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I have written the following coade in Maple:
r := 50;
l1 := 0.2742e-10;
s := I*w;
l := (-1.342110665*10^22*c^2*(Pi^4)-4.225000000*10^25*c^2*(Pi^2)+2.316990000*10^11*c1*(Pi^2)-1)/(-1.342110665*10^22*c^2*c1*(Pi^4)-7.140250000*10^43*c^2*c1*r^2*(Pi^4)+1.957856550*10^33*c^2*(Pi^4)+9.789282750*10^32*c*c1*(Pi^4)-1.690000*10^22*c*(Pi^2)-4.22500*10^21*c1*(Pi^2));
z1 := (c*l1*s^2+1)/(c*s);
z2 := l*s/(c1*l*s^2+1);
h := (z1+2*z2)*((z1+2*r)*(z1+3*z2)/(2*r)-2*z2)/z2-(1/2)*z2*(z1+2*r)*r;
f := h*(z1+3*z2)/z2-(z1+2*r)(2*r)*(z1+3*z2)+2*z2;
gain := 2*z2/f;
a := abs(gain);
d := diff(a, w);
s := subs(w = 2*pi*0.325e11, d)
Now, I have a function named "s" which I want to set to zero, and calculate the relationship between variables c & c1 in order to achieve this. How should it be done?
Thanks.


The year 2015 has been one with interesting and relevant developments in the MathematicalFunctions  and FunctionAdvisor projects.

• 

Gaps were filled regarding mathematical formulas, with more identities for all of BesselI, BesselK, BesselY, ChebyshevT, ChebyshevU, Chi, Ci, FresnelC, FresnelS, GAMMA(z), HankelH1, HankelH2, InverseJacobiAM, the twelve InverseJacobiPQ for P, Q in [C,D,N,S], KelvinBei, KelvinBer, KelvinKei, KelvinKer, LerchPhi, arcsin, arcsinh, arctan, ln;

• 

Developments happened in the Mathematical function package, to both compute with symbolic sequences and symbolic nth order derivatives of algebraic expressions and functions;

• 

The input FunctionAdvisor(differentiate_rule, mathematical_function) now returns both the first derivative (old behavior) and the nth symbolic derivative (new behavior) of a mathematical function;

• 

A new topic, plot, used as FunctionAdvisor(plot, mathematical_function), now returns 2D and 3D plots for each mathematical function, following the NIST Digital Library of Mathematical Functions;

• 

The previously existing FunctionAdvisor(display, mathematical_function) got redesigned, so that it now displays more information about any mathematical function, and organized into a Section with subsections for each of the different topics, making it simpler to find the information one needs without getting distracted by a myriad of formulas that are not related to what one is looking for.

More mathematics

 

More mathematical knowledge is in place, more identities, differentiation rules of special functions with respect to their parameters, differentiation of functions whose arguments involve symbolic sequences with an indeterminate number of operands, and sum representations for special functions under different conditions on the functions' parameters.

Examples

   

More powerful symbolic differentiation (nth order derivative)

 

Significative developments happened in the computation of the nth order derivative of mathematical functions and algebraic expressions involving them.

Examples

   

Mathematical handling of symbolic sequences

 

Symbolic sequences enter various formulations in mathematics. Their computerized mathematical handling, however, was never implemented - only a representation for them existed in the Maple system. In connection with this, a new subpackage, Sequences , within the MathematicalFunctions package, has been developed.

Examples

   

Visualization of mathematical functions

 

When working with mathematical functions, it is frequently desired to have a rapid glimpse of the shape of the function for some sampled values of their parameters. Following the NIST Digital Library of Mathematical Functions, a new option, plot, has now been implemented.

Examples

   

Section and subsections displaying properties of mathematical functions

 

Until recently, the display of a whole set of mathematical information regarding a function was somehow cumbersome, appearing all together on the screen. That display was and is still available via entering, for instance for the sin function, FunctionAdvisor(sin) . That returns a table of information that can be used programmatically.

With time however, the FunctionAdvisor evolved into a consultation tool, where a better organization of the information being displayed is required, making it simpler to find the information we need without being distracted by a screen full of complicated formulas.

To address this requirement, the FunctionAdvisor now returns the information organized into a Section with subsections, built using the DocumentTools package. This enhances the presentation significantly.

Examples

   

These developments can be installed in Maple 2015 as usual, by downloading the updates (bundled with the Physics and Differential Equations updates) from the Maplesoft R&D webpage for Mathematical Functions and Differential Equations


Download MathematicalFunctionsAndFunctionAdvisor.mw

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

The well known William Lowell Putnam Mathematical Competition (76th edition)  took place this month.
Here is a Maple approach for two of the problems.

1. For each real number x, 0 <= x < 1, let f(x) be the sum of  1/2^n  where n runs through all positive integers for which floor(n*x) is even.
Find the infimum of  f.
(Putnam 2015, A4 problem)

f:=proc(x,N:=100)
local n, s:=0;
for n to N do
  if type(floor(n*x),even) then s:=s+2^(-n) fi;
  #if floor(n*x) mod 2 = 0  then s:=s+2^(-n) fi;
od;
evalf(s);
#s
end;

plot(f, 0..0.9999);

 

min([seq(f(t), t=0.. 0.998,0.0001)]);

        0.5714285714

identify(%);

So, the infimum is 4/7.
Of course, this is not a rigorous solution, even if the result is correct. But it is a valuable hint.
I am not sure if in the near future, a CAS will be able to provide acceptable solutions for such problems.

2. If the function f  is three times differentiable and  has at least five distinct real zeros,
then f + 6f' + 12f'' + 8f''' has at least two distinct real zeros.
(Putnam 2015, B1 problem)

restart;
F := f + 6*D(f) + 12*(D@@2)(f) + 8*(D@@3)(f);

dsolve(F(x)=u(x),f(x));

We are sugested to consider

g:=f(x)*exp(x/2):
g3:=diff(g, x$3);

simplify(g3*8*exp(-x/2));

So, F(x) = k(x) * g3 = k(x) * g'''
g  has 5 distinct zeros implies g''' and hence F have 5-3=2 distinct zeros, q.e.d.

 

How do I construct the seuqence 1/16 , 1/32 , 1/64 , 1/128 , 1/256 in maple?

 

What's the syntax?

I looked at the examples in here:

http://www.maplesoft.com/support/help/maple/view.aspx?path=seq

 

But didn't find something similar.

