Comment: | Scripts by Dieter (Olli) Egger (November 30th 2019) |
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Downloads: | Tarball | ZIP archive | SQL archive |
Timelines: | family | descendants | 2009.11.30 | trunk |
Files: | files | file ages | folders |
SHA3-256: |
5ce5316a33430f7c474932758c3e4001 |
User & Date: | jeff@gridfinity.com on 2021-03-01 07:23:02 |
Other Links: | manifest | tags |
2021-03-01
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07:27:08 | Update README.md: Small update, markdownlint fixes check-in: bc03f92dc3 user: jeff@gridfinity.com tags: master, trunk | |
07:23:02 | Scripts by Dieter (Olli) Egger (November 30th 2019) Leaf check-in: 5ce5316a33 user: jeff@gridfinity.com tags: 2009.11.30, trunk | |
Added .gitattributes version [f9d1f1840e].
> > > > > > | 1 2 3 4 5 6 | # Effectively disable Linguist doc/* linguist-vendored src/* linguist-vendored *.md linguist-vendored *.txt linguist-vendored *.red linguist-vendored |
Added .github/issue-close-app.yml version [9a3b66da03].
> > > > > > > > | 1 2 3 4 5 6 7 8 | comment: "Thanks for your report! Unfortunately, we don't use GitHub Issues to manage bug reports for this repository. Instead, please contact the author via [DL2MIE](https://dl2mie.darc.de/)." issueConfigs: - content: - "a2ee3e2e1e0c1b7891c5a390877c070d88745becb9d42d5ecb9cfd03b4dae593367d42ff120b9a836ec944ad242d05b9cc70cac08f13d0f1497dada9b3dc0cfc" - "4373650124fbaa13f0eebd84723ab2e726dd86284bbee61f93cb22426ed2f1e0643b1765f25c68927d0ee3dcb5ccf9b0c4fbe18b9cb95f8a854e66415810841c" - "8156afa9af9e68a82b12c3a33cf6110c2376f05be913d0826da9e56363464d8824b66099ece229fabf88cac5a50bab3ee1c2f5ce491c5bc2ca43f4507ede52f9" - "7c5dc57df185d19fe8656621fa6b5ceaedf04cf9b09968973f8af70095a4b8e4f086a8e75709dc6b28e220d35469c7c1a845c537807f6c98ba6464751f2d051a" |
Added .github/mergeable.yml version [7fb133d986].
> > > > > > > > > > > | 1 2 3 4 5 6 7 8 9 10 11 | version: 2 mergeable: - when: pull_request.opened name: "Greet a contributor" validate: [] pass: - do: comment payload: body: > Thanks for your contribution! Unfortunately, we don't use GitHub pull requests to manage code contributions to this repository. Instead, please contact the author via [DL2MIE](https://dl2mie.darc.de/). - do: close |
Added .gitignore version [a7ffc6f8bf].
Added README.md version [a1a82c2af5].
> > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 | # REDUCE/Symbolic Scripts ## by Dieter Egger ### Script Descriptions - Solving equations: `algebra.txt` - Analyzing functions: `analyze.red` - Decide on the function to be analyzed, for example `"f:=x**3 - 2*x**2;"` - Properties: `boolean.txt` - Analysis: `calculus.txt` - Physical constants: establish definitions: constants.red; delete definitions: `clear_constants.red` - Sine, cosine, tangent: `example01.red` - Binomials, trigonometry, computational accuracy: `example02.red` - Simple function analysis: `example03.red` - Simple derivatives: `example04.red` - Binomials, rules: `example05.red` - Nested parentheses: `heron.txt` - Hypergeometry and MeijerG: `hypermeijerg.txt` - Without LaTeX formatting OK. - Integration, limits: `integLimit.red` - Integration: `integral.txt` - Introduction: `introReduce.txt` - Linear algebra: `linalg.txt` - Matrix inversion: `mat.red` - Computation of space-time metrics - Metric-tensor from the equation of the hypersurface of a hypersphere, 2-dim: `metric2calc.red`, 3-dim: `metric3calc.red`, 4-dim: `metric4calc.red` - Riemann, Ricci, Einstein tensors and solution of the field equations, 2-dim: `metric2d.red`, 3-dim: `metric3d.red`, 4-dim: `metric4d.red` - Dynamic variable names: `mkid.txt` - Partial fractions: `partialFraction.txt` - Energy of a photon (requires `constants.red` and `clear_constants.red`): `photonenergy.txt` - Prefix operators: `prefix.txt` - Programming: `programming.txt` - Simple rules: `rules.red` - Definition of a function: `scalprod.red` - Speed of light: `speedoflight.txt` ### Homepage - [DL2MIE](https://dl2mie.darc.de/) ### Links - [Symbolic for Android](https://play.google.com/store/apps/details?id=de.dieteregger.symbolic) - [German Script Homepage](https://reduce-algebra.sourceforge.io/tutorials/EggerScripts.php) - [English Script Homepage](https://reduce-algebra.sourceforge.io/tutorials/EggerScripts.en.php) |
Added SECURITY.md version [4bf8ab023a].
> > > > > | 1 2 3 4 5 | # Security Information ## Reporting a Vulnerability Please visit the author's homepage, [DL2MIE](https://dl2mie.darc.de/), for contact details. |
Added doc/MetricUniverse.pdf version [bdfb9288e6].
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Added doc/Metrik.pdf version [2a57d66e37].
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Added src/algebra.txt version [4a09a9e43d].
> > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 | %%%%%%%%%%%%%%%%%%%%% % ALGEBRA (SOLVE) %%%%%%%%%%%%%%%%%%%%% % Solve quadratic equation solve(x^2+8x+15=0, x); % Solve for expression solve(a*log(sin(x+3))^2 - b, sin(x+3)); % Solve simultaneous equations solve({x+3y=7, y-x=1},{x,y}); % Solve a system with parameters solve({x=a*z+1, y=b*z},{z,x}); % Simplify expression ((((-r1*(1+k1))/(r2*(1+k2)))+((r1)/(r2)))/(((r1)/(r2)))); % Another solve example % Note the use of $ as the line termination % character to suppress output from % intermediate computations x1 := sqrt(h^2 + p1^2)$ x2 := sqrt((h/2)^2 + (p-p1)^2)$ x3 := x1 + x2$ dx := df(x3, p1)$ solve(dx, p1); % Suppose you are given the equation % x^2+x+1=0 and wish to determine the % value of x^3. The following simple % substitution achieves this. rule := solve(x^2+x+1=0,x)$ y := (x^3 where rule); % Then y=1, because % x^3=x*(x^2)=-x*(x+1)=-x^2-x=1. end; |
Added src/analyze.red version [5edece8dce].
> > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 | % function analysis % Function f (x) is defined ? if (freeof(f,x)) then << write "first define function f(x)"; end; >> fp:=df (f, x); fpp:=df (fp, x); % zeroes xz:=solve (f, x); % extremes xe:=solve (fp, x); % reversal points xr:=solve (fpp, x); % extreme values x1:=first (xe); y1:=sub (x1, f); y2:=sub (x1, fpp); on rounded; if numberp(y2) then if y2<0 then write "local maximum" else if y2=0 then write "reversal point" else write "local minimum"; off rounded; % integration of 2nd derivative f1:=int (fpp, x); % integration of 1st derivative f0:=int (f1, x); f0:=int (fp, x); end; |
Added src/boolean.txt version [62850ba20a].
> > > > > > > > > > > > > > > > > > > > > > > > > > | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 | n:=3; if (evenp(n)) then write n," is even" else write n, " is odd"; if (fixp(n)) then write n," is Integer" else write n, " is not Integer"; n:=pi; if (fixp(n)) then write n," is Integer" else write n, " is not Integer"; n:=3.1; if (fixp(n)) then write n," is Integer" else write n, " is not Integer"; % a:=(b+c)^2; if (freeof(a,b)) then write b," not in ",a else write b," is in ",a; a:=(c+d)^2; if (freeof(a,b)) then write b," not in ",a else write b," is in ",a; % n:=pi; if (numberp(n)) then write n," is a number" else write n, " is not a number"; n:=3.1; if (numberp(n)) then write n," is a number" else write n, " is not a number"; % if (ordp(n,c)) then write n," before ",c else write n, " after ",c; n:=z; if (ordp(n,c)) then write n," before ",c else write n, " after ",c; % n:=10; if (primep(n)) then write n," is prime" else write n, " is not prime"; n:=11; if (primep(n)) then write n," is prime" else write n, " is not prime"; |
Added src/calculus.txt version [4984952ae5].
> > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 | %%%%%%%%%%%%%%%%%%%%% % CALCULUS %%%%%%%%%%%%%%%%%%%%% % Specify blue for echoed input %color("Blue"); % Turn on fancy output %fancy_output; % Turn input echo on %on echo; % Differentiation % df/dx df(x^x,x); % Multivariate Differentiation % df/((dx)(dy^2)(dz)) df(x*exp(i*y)*log(z), x, 1, y, 2, z, 1); % Integration % indefinite integral with respect to x int(x^2 + x*sin(x), x); % SI(x) int(sin(x)/x,x); % Integral in interval [-oo, oo] int(exp(-x^2), x,-infinity,infinity); % Integral with logarithms int(log(log(x)),x); % Turn off echo %off echo; |
Added src/clear_constants.red version [366dfd4dfc].
> > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 | % speed of light clear(c)$ % Planck's constant clear(h)$ % elementary charge clear(el)$ % newtonian constant of gravity clear(g)$ % Avogadro's number clear(a)$ % Boltzmann's constant clear(b)$ % mass of electron clear(me)$ % mass of neutron clear(mn)$ % mass of proton clear(mp)$ % dielectric constant clear(epsilon0)$ % permeability coefficient mue_0 = 4*pi*1e-7 clear(mu0)$ % Einstein's gravitational constant kappa = 8*pi*G / c^4 clear(kappa)$ % radius of universe [m] clear(ru)$ % age of universe [s] clear(tu)$ % critical mass density for a closing universe [cmd] clear(cmd)$ % astronomical unit clear(au)$ % Parsec clear(pc)$ % seconds per sidereal day clear(sidd)$ % days per tropical year clear(tropy)$ % Planck mass clear(pm)$ % Planck time clear(pt)$ off rounded; end; |
Added src/constants.red version [3bd159fe32].
> > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 | on rounded; % speed of light c:=299792458.0$ % Planck's constant h:=6.62606957E-34$ % elementary charge el:=1.602176565E-19$ % newtonian constant of gravity g:=6.67384E-11$ % Avogadro's number a:=6.02214129E23$ % Boltzmann's constant b:=1.3806488E-23$ % mass of electron me:=9.10938291e-31$ % mass of neutron mn:=1.674927351e-27$ % mass of proton mp:=1.672621777e-27$ % dielectric constant epsilon0:=8.85418781762039E-12$ % permeability coefficient mue_0 = 4*pi*1e-7 mu0:=1.25663706143592E-06$ % Einstein's gravitational constant kappa = 8*pi*G / c^4 kappa:=2.076504e-43$ % radius of universe [m] ru:=1.296120e26$ % age of universe [s] tu:=4.323391e17$ % critical mass density for a closing universe [cmd] cmd:=9.568779e-27$ % astronomical unit au:=149597870700$ % Parsec pc:=3.08567758149136E+16$ % seconds per sidereal day sidd:=86164.0991483654$ % days per tropical year tropy:=365.24219879$ % Planck mass pm:=2.17650925244531E-08$ % Planck time pt:=5.3910604238861E-44$ end; |
Added src/example01.red version [26599d6c9f].
> > > > > > > > | 1 2 3 4 5 6 7 8 | a:=sin (x); b:=cos (x); c:=a^2+b^2; d:=a/b; f:=tan (x)-d; trigsimp c; trigsimp f; end; |
Added src/example02.red version [db6f1a63d4].
> > > > > > > > > > > > > | 1 2 3 4 5 6 7 8 9 10 11 12 13 | u:=(x+y+z)^3; on factor; u; df(x^x,x); int(ws,x); sin(pi/4); sin(x)^2+cos(x)^2; trigsimp ws; v:=sqrt(pi); on rounded; v; precision(24); v; off rounded; end; |
Added src/example03.red version [f5853dc345].
> > > > > > > > > > > > > > > > > > > > | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 | % function analysis f:=3*x^3-7*x^2; fp:=df (f, x); fpp:=df (fp, x); % zeroes solve (f, x); % extremes solve (fp, x); % reversal points solve (fpp, x); % integration of 2nd derivative f1:=int (fpp, x); % integration of 1st derivative f0:=int (f1, x); f0:=int (fp, x); end; |
Added src/example04.red version [0addcc0385].
> > > > > > > > > | 1 2 3 4 5 6 7 8 9 | a:=sin(x); b:=cos(x); c:=a+b; d:=c^2; % derivative df(d, x); % manual derivative 2*(a+b)*(df(a,x)+df(b,x)); end; |
Added src/example05.red version [f430f73384].
> > > > > > > > > > > > > > > > > > > > > > > | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 | (a+b)^2; (a+b)^3; (a+b)^4; (a+b)^5; (a+b)^6; c:=(a+b)^6; d:=(a+b)^3; c/d; a:=sin (x); b:=cos (x); c:=a^2+b^2; d:=a/b; f:=tan (x)-d; % Trigo Rules; trig1:={sin(~x)^2=>(1-cos(x)^2)}; let trig1; trig2:={tan (~x)=>(sin (x)/cos (x))}; let trig2; % now with rules; c; d; f; end; |
Added src/heron.txt version [3e5894bb3b].
> > > > > > > | 1 2 3 4 5 6 7 | a*(b+c*(d+e*(f+g*(h+j*(k+l*(m+n*(o+p*(q+r)))))))); z:=ws; off factor; z; on factor; z; end; |
Added src/hypermeijerg.txt version [1bd35ffc38].
> > > > > > > > > > > > > > > > > > > > > > > > > > > > | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 | %%%%%%%%%%%%%%%%%%%%% % HYPERGEOMETRY % & % MEIJERG %%%%%%%%%%%%%%%%%%%%% % Load special functions 2 load_package specfn2$ % Load generalized hypergeometric package load_package ghyper$ % Hypergeometric function hypergeometric({1/2,1},{2},z); % Load meijerg package load_package meijerg$ % MeijerG function % Latex does not work MeijerG({{}},{{5/4},1},x^2/2); % without Latex OK off nat; MeijerG({{}},{{5/4},1},x^2/2); on nat; end; |
Added src/integLimit.red version [849ec37768].
> > > > > > > > > > | 1 2 3 4 5 6 7 8 9 10 | f:=e^(-x)*sin (x); int (f, x); int (f, x, 0, infinity); int (f, x, 0, b); sin (b)+cos (b); c:=sin (b)+cos (b); limit (c, b,infinity); limit ((2*x+5)/(3*x-2), x, infinity); int (e^(-x)*sin (x), x, 0, infinity); end; |
Added src/integral.txt version [9ab6dc4c0e].
> > > > > > > > > > > | 1 2 3 4 5 6 7 8 9 10 11 | y:=e^(-x)*sin (x); int (y, x, 0, infinity); int (e^(-x)*sin (x), x, 0, infinity); y; df (y, x); int (y, x); g:=ws; gup:=limit (g, x, infinity); glw:=sub (x=0, g); f:=gup-glw; end; |
Added src/introReduce.txt version [938a1ca51e].
> > > > > > > > > > > > > > > > > > > > | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 | % Introduction to Reduce % (x+y+z)^2; for i:= 1:40 product i; factorial 40; u := (x+y+z)^2; df(ws,x); int(ws,y); matrix m(2,2); m := mat((a,b),(c,d)); % (sin(a+b)+cos(a+b))*(sin(a-b)-cos(a-b)) where cos(~x)*cos(~y) => (cos(x+y)+cos(x-y))/2, cos(~x)*sin(~y) => (sin(x+y)-sin(x-y))/2, sin(~x)*sin(~y) => (cos(x-y)-cos(x+y))/2; % on fort; df(log(x)*(sin(x)+cos(x))/sqrt(x),x,2); off fort; % |
Added src/linalg.txt version [e9c1c226ce].
> > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 | %%%%%%%%%%%%%%%%%%%%% % LINEAR ALGEBRA %%%%%%%%%%%%%%%%%%%%% % Enable fancy output for easier viewing of % matrix output %fancy_output$ % Load linear algebra package load_package linalg$ % Define a complex 3x3 matrix m1 := mat((1+i*3, 2-i*5, 7-i), (4-i*2, 6+i*9,-8+i*4), (-3-i*7, 3+i*2, -1+i*6)); % Determinant of matrix write "|m1| = ", det(m1)$ % Trace of matrix write "trace(m1) = ", trace(m1)$ % Characteristic polynomial write "characteristic polynomial of m1:"; char_poly(m1,eta); % Enable real arithmetic on rounded$ % Singular value decomposition of a matrix. a := mat((1,3),(-4,3)); write "Singular Value Decomposition of a:"$ svd(a); off rounded; end; |
Added src/mat.red version [24a6c2b757].
> > > > > > > | 1 2 3 4 5 6 7 | % example for matrix inversion mm:=mat ((1, 19, 9, 5), (5, 17, 7, 11), (9, 13, 11, 17), (13, 7, 15, 19)); mminv:=1/mm; end; |
Added src/metric2calc.red version [027137da09].
> > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 | % calculation of metric tensor (2-dim space-time) off echo; on revpri; n:=2; operator x$ x(0):=t; x(1):=lambda0; % metric array g(n,n)$ % rules trig1:={sin(~x)^2=>(1-cos(x)^2)}$ let trig1$ % procedures procedure scalprod(a,b); begin integer n; n:=first(length(a))-1; result:=for i:=0:n-1 sum a(i)*b(i); return result end; procedure showmatrix(mm); begin integer m,n;l:=length(mm);m:=first(l)-1;n:=second(l)-1; matrix hhm(m,n); for i:=0:m-1 do for j:=0:n-1 do hhm(i+1,j+1):=mm(i,j); write hhm end; procedure showvector(vv); begin integer n;n:=first(length(vv))-1; matrix hhv(n,1); for i:=0:n-1 do hhv(i+1,1):=vv(i); write hhv end; array f(n+1), dfdt(n+1), dfdl(n+1)$ % current radius a:=a0*sqrt(1-t^2); % surface of hyper sphere in t and lambda f(0):=a*cos(lambda0); f(1):=a*sin(lambda0); f(2):=a0*t; for i:=0:n do dfdt(i):=df(f(i),x(0)); for i:=0:n do dfdl(i):=df(f(i),x(1)); g(0,0):=scalprod(dfdt,dfdt)$ g(0,1):=scalprod(dfdt,dfdl)$ g(1,0):=scalprod(dfdl,dfdt)$ g(1,1):=scalprod(dfdl,dfdl)$ write "f = "; showvector(f); write "df/dt = "; showvector(dfdt); write "df/dl = "; showvector(dfdl); write "g = "; showmatrix(g); off revpri; on echo; end; |
Added src/metric2d.red version [041b81ff2e].
> > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 | % Calculations concerning the special metric of Dieter Egger % Small and capital letters are treated as being equivalent % Dimension of space-time n:=2; % turn off extra echoes off echo; % smaller exponents first on revpri; % Coordinates OPERATOR X$ X(0):=t$ X(1):=lambda0$ % lambda0 depends on t DEPEND lambda0,t$ % Rules trig1:={sin(~x)^2=>(1-cos(x)^2)}$ let trig1$ % Procedures procedure kovab(aa,bb); begin FOR I:=0:n-1 DO FOR J:=0:n-1 DO aa(I,J):=DF(bb(I),X(J))+FOR M:=0:n-1 SUM CHRIST(I,J,M)*bb(M)$ end; procedure showMatrix(mm); begin MATRIX hh(n,n)$ FOR I:=0:n-1 DO FOR J:=0:n-1 DO hh(I+1,J+1):=mm(I,J)$ write hh; end; procedure showVector(vv); begin MATRIX hh(n,1)$ FOR I:=0:n-1 DO hh(I+1,1):=vv(I)$ write hh; end; % Vectors (1-dim arrays start with index 0) ARRAY U(n), V(n), LV(n), B(n), LB(n), BG(n)$ % Arrays (2-dim arrays start with indices (0,0)) ARRAY G(n,n), GINV(n,n), CHRIST(n,n,n), RIEM(n,n,n,n), RICCI(n,n), EINST(n,n)$ ARRAY UKV(n,n)$ % Calculations % optionally set maximum radius to 1 % a0:=1$ % or leave it open a:=a0*sqrt(1-t^2)$ % Place u(0):=a0*asin(t)$ u(1):=a*lambda0$ % Metric (cellar indices, covariant, default is zero) G(0,0):=a0^2/(1-t^2)$ G(1,1):=a0^2*(1-t^2)$ % Inverse Metric (roof indices, contravariant) MATRIX MG(n,n), MGINV(n,n)$ FOR I:=0:n-1 DO FOR J:=0:n-1 DO MG(I+1,J+1):=G(I,J)$ MGINV:=1/MG$ FOR I:=0:n-1 DO FOR J:=0:n-1 DO GINV(I,J):=MGINV(I+1,J+1)$ % show metric write "g = ",mg; write "ginv = ",mginv; write "g*ginv = ",mg*mginv; % Christoffel symbols for k:=0:n-1 do for l:=0:n-1 do for m:=0:n-1 do CHRIST(k,l,m):=for i:=0:n-1 sum GINV(k,i)/2 * (DF(G(m,i),X(l)) + DF(G(l,i),X(m)) - DF(G(m,l),X(i))); % curvature tensor for m:=0:n-1 do for i:=0:n-1 do for k:=0:n-1 do for p:=0:n-1 do RIEM(m,i,k,p) :=DF(CHRIST(m,i,k),X(p)) - DF(CHRIST(m,i,p),X(k)) + FOR r:=0:n-1 SUM CHRIST(r,i,k)*CHRIST(m,r,p) - CHRIST(r,i,p)*CHRIST(m,r,k)$ % Ricci tensor FOR I:=0:n-1 DO FOR J:=0:n-1 DO RICCI(I,J):=FOR M:=0:n-1 SUM RIEM(M,I,J,M)$ write "ricci = "; showMatrix(ricci); % curvature scalar R:=FOR I:=0:n-1 SUM FOR J:=0:n-1 SUM GINV(I,J)*RICCI(I,J)$ write "curvature scalar r = ",r; % Einstein tensor FOR I:=0:n-1 DO FOR J:=0:n-1 DO EINST(I,J):=RICCI(I,J)-R/2*G(I,J)$ write "einstein = "; showMatrix(einst); % show place write "place u = "; showVector(u); % covariant derivative of place u kovab(ukv,u)$ write "cov. deriv. of u = "; showMatrix(ukv); % classical velocity for k:=0:n-1 do v(k):=df(U(k),X(0))$ write "v = du/dt = "; showVector(v); % local velocity with respect