4, 3], 0] +[[4, 4], 9] (%o21) done (%i22) ratsimp(ein[1,1]); 2 2 4 2 2 @@ -592,7 +597,7 @@ field limit far from a gravitational source. @end deffn - + @subsection Frame fields @@ -685,7 +690,7 @@ (%i5) weyl(false); (%o5) done (%i6) nptetrad(true); -(%t6) np = +(%t6) np = [ sqrt(r - 2 m) sqrt(r) ] [ --------------- --------------------- 0 0 ] @@ -704,15 +709,15 @@ [ sqrt(2) sqrt(2) ] sqrt(r) sqrt(r - 2 m) -(%t7) npi = matrix([- ---------------------, ---------------, 0, 0], +(%t7) npi = matrix([- ---------------------, ---------------, 0, 0], sqrt(2) sqrt(r - 2 m) sqrt(2) sqrt(r) sqrt(r) sqrt(r - 2 m) -[- ---------------------, - ---------------, 0, 0], +[- ---------------------, - ---------------, 0, 0], sqrt(2) sqrt(r - 2 m) sqrt(2) sqrt(r) 1 %i -[0, 0, ---------, --------------------], +[0, 0, ---------, --------------------], sqrt(2) r sqrt(2) r sin(theta) 1 %i |