 

 

 

Consider a taper steel plate of uniform thickness t := 25mm as shown in the Fig. In addition to its self weight, the plate is subjected to a point load P := 100N at its mid point. Find the global force vector [F] , global stiffness matrix [K] , displacement in each element (1 and 2) , stresses in each element  (1 and 2) and reaction force at the support.Take E := 2*10^5N/mm2; rho := 8.2*10^(-5)kg/m3;

restart

t__1 := 150:

t__3 := 75:

w := 25:

l := 600:

t__2 := (t__1-t__3)/l*((1/2)*l)+t__3 = 225/2

A__1 := t__1*w = 3750``

A__2 := t__2*w = 5625/2``

A__3 := t__3*w = 1875``

Revised areas:

A__1e := (A__1+A__2)*(1/2) = 13125/4``

A__2e := (A__2+A__3)*(1/2) = 9375/4``

  E := 2*10^11:m2; F__1 := R__1:is support reaction N; F__2 := 100:N;``

rho__1 := 82*10^(-6) = 41/500000  N/mm2

rho__2 := 82*10^(-6) = 41/500000 N/mm2

l := 600:``

Number of elements,

n__e := 2:

l__e := 300 = 300````

q__0 := 100:N/m ; l := 1: m; n__e := 4:  elementsl  l__e := l/n__e: m;

We shall consider a two element system as shown in the Fig.
For element 1 Stiffness matrix K is

                                           Vector[row](2, {(1) = 1, (2) = 2})
K__1 := A__1e*E/l__e.(Matrix(2, 2, {(1, 1) = 1, (1, 2) = -1, (2, 1) = -1, (2, 2) = 1})) = Matrix([[2625000000000000, -2625000000000000], [-2625000000000000, 2625000000000000]])  Vector(2, {(1) = 1, (2) = 2})

For element 2 Stiffness matrix K is

                                         Vector[row](2, {(1) = 2, (2) = 3})
K__2 := A__2e*E/l__e.(Matrix(2, 2, {(1, 1) = 1, (1, 2) = -1, (2, 1) = -1, (2, 2) = 1})) = Matrix([[1875000000000000, -1875000000000000], [-1875000000000000, 1875000000000000]])  Vector(2, {(1) = 2, (2) = 3})

Global stiffness matrix obtained by adding all the elemental stiffness matrices and given b

           Vector[row](3, {(1) = 0, (2) = 0, (3) = 0})

K__g := Matrix(3, 3, {(1, 1) = K__1[1, 1], (1, 2) = K__1[1, 2], (1, 3) = 0, (2, 1) = K__1[2, 1], (2, 2) = K__1[1, 2]+K__2[1, 1], (2, 3) = K__2[1, 2], (3, 1) = 0, (3, 2) = K__2[2, 1], (3, 3) = K__2[2, 2]}) = Matrix([[K__1[1, 1], K__1[1, 2], 0], [K__1[2, 1], K__1[1, 2]+K__2[1, 1], K__2[1, 2]], [0, K__2[2, 1], K__2[2, 2]]])  Vector(3, {(1) = 0, (2) = 0, (3) = 0})

For element 1 Load matrix F is

  F__1e := (1/2)*`&rho;__1`*A__1e*l__e*(Vector(2, {(1) = 1, (2) = 1})) = Vector[column]([[861/25600], [861/25600]]) Vector(2, {(1) = 1, (2) = 2})

``

For element 2 Load matrix F isNULL

F__2e := (1/2)*A__2e*l__e*`&rho;__2`*(Vector(2, {(1) = 1, (2) = 1})) = Vector[column]([[123/5120], [123/5120]]) 

``

 

Download wrong_answers.mwwrong_answers.mwwrong_answers.mw

Ramakrishnan V

rukmini_ramki@hotmail.com

 

``

 

I would appreciate if anyone lets me know how to write circular references  (say 1 inside a circle to refer element 1. At present i do a drawing insert text and using.

 

Also i do not know how to remove the boundary of the overall drawing.

NULL

 

Download A_DOUBT_to_be_sent_to_prime_community.mw

Ramakrishnan V

rukmini_ramki@hotmail.com

There have come unwanted lines and marks . I donot know how to remove them. Using doc.block, remove block seems to be little tough to incorporate! Please enlighten me. Modified doc. is most welcome. Thanks. Ramakrishnan V 

Gaussian Elimination Method

 

 

Given*the*equations

  restartreset:

with(Student[LinearAlgebra])``

(1)
Coefficient Tanle

Equation 1

Equation 2

Equation 3

Equations

`m__1,1` := 3:
`` 

`m__2,1` := 2:
``

`m__3,1` := 1:
``

`m__1,1`*x__1+`m__1,2`*y+`m__1,3`*z = `m__1,4`; = 3*x__1+y-z = 3

`m__2,1`*x__1+`m__2,2`*y+`m__2,3`*z = `m__2,4`; = 2*x__1-8*y+z = -5

```m__3,1`*x__1+`m__3,2`*y+`m__3,3`*z = `m__3,4`; = x__1-2*y+9*z = 8

The equations in matrix form is given by

Matrix([[3, 1, -1, 3], [2, -8, 1, -5], [1, -2, 9, 8]])

(2)

The Gaussian Elimination gives the simplified natrix equation as given below:

Matrix([[3, 1, -1, 3], [0, -26/3, 5/3, -7], [0, 0, 231/26, 231/26]])

(3)

``The equations in simplified form are:

3*x+y-z = 3

(4)

-(26/3)*y+(5/3)*z = -7

(5)

(231/26)*z = 231/26

(6)

``

The aolution ia obtained by solving the above equations in reverse order

{x = 1, y = 1, z = 1}

(7)

 

``

 

Download GausianFinal15Nov2015.mwGausianFinal15Nov2015.mw

We are looking for enthusiastic Maple users to become Maple Ambassadors, to inspire and educate others about the benefits that Maple brings to education.

 

As an Ambassador, you will have the opportunity to influence the development of Maple through regular meetings with Maplesoft developers, get advance news of upcoming features and products, get assistance with Maple events on your campus, and more. In return, we ask that you do what you are probably already doing – sharing your experiences with Maple, answer questions on forums (like this one!), sharing your Maple applications, providing us with feedback, etc.

 

You can find more information and an application form at Maple Ambassador Program. We’re looking forward to hearing from you!

 

Daniel

Maple Product Management

Are there any commands in maple that will help me find a suitable function that approximates the numerical solution of:



  restart;
  PDE := diff(v(x, t), t) = diff(v(x, t), x, x);
  JACOBIINTEGRAL := int(JacobiTheta3(0, exp(-Pi^2*s))*v(1, t-s)^4, s = 0 .. t);
  IBC:= D[1](v)(0,t)=0,
        D[1](v)(1,t)=-0.000065*v(1, t)^4,
        v(x,0)=1;
#
# For x=0..1, t=0..1, the solution varies only very slowly
# so I have increased the timestep/spacestep, just to speed
# up results generation for diagnostic purposes
#
  pds := pdsolve( PDE, [IBC], numeric, time = t, range = 0 .. 1,
                  spacestep = 0.1e-1, timestep = 0.1e-1,
                  errorest=true
                )

diff(v(x, t), t) = diff(diff(v(x, t), x), x)

 

int(JacobiTheta3(0, exp(-Pi^2*s))*v(1, t-s)^4, s = 0 .. t)

 

(D[1](v))(0, t) = 0, (D[1](v))(1, t) = -0.65e-4*v(1, t)^4, v(x, 0) = 1

 

_m649569600

(1)

#
# Plot the solution over the ranges x=0..1,
# time=0..1. Not a lot happens!
#
  pds:-plot(x=1, t=0..1);

 

#
# Plot the estimated error over the ranges x=0..1,
# time=0..1
#
  pds:-plot( err(v(x,t)), x=1,t=0..1);

 

#
# Get some numerical solution values
#
  pVal:=pds:-value(v(x,t), output=procedurelist):
  for k from 0 by 0.1 to 1 do
      pVal(1, k)[2], pVal(1, k)[3];
  od;

 

t = 0., v(x, t) = Float(undefined)

 

t = .1, v(x, t) = .999977377613528229

 

t = .2, v(x, t) = .999967577518313666

 

t = .3, v(x, t) = .999959874331053822

 

t = .4, v(x, t) = .999952927885405241

 

t = .5, v(x, t) = .999946262964885979

 

t = .6, v(x, t) = .999939702966688881

 

t = .7, v(x, t) = .999933182128311282

 

t = .8, v(x, t) = .999926675964661227

 

t = .9, v(x, t) = .999920175361791563

 

t = 1.0, v(x, t) = .999913676928735229

(2)

 

 

 

Download PDEprob2_(2).mw

 

I am refering to the first graph, is there a way in maple to find an explicit suitable approximating function?

I.e, I want the function to have the same first graph obviously, it seems like addition of exponent and a line function, I tried plotting exp(-t)-0.3*t, it doesn't look like it approximates it very well. Any suggestion on how to implement this task in maple?

Thanks.

 

Dear collagues

Hi,

I write a code to solve a system of ODE. It solve the ODES in a wide range of parameters but as I decrease NBT below 0.5, it doesnt converge. I do my best but I couldn't find the answer. Would you please help me? Thank you

Here is my code and it should be run for 0.1<NBT<10. the value of NBT is input directly in res1.

restart:
EPSILONE:=1000:
Digits:=15:

a[mu1]:=5.45:
b[mu1]:=108.2:
a[k1]:=1.292:
b[k1]:=-11.99:




rhop:=4175:
rhobf:=998.2:
mu1[bf]:=9.93/10000:
k1[bf]:=0.597:

rhost(eta):=1-phi(eta)+phi(eta)*rhop/rhobf;
k:=unapply(k1[bf]*(1+a[k1]*phi(eta)+b[k1]*phi(eta)^2),eta);


eq1:=(diff(u(eta), eta))*a[mu1]*(diff(phi(eta), eta))+2*(diff(u(eta), eta))*b[mu1]*phi(eta)*(diff(phi(eta), eta))+((diff(u(eta), eta))+(diff(u(eta), eta))*a[mu1]*phi(eta)+(diff(u(eta), eta))*b[mu1]*phi(eta)^2)/(eta+EPSILONE)+diff(u(eta), eta, eta)+(diff(u(eta), eta, eta))*a[mu1]*phi(eta)+(diff(u(eta), eta, eta))*b[mu1]*phi(eta)^2+1-phi(eta)+phi(eta)*rhop/rhobf:
eq2:=(1+a[k1]*phi(eta)+b[k1]*phi(eta)^2)*(diff(T(eta), eta))/(eta+EPSILONE) + (a[k1]*(diff(phi(eta), eta))+2*b[k1]*phi(eta)*(diff(phi(eta), eta)))*(diff(T(eta), eta))+(1+a[k1]*phi(eta)+b[k1]*phi(eta)^2)*(diff(T(eta), eta, eta));
eq3:=diff(phi(eta),eta)+phi(eta)/(N_bt*(1+gama1*T(eta))^2)*diff(T(eta),eta):

eq2:=subs(phi(0)=phi0,eq2):
eq3:=subs(phi(0)=phi0,eq3):


eval(dsolve({eq3,phi(1)=phiv},phi(eta)),(T)(1)=1):
Phi:=normal(combine(%)):
Teq:=isolate(eval(eq2,Phi),diff(T(eta),eta)):
ueq1:=eval(eq1,Phi)=0:
ueq2:=subs(Teq,ueq1):


lambda:=0;
Ha:=0;
N_bt:=cc*NBT+(1-cc)*0.8;
kratio:=k1[p]/k1[bf]:






GUESS:=[T(eta) =0.0001*eta, u(eta) =0.1*eta, phi(eta) = 0.3*(eta-1)^4];
res1 := dsolve(subs(NBT=0.48,gama1=0.2,phiv=0.06,{eq1,eq2,eq3,u(0)=lambda*D(u)(0),D(u)(1)=0,T(0)=0,phi(1)=phiv,T(1)=1}), numeric,method=bvp[midrich],maxmesh=4000,approxsoln=GUESS, output=listprocedure,continuation=cc):
G0,G1,G2:=op(subs(subs(res1),[phi(eta),u(eta),diff(T(eta),eta)])):

masst:=evalf(int((1-G0(eta)+G0(eta)*rhop/rhobf)*G1(eta), eta = 0..1));