to (x0,x1) for k:=0:n-1 do LV(k):=V(k)/V(0)$ write "lv = dx1/dx0 = "; showVector(lv); % max. velocity Array vmax(n)$ svmax:=a0/sqrt(1-t^2)$ for i:=0:n-1 do vmax(i):=svmax$ svmaxq:=svmax*svmax$ write "max. velocity = ",svmax; % equation of motion for k:=0:n-1 do BG(k):=-for m:=0:n-1 sum for n:=0:n-1 sum CHRIST(k,m,n)* vmax(m)*vmax(n)$ write "equation of motion = "; showVector(bg); % local acceleration wrt (x0,x1) for k:=0:n-1 do LB(k) :=1/V(0)*df(lv(k),x(0))$ write "la = dlv/dx0 * 1/v0 = "; showVector(lb); %-------------------------------------------------------------- % write results to file OUT "metric2d_results.txt"; off echo; off nat; % Metric write "metric = "; FOR I:=0:n-1 DO FOR J:=0:n-1 DO WRITE "(",I,",",J,") = ", G(I,J)$ % Inverse Metric WRITE "inverse metric = "; FOR I:=0:n-1 DO FOR J:=0:n-1 DO WRITE "(",I,",",J,") = ", GINV(I,J)$ % Christoffel symbols write "christoffel symbols = "; FOR K:=0:n-1 DO FOR I:=0:n-1 DO FOR J:=0:n-1 DO WRITE "(",K,",",I,",",J,") = ", CHRIST(K,I,J)$ % curvature tensor write "curvature tensor = "; FOR I:=0:n-1 DO FOR J:=0:n-1 DO FOR K:=0:n-1 DO FOR L:=0:n-1 DO WRITE "(",I,",",J,",",K,",",L,") = ", RIEM(I,J,K,L)$ % Ricci tensor write "ricci tensor = "; FOR I:=0:n-1 DO FOR J:=0:n-1 DO WRITE "(",I,",",J,") = ", RICCI(I,J)$ % curvature scalar write "curvature scalar = ",R$ % Einstein tensor write "einstein tensor = "; FOR I:=0:n-1 DO FOR J:=0:n-1 DO WRITE "(",I,",",J,") = ",EINST(I,J)$ % place U write "place u = "; FOR I:=0:n-1 DO WRITE "(",I,") = ", U(I)$ % covariant derivative of U write "covariant derivative of u = "; FOR I:=0:n-1 DO FOR J:=0:n-1 DO WRITE "(",I,",",J,") = ",Ukv(I,J)$ % velocity V write "velocity v = "; FOR I:=0:n-1 DO WRITE "(",I,") = ", V(I)$ % local velocity wrt (x0,x1) write "local velocity wrt (x0,x1) = "; FOR I:=0:n-1 DO WRITE "(",I,") = ", LV(I)$ % acceleration write "acceleration = "; FOR I:=0:n-1 DO WRITE "(",I,") = ", B(I)$ % local acceleration wrt (x0,x1) write "local acceleration wrt (x0,x1) = "; FOR I:=0:n-1 DO WRITE "(",I,") = ", LB(I)$ % equation of motion write "equation of motion = "; FOR I:=0:n-1 DO WRITE "(",I,") = ", BG(I)$ % equation of motion on factor; write "equation of motion = "; FOR I:=0:n-1 DO WRITE "(",I,") =", BG(I)$ off factor; SHUT "metric2d_results.txt"; off revpri; on nat; END; |
Added src/metric3calc.red version [3b9fd98e09].
> > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 | % calculation of metric tensor (3-dim space-time) off echo; on revpri; n:=3; operator x$ x(0):=t; x(1):=lambda0; x(2):=lambda1; % rules trig1:={sin(~x)^2=>(1-cos(x)^2)}$ let trig1$ % procedures procedure scalprod(a,b); begin integer n; n:=first(length(a))-1; result:=for i:=0:n-1 sum a(i)*b(i); return result end; procedure showmatrix(mm);begin integer m,n;l:=length(mm);m:=first(l)-1;n:=second(l)-1;matrix hm(m,n);for i:=0:m-1 do for j:=0:n-1 do hm(i+1,j+1):=mm(i,j); write hm end; procedure showvector(vv);begin scalar n;n:=first(length(vv))-1;matrix hv(n,1);for i:=0:n-1 do hv(i+1,1):=vv(i);write hv end; array f(n+1), dfdt(n+1), dfdl0(n+1), dfdl1(n+1)$ % current radius a:=a0*sqrt(1-t^2); % surface of hyper sphere in t and lambda f(0):=a*cos(lambda0)*cos(lambda1); f(1):=a*cos(lambda0)*sin(lambda1); f(2):=a*sin(lambda0); f(3):=a0*t; for i:=0:n do dfdt(i):=df(f(i),x(0))$ for i:=0:n do dfdl0(i):=df(f(i),x(1))$ for i:=0:n do dfdl1(i):=df(f(i),x(2))$ array g(n,n)$ g(0,0):=scalprod(dfdt,dfdt)$ g(0,1):=scalprod(dfdt,dfdl0)$ g(0,2):=scalprod(dfdt,dfdl1)$ g(1,0):=scalprod(dfdl0,dfdt)$ g(1,1):=scalprod(dfdl0,dfdl0)$ g(1,2):=scalprod(dfdl0,dfdl1)$ g(2,0):=scalprod(dfdl1,dfdt)$ g(2,1):=scalprod(dfdl1,dfdl0)$ g(2,2):=scalprod(dfdl1,dfdl1)$ write "f = "; showvector(f); write "df/dt = "; showvector(dfdt); write "g = "; showmatrix(g); off revpri; on echo; end; |
Added src/metric3d.red version [87125fc9f6].
> > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 | % Calculations concerning the special metric of Dieter Egger % Small and capital letters are treated as being equivalent % Dimension of space-time n:=3; % turn off extra echoes off echo; % smaller exponents first on revpri; % Coordinates OPERATOR X$ X(0):=t$ X(1):=lambda0$ X(2):=lambda1$ % Vectors (1-dim arrays start with index 0) ARRAY U(n), V(n)$ % place (fixed to origin) U(0):=a0*asin(t)$ U(1):=0$ U(2):=0$ % Rule trig1:={sin(~x)^2=>(1-cos(x)^2)}$ let trig1$ % Procedure procedure showMatrix(mm); begin MATRIX hh(n,n)$ FOR I:=0:n-1 DO FOR J:=0:n-1 DO hh(I+1,J+1):=mm(I,J)$ write hh; end; % Arrays (2-dim arrays start with indices (0,0)) ARRAY G(n,n), GINV(n,n), CHRIST(n,n,n), RIEM(n,n,n,n), RICCI(n,n), EINST(n,n)$ ARRAY EIT(n,n), ENI(n,n)$ % Metric (cellar indices) G(0,0):=a0^2/(1-t^2)$ G(1,1):=a0^2*(1-t^2)$ G(2,2):=a0^2*(1-t^2)*cos(lambda0)^2$ % Inverse Metric (roof indices) MATRIX MG(n,n), MGINV(n,n)$ FOR I:=0:n-1 DO FOR J:=0:n-1 DO MG(I+1,J+1):=G(I,J)$ MGINV:=1/MG$ FOR I:=0:n-1 DO FOR J:=0:n-1 DO GINV(I,J):=MGINV(I+1,J+1)$ write "g = ",mg; write "ginv = ",mginv; write "g*ginv = ",mg*mginv; % velocity for i:=0:n-1 do v(i):=df(u(i),t)$ % max. velocity Array vmax(n)$ svmax:=a0/sqrt(1-t^2)$ for i:=0:n-1 do vmax(i):=svmax$ svmaxq:=svmax*svmax$ write "max. velocity = ",svmax; % energy impulse tensor (eit, roof indices) for i:=0:n-1 do for j:=0:n-1 do eit(i,j):=v(i)*v(j)*(p/svmaxq + rho) - p * ginv(i,j)$ write "eit roof = "; showMatrix(eit); % energy impulse tensor (eni, cellar indices, including kappa) for i:=0:n-1 do for j:=0:n-1 do eni(i,j) := - kappa * for k:=0:n-1 sum g(i,k)* for l:=0:n-1 sum g(j,l)*eit(k,l)$ write "eni = -kappa*(eit cellar) = "; showMatrix(eni); % Christoffel symbols (Fliessbach) for k:=0:n-1 do for l:=0:n-1 do for m:=0:n-1 do CHRIST(k,l,m):= for n:=0:n-1 sum GINV(k,n)/2 * (DF(G(m,n),X(l)) + DF(G(l,n),X(m)) - DF(G(m,l),X(n)))$ % curvature tensor (Fliessbach) for m:=0:n-1 do for i:=0:n-1 do for k:=0:n-1 do for p:=0:n-1 do RIEM(m,i,k,p) := DF(CHRIST(m,i,k),X(p)) - DF(CHRIST(m,i,p),X(k)) + FOR r:=0:n-1 SUM CHRIST(r,i,k)*CHRIST(m,r,p) - CHRIST(r,i,p)*CHRIST(m,r,k)$ % Ricci tensor (Fliessbach) FOR I:=0:n-1 DO FOR J:=0:n-1 DO RICCI(I,J):= FOR M:=0:n-1 SUM RIEM(M,I,M,J)$ write "ricci = "; showMatrix(ricci); % curvature scalar R:= FOR I:=0:n-1 SUM FOR J:=0:n-1 SUM GINV(I,J)*RICCI(I,J)$ write "curvature scalar r = ",r; % Einstein tensor FOR I:=0:n-1 DO FOR J:=0:n-1 DO EINST(I,J):=RICCI(I,J)-R/2*G(I,J)$ write "einstein = "; showMatrix(einst); % solving field equations write "solving field equations ..."; on factor; erho:=solve(eni(0,0)=einst(0,0),rho)$ write "mass density = ", erho; ep:=solve(eni(1,1)=einst(1,1),p)$ write "pressure = ", ep; off factor; ferho:=sub(ep,erho)$ write "final mass density = ", ferho; %-------------------------------------------------------------- % write results to file OUT "metric3d_results.txt"; off echo; off nat; % Metric write "metric = "; FOR I:=0:n-1 DO FOR J:=0:n-1 DO WRITE "(",I,",",J,") = ", G(I,J)$ % Inverse Metric WRITE "inverse metric = "; FOR I:=0:n-1 DO FOR J:=0:n-1 DO WRITE "(",I,",",J,") = ", GINV(I,J)$ % Christoffel symbols write "christoffel symbols = "; FOR K:=0:n-1 DO FOR I:=0:n-1 DO FOR J:=0:n-1 DO WRITE "(",K,",",I,",",J,") = ", CHRIST(K,I,J)$ % curvature tensor write "curvature tensor = "; FOR I:=0:n-1 DO FOR J:=0:n-1 DO FOR K:=0:n-1 DO FOR L:=0:n-1 DO WRITE "(",I,",",J,",",K,",",L,") = ", RIEM(I,J,K,L)$ % Ricci tensor write "ricci tensor = "; FOR I:=0:n-1 DO FOR J:=0:n-1 DO WRITE "(",I,",",J,") = ", RICCI(I,J)$ % curvature scalar write "curvature scalar = ",R$ % Einstein tensor write "einstein tensor = "; FOR I:=0:n-1 DO FOR J:=0:n-1 DO WRITE "(",I,",",J,") = ",EINST(I,J)$ % energy impulse tensor write "energy impulse tensor eit (roof) = "; FOR I:=0:n-1 DO FOR J:=0:n-1 DO WRITE "(",I,",",J,") = ",eit(I,J); % energy impulse tensor write "energy impulse tensor eni (with kappa, cellar) = "; FOR I:=0:n-1 DO FOR J:=0:n-1 DO WRITE "(",I,",",J,") = ",eni(I,J); % solving field equations write "solving field equations ..."; on factor; write "mass density = ", erho; write "pressure = ", ep; off factor; write "final mass density = ", ferho; SHUT "metric3d_results.txt"; off revpri; on nat; END; |
Added src/metric4calc.red version [12312f33f4].
> > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 | % calculation of metric tensor (4-dim space-time) off echo; on revpri; n:=4; array f(n+1), dfdt(n+1), dfdl0(n+1), dfdl1(n+1), dfdl2(n+1)$ array g(n,n)$ operator x$ x(0):=t; x(1):=lambda0; x(2):=lambda1; x(3):=lambda2; % rules trig1:={sin(~x)^2=>(1-cos(x)^2)}$ let trig1$ % procedures procedure scalprod(a,b); begin integer n; n:=first(length(a))-1; result:=for i:=0:n-1 sum a(i)*b(i); return result end; procedure showmatrix(mm);begin integer m,n;l:=length(mm);m:=first(l)-1;n:=second(l)-1;matrix hm(m,n);for i:=0:m-1 do for j:=0:n-1 do hm(i+1,j+1):=mm(i,j); write hm end; procedure showvector(vv);begin integer n;n:=first(length(vv))-1; matrix hv(n,1); for i:=0:n-1 do hv(i+1,1):=vv(i); write hv end; % current radius a:=a0*sqrt(1-t^2); % surface of hyper sphere in t and lambda f(0):=a*cos(lambda0)*cos(lambda1)*cos(lambda2); f(1):=a*cos(lambda0)*cos(lambda1)*sin(lambda2); f(2):=a*cos(lambda0)*sin(lambda1); f(3):=a*sin(lambda0); f(4):=a0*t; for i:=0:n do dfdt(i):=df(f(i),x(0))$ for i:=0:n do dfdl0(i):=df(f(i),x(1))$ for i:=0:n do dfdl1(i):=df(f(i),x(2))$ for i:=0:n do dfdl2(i):=df(f(i),x(3))$ g(0,0):=scalprod(dfdt,dfdt)$ g(0,1):=scalprod(dfdt,dfdl0)$ g(0,2):=scalprod(dfdt,dfdl1)$ g(0,3):=scalprod(dfdt,dfdl2)$ g(1,0):=scalprod(dfdl0,dfdt)$ g(1,1):=scalprod(dfdl0,dfdl0)$ g(1,2):=scalprod(dfdl0,dfdl1)$ g(1,3):=scalprod(dfdl0,dfdl2)$ g(2,0):=scalprod(dfdl1,dfdt)$ g(2,1):=scalprod(dfdl1,dfdl0)$ g(2,2):=scalprod(dfdl1,dfdl1)$ g(1,3):=scalprod(dfdl1,dfdl2)$ g(3,0):=scalprod(dfdl2,dfdt)$ g(3,1):=scalprod(dfdl2,dfdl0)$ g(3,2):=scalprod(dfdl2,dfdl1)$ g(3,3):=scalprod(dfdl2,dfdl2)$ write "f = "; showvector(f); write "df/dt = "; showvector(dfdt); write "g = "; showmatrix(g); off revpri; on echo; end; |
Added src/metric4d.red version [105692f385].
> > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 | % Calculations concerning the special metric of Dieter Egger % Small and capital letters are treated as being equivalent % Dimension of space-time n:=4; % turn off extra echoes off echo; % smaller exponents first on revpri; % Coordinates OPERATOR X$ X(0):=t$ X(1):=lambda0$ X(2):=lambda1$ X(3):=lambda2$ % Vectors (1-dim arrays start with index 0) ARRAY U(n), V(n)$ % place (fixed to origin) U(0):=a0*asin(t)$ U(1):=0$ U(2):=0$ U(3):=0$ % Rule trig1:={sin(~x)^2=>(1-cos(x)^2)}$ let trig1$ % Procedure procedure showMatrix(mm); begin MATRIX hh(n,n)$ FOR I:=0:n-1 DO FOR J:=0:n-1 DO hh(I+1,J+1):=mm(I,J)$ write hh; end; % Arrays (2-dim arrays start with indices (0,0)) ARRAY G(n,n), GINV(n,n), CHRIST(n,n,n), RIEM(n,n,n,n), RICCI(n,n), EINST(n,n)$ ARRAY EIT(n,n), ENI(n,n)$ % Metric (cellar indices) G(0,0):=a0^2/(1-t^2)$ G(1,1):=a0^2*(1-t^2)$ G(2,2):=a0^2*(1-t^2)*cos(lambda0)^2$ G(3,3):=a0^2*(1-t^2)*cos(lambda0)^2*cos(lambda1)^2$ % Inverse Metric (roof indices) MATRIX MG(n,n), MGINV(n,n)$ FOR I:=0:n-1 DO FOR J:=0:n-1 DO MG(I+1,J+1):=G(I,J)$ MGINV:=1/MG$ FOR I:=0:n-1 DO FOR J:=0:n-1 DO GINV(I,J):=MGINV(I+1,J+1)$ write "g = ",mg$ write "ginv = ",mginv$ write "g*ginv = ",mg*mginv$ % velocity for i:=0:n-1 do v(i):=df(u(i),t)$ % max. velocity Array vmax(n)$ svmax:=a0/sqrt(1-t^2)$ for i:=0:n-1 do vmax(i):=svmax$ svmaxq:=svmax*svmax$ write "max. velocity = ",svmax$ % energy impulse tensor (eit, roof indices) for i:=0:n-1 do for j:=0:n-1 do eit(i,j):=v(i)*v(j)*(p/svmaxq + rho) - p * ginv(i,j)$ write "eit roof = "$ showMatrix(eit)$ % energy impulse tensor (eni, cellar indices, including kappa) for i:=0:n-1 do for j:=0:n-1 do eni(i,j) := - kappa * for k:=0:n-1 sum g(i,k)* for l:=0:n-1 sum g(j,l)*eit(k,l)$ write "eni = -kappa*(eit cellar) = "$ showMatrix(eni)$ % Christoffel symbols (Fliessbach) for k:=0:n-1 do for l:=0:n-1 do for m:=0:n-1 do CHRIST(k,l,m):= for n:=0:n-1 sum GINV(k,n)/2 * (DF(G(m,n),X(l)) + DF(G(l,n),X(m)) - DF(G(m,l),X(n)))$ % curvature tensor (Fliessbach) for m:=0:n-1 do for i:=0:n-1 do for k:=0:n-1 do for p:=0:n-1 do RIEM(m,i,k,p) := DF(CHRIST(m,i,k),X(p)) - DF(CHRIST(m,i,p),X(k)) + FOR r:=0:n-1 SUM CHRIST(r,i,k)*CHRIST(m,r,p) - CHRIST(r,i,p)*CHRIST(m,r,k)$ % Ricci tensor (Fliessbach) FOR I:=0:n-1 DO FOR J:=0:n-1 DO RICCI(I,J):= FOR M:=0:n-1 SUM RIEM(M,I,M,J)$ write "ricci = "$ showMatrix(ricci)$ % curvature scalar R:= FOR I:=0:n-1 SUM FOR J:=0:n-1 SUM GINV(I,J)*RICCI(I,J)$ write "curvature scalar r = ",r; % Einstein tensor FOR I:=0:n-1 DO FOR J:=0:n-1 DO EINST(I,J):=RICCI(I,J)-R/2*G(I,J)$ write "einstein = "$ showMatrix(einst)$ % solving field equations write "solving field equations ..."; on factor; erho:=solve(eni(0,0)=einst(0,0),rho)$ write "mass density = ", erho$ ep:=solve(eni(1,1)=einst(1,1),p)$ write "pressure = ", ep$ off factor; ferho:=sub(ep,erho)$ write "final mass density = ", ferho$ %-------------------------------------------------------------- % write results to file OUT "metric4d_results.txt"; off echo; off nat; % Metric write "metric = "; FOR I:=0:n-1 DO FOR J:=0:n-1 DO WRITE "(",I,",",J,") = ", G(I,J)$ % Inverse Metric WRITE "inverse metric = "; FOR I:=0:n-1 DO FOR J:=0:n-1 DO WRITE "(",I,",",J,") = ", GINV(I,J)$ % Christoffel symbols write "christoffel symbols = "; FOR K:=0:n-1 DO FOR I:=0:n-1 DO FOR J:=0:n-1 DO WRITE "(",K,",",I,",",J,") = ", CHRIST(K,I,J)$ % curvature tensor write "curvature tensor = "; FOR I:=0:n-1 DO FOR J:=0:n-1 DO FOR K:=0:n-1 DO FOR L:=0:n-1 DO WRITE "(",I,",",J,",",K,",",L,") = ", RIEM(I,J,K,L)$ % Ricci tensor write "ricci tensor = "; FOR I:=0:n-1 DO FOR J:=0:n-1 DO WRITE "(",I,",",J,") = ", RICCI(I,J)$ % curvature scalar write "curvature scalar = ",R$ % Einstein tensor write "einstein tensor = "; FOR I:=0:n-1 DO FOR J:=0:n-1 DO WRITE "(",I,",",J,") = ",EINST(I,J)$ % energy impulse tensor write "energy impulse tensor eit (roof) = "; FOR I:=0:n-1 DO FOR J:=0:n-1 DO WRITE "(",I,",",J,") = ",eit(I,J); % energy impulse tensor write "energy impulse tensor eni (with kappa, cellar) = "; FOR I:=0:n-1 DO FOR J:=0:n-1 DO WRITE "(",I,",",J,") = ",eni(I,J); % solving field equations write "solving field equations ..."; on factor; write "mass density = ", erho; write "pressure = ", ep; off factor; write "final mass density = ", ferho; SHUT "metric4d_results.txt"; off revpri; on nat; END; |
Added src/mkid.txt version [6aa00a225d].
> > > > > | 1 2 3 4 5 | % dynamical identifiers for i:=1:5 do set (mkid (z, i), i); for i:=1:5 do write mkid (z, i); for i:=1:5 sum mkid (z, i); end; |
Added src/partialFraction.txt version [f1283d1b2c].
> > > > > > > > > > | 1 2 3 4 5 6 7 8 9 10 | % partial fractions b:=(x+1)^2*(x+2); a:=2/b; pf (a, x); c:=ws; first (c)+second (c)+third (c); off exp; ws; first (c)+second (c)+third (c); end; |
Added src/photonenergy.txt version [4d32c62a43].
> > > > > > > > > > | 1 2 3 4 5 6 7 8 9 10 | % define full path to desired Reduce input file in "/storage/extSdCard/Reduce/constants.red"; lambda0:=5*10^(-6); nu:=c/lambda0; lambda0:=5*10^(-7); nu:=c/lambda0; h*nu/c; h*nu; h*nu/el; end; |
Added src/prefix.txt version [9a0ec2ab9b].
> > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 | abs (-3/4); abs (2a); abs (i); abs (-x); ceiling (-5/4); ceiling (-3/4); ceiling (3/4); ceiling (5/4); ceiling (-a); conj (1+i); conj (a+i*b); factorial 5; factorial a; fix (-5/4); fix (-5/4); fix (-3/4); fix (3/4); fix (5/4); floor (-5/4); floor (-3/4); floor (3/4); floor (5/4); floor (a); impart (1+i); impart (1+2i); impart (a+i*b); max (2,-3, 4, 5); min (2,-3, 4, 5); min (2,-3, 4, 5, a); max (2,-3, 4, 5, a); nextprime 1111; nextprime 0; nextprime 11; nextprime 111; random 49; random 49; random 49; random 49; random 49; random 49; random_new_seed (1000); random 49; random 49; random_new_seed (1000); random 49; random 49; repart (4+2i); repart (4a+2i); repart (a+i*b); round (-5/4); round (-3/4); round (3/4); round (5/4); round (7/4); sign (-5); sign (-5/4); sign (5/4); sign (0); end; |
Added src/programming.txt version [e35a9918fc].
> > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 | %%%%%%%%%%%%%%%%%%%%% % PROGRAMMING %%%%%%%%%%%%%%%%%%%%% % Define number to factorize x:=42; % Factorize x and write out each individual % factor factors:=factorize(fix(x))$ x:=0$ for i:=1:length(factors) do begin q:=part(factors,i); for j:=1:part(q,2) do begin x:=x+1; write "factor ", x, ": ", part(q,1); end; end; % Procedure to calculate Legendre polynomial % using recursion procedure p(n,x); if n<0 then rederr "Invalid argument to p(n,x)" else if n=0 then 1 else if n=1 then x else ((2*n-1)*x*p(n-1,x)-(n-1)*p(n-2,x))/n$ % Enable fancy output %fancy_output$ % Calculate p(2,w) write "P(2,w) = ", p(2,w); % Incidentally, p(n,x) can be calculated more % efficiently as follows procedure p(n,x); sub(y=0,df(1/(y^2-2*x*y+1)^(1/2),y,n))/(for i:=1:n product i)$ write "P(3,w) = ", p(3,w); end; |
Added src/rules.red version [e266788fa7].
> > > > > > > | 1 2 3 4 5 6 7 | % Rules %trig1:={sin(~x)^2+cos(~x)^2=>1, cos(~x)^2+sin(~x)^2=>1}; trig1:={sin(~x)^2=>(1-cos(x)^2)}; let trig1; trig2:={tan (~x)=>(sin (x)/cos (x))}; let trig2; end; |
Added src/scalprod.red version [085ebbfed8].
> > > > > > > | 1 2 3 4 5 6 7 | procedure scalprod(a,b); begin; scalar n; n:=first (length(a))-1; result:=for i:=0:n sum a(i)*b(i); return result; end; end; |
Added src/speedoflight.txt version [8a84a36ba8].
> > | 1 2 | c:=1/sqrt (epsilon0*mu0); end; |