heatt:=(1+a[k1]*G0(0)+b[k1]*G0(0)^2)*G2(0);

plots:-odeplot(res1,[[eta,T(eta)]],0..1,legend=[T],color=["Black"],linestyle=Solid,axes=boxed,thickness=3);
plots:-odeplot(res1,[[eta,u(eta)]],0..1,legend=[u],color=["Black"],linestyle=Solid,axes=boxed,thickness=3);
plots:-odeplot(res1,[[eta,phi(eta)]],0..1,legend=[phi],color=["Black"],linestyle=Solid,axes=boxed,thickness=3);

>
>

 

Thank you

 

Amir

Hello all

I am trying to write  a tutorial about systems of linear equations, and I want to demonstrate the idea that when you have a system of 3 euqtions with 3 unknowns, the solution is the intersection point between these planes. Plotting 3 planes in Maple 2015 is fairly easy (you plot one and just drag the others in), but I don't know how to plot the intersection point. Can you help please ?

 

My equations are:

x-2y+z=0

2y-8z=8

-4x+5y+9z=-9

The intersection point is (29,16,3)

 

Thank you !

Dear Collegues

I have a system of odes as follows

restart:
#gama1:=0.2:

#rhop:=5180:
#rhobf:=998.2:
#a[mu1]:=39.11:
#b[mu1]:=533.9:
#a[k1]:=7.47:
#b[k1]:=0:

Teq := N_bt*T(eta)^2*(exp(-(T(eta)-1)/(N_bt*(1+gama1)*(1+gama1*T(eta)))))^2*(diff(T(eta), eta, eta))*gama1^2*b[k1]+N_bt*T(eta)^2*exp(-(T(eta)-1)/(N_bt*(1+gama1)*(1+gama1*T(eta))))*(diff(T(eta), eta, eta))*gama1^2*a[k1]+2*N_bt*T(eta)*(exp(-(T(eta)-1)/(N_bt*(1+gama1)*(1+gama1*T(eta)))))^2*(diff(T(eta), eta, eta))*gama1*b[k1]+N_bt*T(eta)^2*(diff(T(eta), eta, eta))*gama1^2;


UEQ:=(a[mu1]*(-(diff(T(eta), eta))/(N_bt*(1+gama1)*(1+gama1*T(eta)))+(T(eta)-1)*gama1*(diff(T(eta), eta))/(N_bt*(1+gama1)*(1+gama1*T(eta))^2))*exp(-(T(eta)-1)/(N_bt*(1+gama1)*(1+gama1*T(eta))))+2*b[mu1]*(exp(-(T(eta)-1)/(N_bt*(1+gama1)*(1+gama1*T(eta)))))^2*(-(diff(T(eta), eta))/(N_bt*(1+gama1)*(1+gama1*T(eta)))+(T(eta)-1)*gama1*(diff(T(eta), eta))/(N_bt*(1+gama1)*(1+gama1*T(eta))^2)))*(diff(u(eta), eta))+(1+a[mu1]*exp(-(T(eta)-1)/(N_bt*(1+gama1)*(1+gama1*T(eta))))+b[mu1]*(exp(-(T(eta)-1)/(N_bt*(1+gama1)*(1+gama1*T(eta)))))^2)*(diff(u(eta), eta, eta))+1-exp(-(T(eta)-1)/(N_bt*(1+gama1)*(1+gama1*T(eta))))+exp(-(T(eta)-1)/(N_bt*(1+gama1)*(1+gama1*T(eta))))*rhop/rhobf;

I want to solve them with the following boundary conditions

T(0)=0, T(1)=1

u(0)=L*D(u)(0), D(u)(1)=0

However I tried, I cannot find the solution in a closed form. I want to know that is there a closed form solution for the above odes?

Thank you

Amir


Exact solutions to Einstein’s equations” is one of those books that are difficult even to imagine: the authors reviewed more than 4,000 papers containing solutions to Einstein’s equations in the general relativity literature, collecting, classifying, discarding repetitions in disguise, and organizing the whole material into chapters according to the physical properties of these solutions. The book is already in its second edition and it is a monumental piece of work.

 

As good as it is, however, the project resulted only in printed material, a textbook constituted of paper and ink. In 2006, when the DifferentialGeometry package was rewritten to enter the Maple library, one of the first things that passed through our minds was to bring the whole of “Exact solutions to Einstein’s equations” into Maple.

 

It took some time to start but in 2010, for Maple 14, we featured the first 26 solutions from this book. In Maple 15 this number jumped to 61. For Maple 17 we decided to emphasize the general relativity functionality of the DifferentialGeometry package, and Maple 18 added 50 more, featuring in total 225 of these solutions - great! but still far from the whole thing …

 

And this is when we decided to “step on the gas” - go for it, the whole book. One year later, working in collaboration with Denitsa Staicova from Bulgarian Academy of Sciences, Maple 2015 appeared with 330 solutions to Einstein’s equations. Today we have already implemented 492 solutions, and for the first time we can see the end of the tunnel: we are targeting finishing the whole book by the end of this year.

 

Wow2! This is a terrific result. First, because these solutions are key in the area of general relativity, and at this point what we have in Maple is already the most thorough digitized database of solutions to Einstein’s equations in the world. Second, and not any less important, because within Maple this knowledge comes alive. The solutions are fully searcheable and are set by a simple call to the Physics:-g_  spacetime metric command, and that automatically sets the related coordinates, Christoffel symbols , Ricci  and Riemann  tensors, orthonormal and null tetrads , etc. All of this happens on the fly, and all the mathematics within the Maple library are ready to work with these solutions. Having everything come alive completely changes the game. The ability to search the database according to the physical properties of the solutions, their classification, or just by parts of keywords also makes the whole book concretely more useful.

 

And, not only are these solutions to Einstein’s equations brought to life in a full-featured way through the Physics  package: they can also be reached through the DifferentialGeometry:-Library:-MetricSearch  applet. Almost all of the mathematical operations one can perform on them are also implemented as commands in DifferentialGeometry .

 

Finally, in the Maple PDEtools package , we already have all the mathematical tools to start resolving the equivalence problem around these solutions. That is: to answer whether a new solution is or not new, or whether it can be obtained from an existing solution by transformations of coordinates of different kinds. And we are going for it.

 

What follows is a basic illustration of what has already been implemented. As usual, in order to reproduce these results, you need to update your Physics library from the Maplesoft R&D Physics webpage.

 

Load Physics , set the metric to Schwarzschild (and everything else automatically) in one go

with(Physics)

g_[sc]

`Systems of spacetime Coordinates are: `*{X = (r, theta, phi, t)}

 

`Default differentiation variables for d_, D_ and dAlembertian are: `*{X = (r, theta, phi, t)}

 

`The Schwarzschild metric in coordinates `[r, theta, phi, t]

 

`Parameters: `[m]

 

g[mu, nu] = (Matrix(4, 4, {(1, 1) = r/(-r+2*m), (1, 2) = 0, (1, 3) = 0, (1, 4) = 0, (2, 1) = 0, (2, 2) = -r^2, (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = -r^2*sin(theta)^2, (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = 0, (4, 4) = (r-2*m)/r}))

(1)

And that is all we do :) Although the strength in Physics  is to compute with tensors using indicial notation, all of the tensor components and related properties of this metric are also derived on the fly (and no, they are not in any database). For instance these are the definition in terms of Christoffel symbols , and the covariant components of the Ricci tensor

Ricci[definition]

Physics:-Ricci[mu, nu] = Physics:-d_[alpha](Physics:-Christoffel[`~alpha`, mu, nu], [X])-Physics:-d_[nu](Physics:-Christoffel[`~alpha`, mu, alpha], [X])+Physics:-Christoffel[`~beta`, mu, nu]*Physics:-Christoffel[`~alpha`, beta, alpha]-Physics:-Christoffel[`~beta`, mu, alpha]*Physics:-Christoffel[`~alpha`, nu, beta]

(2)

Ricci[]

Physics:-Ricci[mu, nu] = Matrix(%id = 18446744078179871670)

(3)

These are the 16 Riemann invariants  for Schwarzschild solution, using the formulas by Carminati and McLenaghan

Riemann[invariants]

r[0] = 0, r[1] = 0, r[2] = 0, r[3] = 0, w[1] = 6*m^2/r^6, w[2] = 6*m^3/r^9, m[1] = 0, m[2] = 0, m[3] = 0, m[4] = 0, m[5] = 0

(4)

The related Weyl scalars  in the context of the Newman-Penrose formalism

Weyl[scalars]

psi__0 = 0, psi__1 = 0, psi__2 = -m/r^3, psi__3 = 0, psi__4 = 0

(5)

 

These are the 2x2 matrix components of the Christoffel symbols of the second kind (that describe, in coordinates, the effects of parallel transport in curved surfaces), when the first of its three indices is equal to 1

"Christoffel[~1,alpha,beta,matrix]"

Physics:-Christoffel[`~1`, alpha, beta] = Matrix(%id = 18446744078160684686)

(6)

In Physics, the Christoffel symbols of the first kind are represented by the same object (not two commands) just by taking the first index covariant, as we do when computing with paper and pencil

Christoffel[1, alpha, beta, matrix]

Physics:-Christoffel[1, alpha, beta] = Matrix(%id = 18446744078160680590)

(7)

One could query the database, directly from the spacetime metrics, about the solutions (metrics) to Einstein's equations related to Levi-Civita, the Italian mathematician

g_[civi]

____________________________________________________________

 

[12, 16, 1] = ["Authors" = ["Bertotti (1959)", "Kramer (1978)", "Levi-Civita (1917)", "Robinson (1959)"], "PrimaryDescription" = "EinsteinMaxwell", "SecondaryDescription" = ["Homogeneous"]]

 

____________________________________________________________

 

[12, 18, 1] = ["Authors" = ["Bertotti (1959)", "Kramer (1978)", "Levi-Civita (1917)", "Robinson (1959)"], "PrimaryDescription" = "EinsteinMaxwell", "SecondaryDescription" = ["Homogeneous"]]

 

____________________________________________________________

 

[12, 19, 1] = ["Authors" = ["Bertotti (1959)", "Kramer (1978)", "Levi-Civita (1917)", "Robinson (1959)"], "PrimaryDescription" = "EinsteinMaxwell", "SecondaryDescription" = ["Homogeneous"]]

(8)

These solutions can be set in one go from the metrics command, just by indicating the number with which it appears in "Exact Solutions to Einstein's Equations"

g_[[12, 16, 1]]

`Systems of spacetime Coordinates are: `*{X = (t, x, theta, phi)}

 

`Default differentiation variables for d_, D_ and dAlembertian are: `*{X = (t, x, theta, phi)}

 

`The Bertotti (1959), Kramer (1978), Levi-Civita (1917), Robinson (1959) metric in coordinates `[t, x, theta, phi]

 

`Parameters: `[k, kappa0, beta]

 

g[mu, nu] = (Matrix(4, 4, {(1, 1) = -k^2*sinh(x)^2, (1, 2) = 0, (1, 3) = 0, (1, 4) = 0, (2, 1) = 0, (2, 2) = k^2, (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = k^2, (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = 0, (4, 4) = k^2*sin(theta)^2}))

(9)

Automatically, everything gets set accordingly; these are the contravariant components of the related Ricci tensor

"Ricci[~]"

Physics:-Ricci[`~mu`, `~nu`] = Matrix(%id = 18446744078179869750)

(10)

One works with the Newman-Penrose formalism frequently using tetrads (local system of references); the Physics subpackage for this is Tetrads

with(Tetrads)

`Setting lowercaselatin letters to represent tetrad indices `

 

0, "%1 is not a command in the %2 package", Tetrads, Physics

 

0, "%1 is not a command in the %2 package", Tetrads, Physics

 

[IsTetrad, NullTetrad, OrthonormalTetrad, SimplifyTetrad, TransformTetrad, e_, eta_, gamma_, l_, lambda_, m_, mb_, n_]

(11)

This is the tetrad related to the book's metric with number 12.16.1

e_[]

Physics:-Tetrads:-e_[a, mu] = Matrix(%id = 18446744078160685286)

(12)

One can check these directly; for instance this is the definition of the tetrad, where the right-hand side is the tetrad metric

e_[definition]

Physics:-Tetrads:-e_[a, mu]*Physics:-Tetrads:-e_[b, `~mu`] = Physics:-Tetrads:-eta_[a, b]

(13)

This shows that, for the components given by (12), the definition holds

TensorArray(Physics:-Tetrads:-e_[a, mu]*Physics:-Tetrads:-e_[b, `~mu`] = Physics:-Tetrads:-eta_[a, b])

Matrix(%id = 18446744078195401422)

(14)

One frequently works with a different signature and null tetrads; set that, and everything gets automatically recomputed for the metric 12.16.1 accordingly

Setup(signature = "+---", tetradmetric = null)

[signature = `+ - - -`, tetradmetric = {(1, 2) = 1, (3, 4) = -1}]

(15)

eta_[]

eta[a, b] = (Matrix(4, 4, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 0, (1, 4) = 0, (2, 1) = 1, (2, 2) = 0, (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0, (3, 4) = -1, (4, 1) = 0, (4, 2) = 0, (4, 3) = -1, (4, 4) = 0}))

(16)

e_[]

Physics:-Tetrads:-e_[a, mu] = Matrix(%id = 18446744078191417574)

(17)

TensorArray(Physics:-Tetrads:-e_[a, mu]*Physics:-Tetrads:-e_[b, `~mu`] = Physics:-Tetrads:-eta_[a, b])

Matrix(%id = 18446744078191319390)

(18)

The related 16 Riemann invariant

Riemann[invariants]

r[0] = 0, r[1] = 1/k^4, r[2] = 0, r[3] = (1/4)/k^8, w[1] = 0, w[2] = 0, m[1] = 0, m[2] = 0, m[3] = 0, m[4] = 0, m[5] = 0

(19)

The ability to query rapidly, set things in one go, change everything again etc. are at this point fantastic. For instance, these are the metrics by Kaigorodov; next are those published in 1962

g_[Kaigorodov]

____________________________________________________________

 

[12, 34, 1] = ["Authors" = ["Kaigorodov (1962)", "Cahen (1964)", "Siklos (1981)", "Ozsvath (1987)"], "PrimaryDescription" = "Einstein", "SecondaryDescription" = ["Homogeneous"], "Comments" = ["All metrics with _epsilon <> 0 are equivalent to the cases _epsilon = +1, -1, _epsilon = 0 is anti-deSitter space"]]

 

____________________________________________________________

 

[12, 35, 1] = ["Authors" = ["Kaigorodov (1962)", "Cahen (1964)", "Siklos (1981)", "Ozsvath (1987)"], "PrimaryDescription" = "Einstein", "SecondaryDescription" = ["Homogeneous", "SimpleTransitive"]]

(20)

g_[`1962`]

____________________________________________________________

 

[12, 13, 1] = ["Authors" = ["Ozsvath, Schucking (1962)"], "PrimaryDescription" = "Vacuum", "SecondaryDescription" = ["Homogeneous", "PlaneWave"], "Comments" = ["geodesically complete, no curvature singularities"]]

 

____________________________________________________________

 

[12, 14, 1] = ["Authors" = ["Petrov (1962)"], "PrimaryDescription" = "Vacuum", "SecondaryDescription" = ["Homogeneous", "SimpleTransitive"]]

 

____________________________________________________________

 

[12, 34, 1] = ["Authors" = ["Kaigorodov (1962)", "Cahen (1964)", "Siklos (1981)", "Ozsvath (1987)"], "PrimaryDescription" = "Einstein", "SecondaryDescription" = ["Homogeneous"], "Comments" = ["All metrics with _epsilon <> 0 are equivalent to the cases _epsilon = +1, -1, _epsilon = 0 is anti-deSitter space"]]

 

____________________________________________________________

 

[12, 35, 1] = ["Authors" = ["Kaigorodov (1962)", "Cahen (1964)", "Siklos (1981)", "Ozsvath (1987)"], "PrimaryDescription" = "Einstein", "SecondaryDescription" = ["Homogeneous", "SimpleTransitive"]]

 

____________________________________________________________

 

[28, 16, 1] = ["Authors" = ["Robinson-Trautman (1962)"], "PrimaryDescription" = "Vacuum", "SecondaryDescription" = ["RobinsonTrautman"], "Comments" = ["The coordinate zeta is changed to xi", "AlternativeOrthonormalTetrad1 and AlternativeNullTetrad1 are adapted to the shear-free null geodesic congruence (Robinson-Trautman tetrads)"]]

 

____________________________________________________________

 

[28, 26, 1] = ["Authors" = ["Robinson, Trautman (1962)"], "PrimaryDescription" = "Vacuum", "SecondaryDescription" = ["RobinsonTrautman"], "Comments" = ["One can use the diffeo r -> -r and u -> -u to make the assumption r > 0", "The case _m = 0 is Stephani, [28, 16,1]", "The metric is type D at points where r = 3*_m/(xi1+xi2) and type II on either side of this hypersurface. For convenience, it is assumed that 3*_m  - r*(xi1 + xi2) > 0", "AlternativeOrthonormalTetrad1 and AlternativeNullTetrad1 are adapted to the shear-free null geodesic congruence (Robinson-Trautman tetrads)"]]

 

____________________________________________________________

 

[28, 26, 2] = ["Authors" = ["Robinson, Trautman (1962)"], "PrimaryDescription" = "Vacuum", "SecondaryDescription" = ["RobinsonTrautman"], "Comments" = ["One can use the diffeo r -> -r and u -> -u to make the assumption r > 0", "The case _m = 0 is Stephani, [28, 16,1].", "AlternativeOrthonormalTetrad1 and AlternativeNullTetrad1 are adapted to the shear-free null geodesic congruence (Robinson-Trautman tetrads)"]]

 

____________________________________________________________

 

[28, 26, 3] = ["Authors" = ["Robinson, Trautman (1962)"], "PrimaryDescription" = "Vacuum", "SecondaryDescription" = ["RobinsonTrautman"], "Comments" = ["One can use the diffeo r -> -r and u -> -u to make the assumption r > 0", "The case _m = 0 is Stephani, [28, 16,1].", "AlternativeOrthonormalTetrad1 and AlternativeNullTetrad1 are adapted to the shear-free null geodesic congruence (Robinson-Trautman tetrads)"]]

 

____________________________________________________________

 

[28, 43, 1] = ["Authors" = ["Robinson, Trautman (1962)"], "PrimaryDescription" = "EinsteinMaxwell", "SecondaryDescription" = ["PureRadiation", "RobinsonTrautman"], "Comments" = ["h1(u) is the conjugate of h(u)"]]

(21)

 

The search can be done visually, by properties; this is the only solution in the database that is a Pure Ratiation solution, of Petrov Type "D", Plebanski-Petrov Type "O" and that has Isometry Dimension equal to 1:

DifferentialGeometry:-Library:-MetricSearch()

 

Set the solution, and everything related to work with it, in one go

g_[[28, 74, 1]]

`Systems of spacetime Coordinates are: `*{X = (u, eta, r, y)}

 

`Default differentiation variables for d_, D_ and dAlembertian are: `*{X = (u, eta, r, y)}

 

`The Frolov and Khlebnikov (1975) metric in coordinates `[u, eta, r, y]

 

`Parameters: `[kappa0, m(u), b, d]

 

"`Comments: `With _m(u) = constant, the metric is Ricci flat and becomes 28.24 in Stephani."

 

g[mu, nu] = (Matrix(4, 4, {(1, 1) = (2*m(u)^3-6*m(u)^2*eta*r-r^2*(-6*eta^2+b)*m(u)+r^3*(-2*eta^3+b*eta+d))/(r*m(u)^2), (1, 2) = -r^2/m(u), (1, 3) = -1, (1, 4) = 0, (2, 1) = -r^2/m(u), (2, 2) = r^2/(-2*eta^3+b*eta+d), (2, 3) = 0, (2, 4) = 0, (3, 1) = -1, (3, 2) = 0, (3, 3) = 0, (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = 0, (4, 4) = r^2*(-2*eta^3+b*eta+d)}))

(22)

 

The related Riemann invariants:

Riemann[invariants]

r[0] = 0, r[1] = 0, r[2] = 0, r[3] = 0, w[1] = 6*m(u)^2/r^6, w[2] = -6*m(u)^3/r^9, m[1] = 0, m[2] = 0, m[3] = 0, m[4] = 0, m[5] = 0

(23)

To conclude, how many solutions from the book have we already implemented?

DifferentialGeometry:-Library:-Retrieve("Stephani", 1)

[[8, 33, 1], [8, 34, 1], [12, 6, 1], [12, 7, 1], [12, 8, 1], [12, 8, 2], [12, 8, 3], [12, 8, 4], [12, 8, 5], [12, 8, 6], [12, 8, 7], [12, 8, 8], [12, 9, 1], [12, 9, 2], [12, 9, 3], [12, 12, 1], [12, 12, 2], [12, 12, 3], [12, 12, 4], [12, 13, 1], [12, 14, 1], [12, 16, 1], [12, 18, 1], [12, 19, 1], [12, 21, 1], [12, 23, 1], [12, 23, 2], [12, 23, 3], [12, 24.1, 1], [12, 24.2, 1], [12, 24.3, 1], [12, 26, 1], [12, 27, 1], [12, 28, 1], [12, 29, 1], [12, 30, 1], [12, 31, 1], [12, 32, 1], [12, 34, 1], [12, 35, 1], [12, 36, 1], [12, 37, 1], [12, 37, 2], [12, 37, 3], [12, 37, 4], [12, 37, 5], [12, 37, 6], [12, 37, 7], [12, 37, 8], [12, 37, 9], [12, 38, 1], [12, 38, 2], [12, 38, 3], [12, 38, 4], [12, 38, 5], [13, 2, 1], [13, 2, 2], [13, 2, 3], [13, 7, 1], [13, 7, 2], [13, 7, 3], [13, 7, 4], [13, 7, 5], [13, 7, 6], [13, 7, 7], [13, 7, 8], [13, 14, 1], [13, 14, 2], [13, 14, 3], [13, 19, 1], [13, 31, 1], [13, 32, 1], [13, 46, 1], [13, 48, 1], [13, 49, 1], [13, 49, 2], [13, 51, 1], [13, 53, 1], [13, 59, 1], [13, 59, 2], [13, 60, 1], [13, 60, 2], [13, 60, 3], [13, 60, 4], [13, 60, 5], [13, 60, 6], [13, 60, 7], [13, 60, 8], [13, 61, 1], [13, 61, 2], [13, 62, 1], [13, 62, 2], [13, 62, 4], [13, 62, 6], [13, 63, 1], [13, 63, 2], [13, 63, 3], [13, 63, 4], [13, 64, 1], [13, 64, 2], [13, 64, 3], [13, 64, 4], [13, 65, 1], [13, 69, 1], [13, 71, 1], [13, 72, 1], [13, 73, 1], [13, 74, 1], [13, 74, 2], [13, 74, 3], [13, 76, 1], [13, 77, 1], [13, 77, 2], [13, 79, 1], [13, 79, 2], [13, 80, 1], [13, 81, 1], [13, 83, 1], [13, 84, 1], [13, 84, 2], [13, 84, 3], [13, 85, 1], [13, 85, 2], [13, 86, 1], [13, 87, 1], [14, 6.1, 1], [14, 6.2, 1], [14, 6.3, 1], [14, 7, 1], [14, 8.1, 1], [14, 8.2, 1], [14, 8.3, 1], [14, 9.1, 1], [14, 9.2, 1], [14, 10, 1], [14, 10, 2], [14, 12, 1], [14, 12, 2], [14, 12, 3], [14, 14, 1], [14, 14, 2], [14, 15, 1], [14, 15.1, 2], [14, 15.2, 2], [14, 15.3, 2], [14, 16, 1], [14, 16, 2], [14, 17, 1], [14, 18, 1], [14, 18, 2], [14, 19, 1], [14, 20, 1], [14, 21, 1], [14, 21, 2], [14, 21, 3], [14, 22, 1], [14, 23, 1], [14, 24, 1], [14, 25, 1], [14, 26, 1], [14, 26, 2], [14, 26, 3], [14, 26, 4], [14, 27, 1], [14, 28, 1], [14, 28, 2], [14, 28, 3], [14, 29, 1], [14, 30, 1], [14, 31, 1], [14, 32, 1], [14, 33, 1], [14, 35, 1], [14, 37, 1], [14, 38, 1], [14, 38, 2], [14, 38, 3], [14, 39, 1], [14, 39, 2], [14, 39, 3], [14, 39, 4], [14, 39, 5], [14, 39, 6], [14, 40, 1], [14, 41, 1], [14, 42, 1], [14, 46, 1], [15, 3, 1], [15, 3, 2], [15, 4, 1], [15, 4, 2], [15, 4, 3], [15, 9, 1], [15, 10, 1], [15, 12, 1], [15, 12, 2], [15, 12, 3], [15, 12, 4], [15, 12, 5], [15, 12, 6], [15, 17, 1], [15, 17, 2], [15, 17, 3], [15, 17, 4], [15, 18, 1], [15, 19, 1], [15, 19, 2], [15, 20, 1], [15, 21, 1], [15, 21, 2], [15, 22, 1], [15, 23, 1], [15, 23, 2], [15, 24, 1], [15, 24, 2], [15, 25, 1], [15, 25, 2], [15, 26, 1], [15, 26, 2], [15, 27, 1], [15, 27, 2], [15, 27, 3], [15, 27, 4], [15, 27, 5], [15, 27, 6], [15, 27, 7], [15, 27, 8], [15, 28, 1], [15, 29, 1], [15, 30, 1], [15, 31, 1], [15, 32, 1], [15, 34, 1], [15, 34, 2], [15, 34, 3], [15, 43, 1], [15, 43, 2], [15, 43, 3], [15, 50, 1], [15, 50, 2], [15, 50, 3], [15, 50, 4], [15, 50, 5], [15, 50, 6], [15, 62, 1], [15, 62, 2], [15, 62, 3], [15, 63, 1], [15, 63, 2], [15, 63, 3], [15, 65, 1], [15, 65, 2], [15, 66, 1], [15, 66, 2], [15, 66, 3], [15, 75, 1], [15, 75, 2], [15, 75, 3], [15, 77, 1], [15, 77, 2], [15, 77, 3], [15, 78, 1], [15, 79, 1], [15, 81, 1], [15, 81, 2], [15, 81, 3], [15, 82, 1], [15, 82, 2], [15, 82, 3], [15, 83, 1.1], [15, 83, 1.2], [15, 83, 2], [15, 83, 3.1], [15, 83, 3.2], [15, 83, 4], [15, 84, 1], [15, 85, 1], [15, 85, 2], [15, 85, 3], [15, 86, 1], [15, 86, 2], [15, 86, 3], [15, 87, 1], [15, 87, 2], [15, 87, 3], [15, 87, 4], [15, 87, 5], [15, 88, 1], [15, 89, 1], [15, 90, 1], [16, 1, 1], [16, 1, 2], [16, 1, 3], [16, 1, 4], [16, 1, 5], [16, 1, 6], [16, 1, 7], [16, 1, 8], [16, 1, 9], [16, 1, 10], [16, 1, 11], [16, 1, 12], [16, 1, 13], [16, 1, 14], [16, 1, 15], [16, 1, 16], [16, 1, 17], [16, 1, 18], [16, 1, 19], [16, 1, 20], [16, 1, 21], [16, 1, 22], [16, 1, 23], [16, 1, 24], [16, 1, 25], [16, 1, 26], [16, 1, 27], [16, 14, 1], [16, 14, 2], [16, 14, 3], [16, 14, 4], [16, 14, 5], [16, 14, 6], [16, 14, 7], [16, 14, 8], [16, 14, 9], [16, 14, 10], [16, 14, 11], [16, 14, 12], [16, 14, 13], [16, 14, 14], [16, 14, 15], [16, 14, 16], [16, 14, 17], [16, 14, 18], [16, 14, 19], [16, 14, 20], [16, 18, 1], [16, 19, 1], [16, 20, 1], [16, 22, 1], [16, 24, 1], [16, 24, 2], [16, 43, 1], [16, 45, 1], [16, 45, 2], [16, 46, 1], [16, 47, 1], [16, 50, 1], [16, 51, 1], [16, 54, 1], [16, 61, 1], [16, 63, 1], [16, 66, 1], [16, 66, 2], [16, 66, 3], [16, 67, 1], [16, 71, 1], [16, 72, 1], [16, 73, 1], [16, 74, 1], [16, 75, 1], [16, 76, 1], [16, 77, 1], [16, 77, 2], [16, 77, 3], [16, 78, 1], [17, 4, 1], [17, 4, 2], [17, 5, 1], [17, 9, 1], [17, 14, 1], [17, 15, 1], [17, 15, 2], [17, 16, 1], [17, 20, 1], [17, 22, 1], [17, 23, 1], [17, 24, 1], [17, 24, 2], [17, 26, 1], [17, 27, 1], [17, 27, 2], [17, 28, 1], [17, 28, 2], [17, 29, 1], [17, 29, 2], [17, 30, 1], [17, 31, 1], [18, 2, 1], [18, 2, 2], [18, 2, 3], [18, 2, 4], [18, 2, 5], [18, 2, 6], [18, 2, 7], [18, 2, 8], [18, 48, 1], [18, 48, 2], [18, 49, 1], [18, 50, 1], [18, 64, 1], [18, 64, 2], [18, 64, 3], [18, 65, 1], [18, 66, 1], [18, 67, 1], [18, 71, 1], [18, 75, 1], [19, 17, 1], [19, 17, 2], [19, 21, 1], [20, 3, 1], [20, 4, 1], [20, 5, 1], [20, 7, 1], [20, 8, 1], [20, 9, 1], [20, 10, 1], [20, 11, 1], [20, 12, 1], [20, 13, 1], [20, 15, 1], [20, 16, 1], [20, 17, 1], [20, 20, 1], [20, 21, 1], [20, 23, 1], [20, 27, 1], [20, 28, 1], [20, 29, 1], [20, 32, 1], [20, 34, 1], [20, 36, 1], [20, 38, 1], [20, 38, 2], [20, 38, 3], [20, 44, 1], [20, 46, 1], [20, 54, 1], [20, 57, 1], [20, 57, 2], [28, 16, 1], [28, 17, 1], [28, 21, 1], [28, 21, 2], [28, 21, 3], [28, 21, 4], [28, 21, 5], [28, 21, 6], [28, 21, 7], [28, 24, 1], [28, 25, 1], [28, 26, 1], [28, 26, 2], [28, 26, 3], [28, 41, 1], [28, 43, 1], [28, 44, 1], [28, 44, 2], [28, 44, 3], [28, 44, 4], [28, 44, 5], [28, 44, 6], [28, 45, 1], [28, 45, 2], [28, 46, 1], [28, 46, 2], [28, 53, 1], [28, 53, 2], [28, 55, 1], [28, 55, 2], [28, 56.1, 1], [28, 56.2, 2], [28, 56.2, 3], [28, 56.3, 1], [28, 56.4, 1], [28, 56.5, 1], [28, 56.6, 1], [28, 58.2, 1], [28, 58.3, 1], [28, 58.3, 2], [28, 58.4, 1], [28, 60, 1], [28, 61, 1], [28, 64, 1], [28, 66, 1], [28, 67, 1], [28, 68, 1], [28, 72, 1], [28, 73, 1], [28, 74, 1]]

(24)

nops([[8, 33, 1], [8, 34, 1], [12, 6, 1], [12, 7, 1], [12, 8, 1], [12, 8, 2], [12, 8, 3], [12, 8, 4], [12, 8, 5], [12, 8, 6], [12, 8, 7], [12, 8, 8], [12, 9, 1], [12, 9, 2], [12, 9, 3], [12, 12, 1], [12, 12, 2], [12, 12, 3], [12, 12, 4], [12, 13, 1], [12, 14, 1], [12, 16, 1], [12, 18, 1], [12, 19, 1], [12, 21, 1], [12, 23, 1], [12, 23, 2], [12, 23, 3], [12, 24.1, 1], [12, 24.2, 1], [12, 24.3, 1], [12, 26, 1], [12, 27, 1], [12, 28, 1], [12, 29, 1], [12, 30, 1], [12, 31, 1], [12, 32, 1], [12, 34, 1], [12, 35, 1], [12, 36, 1], [12, 37, 1], [12, 37, 2], [12, 37, 3], [12, 37, 4], [12, 37, 5], [12, 37, 6], [12, 37, 7], [12, 37, 8], [12, 37, 9], [12, 38, 1], [12, 38, 2], [12, 38, 3], [12, 38, 4], [12, 38, 5], [13, 2, 1], [13, 2, 2], [13, 2, 3], [13, 7, 1], [13, 7, 2], [13, 7, 3], [13, 7, 4], [13, 7, 5], [13, 7, 6], [13, 7, 7], [13, 7, 8], [13, 14, 1], [13, 14, 2], [13, 14, 3], [13, 19, 1], [13, 31, 1], [13, 32, 1], [13, 46, 1], [13, 48, 1], [13, 49, 1], [13, 49, 2], [13, 51, 1], [13, 53, 1], [13, 59, 1], [13, 59, 2], [13, 60, 1], [13, 60, 2], [13, 60, 3], [13, 60, 4], [13, 60, 5], [13, 60, 6], [13, 60, 7], [13, 60, 8], [13, 61, 1], [13, 61, 2], [13, 62, 1], [13, 62, 2], [13, 62, 4], [13, 62, 6], [13, 63, 1], [13, 63, 2], [13, 63, 3], [13, 63, 4], [13, 64, 1], [13, 64, 2], [13, 64, 3], [13, 64, 4], [13, 65, 1], [13, 69, 1], [13, 71, 1], [13, 72, 1], [13, 73, 1], [13, 74, 1], [13, 74, 2], [13, 74, 3], [13, 76, 1], [13, 77, 1], [13, 77, 2], [13, 79, 1], [13, 79, 2], [13, 80, 1], [13, 81, 1], [13, 83, 1], [13, 84, 1], [13, 84, 2], [13, 84, 3], [13, 85, 1], [13, 85, 2], [13, 86, 1], [13, 87, 1], [14, 6.1, 1], [14, 6.2, 1], [14, 6.3, 1], [14, 7, 1], [14, 8.1, 1], [14, 8.2, 1], [14, 8.3, 1], [14, 9.1, 1], [14, 9.2, 1], [14, 10, 1], [14, 10, 2], [14, 12, 1], [14, 12, 2], [14, 12, 3], [14, 14, 1], [14, 14, 2], [14, 15, 1], [14, 15.1, 2], [14, 15.2, 2], [14, 15.3, 2], [14, 16, 1], [14, 16, 2], [14, 17, 1], [14, 18, 1], [14, 18, 2], [14, 19, 1], [14, 20, 1], [14, 21, 1], [14, 21, 2], [14, 21, 3], [14, 22, 1], [14, 23, 1], [14, 24, 1], [14, 25, 1], [14, 26, 1], [14, 26, 2], [14, 26, 3], [14, 26, 4], [14, 27, 1], [14, 28, 1], [14, 28, 2], [14, 28, 3], [14, 29, 1], [14, 30, 1], [14, 31, 1], [14, 32, 1], [14, 33, 1], [14, 35, 1], [14, 37, 1], [14, 38, 1], [14, 38, 2], [14, 38, 3], [14, 39, 1], [14, 39, 2], [14, 39, 3], [14, 39, 4], [14, 39, 5], [14, 39, 6], [14, 40, 1], [14, 41, 1], [14, 42, 1], [14, 46, 1], [15, 3, 1], [15, 3, 2], [15, 4, 1], [15, 4, 2], [15, 4, 3], [15, 9, 1], [15, 10, 1], [15, 12, 1], [15, 12, 2], [15, 12, 3], [15, 12, 4], [15, 12, 5], [15, 12, 6], [15, 17, 1], [15, 17, 2], [15, 17, 3], [15, 17, 4], [15, 18, 1], [15, 19, 1], [15, 19, 2], [15, 20, 1], [15, 21, 1], [15, 21, 2], [15, 22, 1], [15, 23, 1], [15, 23, 2], [15, 24, 1], [15, 24, 2], [15, 25, 1], [15, 25, 2], [15, 26, 1], [15, 26, 2], [15, 27, 1], [15, 27, 2], [15, 27, 3], [15, 27, 4], [15, 27, 5], [15, 27, 6], [15, 27, 7], [15, 27, 8], [15, 28, 1], [15, 29, 1], [15, 30, 1], [15, 31, 1], [15, 32, 1], [15, 34, 1], [15, 34, 2], [15, 34, 3], [15, 43, 1], [15, 43, 2], [15, 43, 3], [15, 50, 1], [15, 50, 2], [15, 50, 3], [15, 50, 4], [15, 50, 5], [15, 50, 6], [15, 62, 1], [15, 62, 2], [15, 62, 3], [15, 63, 1], [15, 63, 2], [15, 63, 3], [15, 65, 1], [15, 65, 2], [15, 66, 1], [15, 66, 2], [15, 66, 3], [15, 75, 1], [15, 75, 2], [15, 75, 3], [15, 77, 1], [15, 77, 2], [15, 77, 3], [15, 78, 1], [15, 79, 1], [15, 81, 1], [15, 81, 2], [15, 81, 3], [15, 82, 1], [15, 82, 2], [15, 82, 3], [15, 83, 1.1], [15, 83, 1.2], [15, 83, 2], [15, 83, 3.1], [15, 83, 3.2], [15, 83, 4], [15, 84, 1], [15, 85, 1], [15, 85, 2], [15, 85, 3], [15, 86, 1], [15, 86, 2], [15, 86, 3], [15, 87, 1], [15, 87, 2], [15, 87, 3], [15, 87, 4], [15, 87, 5], [15, 88, 1], [15, 89, 1], [15, 90, 1], [16, 1, 1], [16, 1, 2], [16, 1, 3], [16, 1, 4], [16, 1, 5], [16, 1, 6], [16, 1, 7], [16, 1, 8], [16, 1, 9], [16, 1, 10], [16, 1, 11], [16, 1, 12], [16, 1, 13], [16, 1, 14], [16, 1, 15], [16, 1, 16], [16, 1, 17], [16, 1, 18], [16, 1, 19], [16, 1, 20], [16, 1, 21], [16, 1, 22], [16, 1, 23], [16, 1, 24], [16, 1, 25], [16, 1, 26], [16, 1, 27], [16, 14, 1], [16, 14, 2], [16, 14, 3], [16, 14, 4], [16, 14, 5], [16, 14, 6], [16, 14, 7], [16, 14, 8], [16, 14, 9], [16, 14, 10], [16, 14, 11], [16, 14, 12], [16, 14, 13], [16, 14, 14], [16, 14, 15], [16, 14, 16], [16, 14, 17], [16, 14, 18], [16, 14, 19], [16, 14, 20], [16, 18, 1], [16, 19, 1], [16, 20, 1], [16, 22, 1], [16, 24, 1], [16, 24, 2], [16, 43, 1], [16, 45, 1], [16, 45, 2], [16, 46, 1], [16, 47, 1], [16, 50, 1], [16, 51, 1], [16, 54, 1], [16, 61, 1], [16, 63, 1], [16, 66, 1], [16, 66, 2], [16, 66, 3], [16, 67, 1], [16, 71, 1], [16, 72, 1], [16, 73, 1], [16, 74, 1], [16, 75, 1], [16, 76, 1], [16, 77, 1], [16, 77, 2], [16, 77, 3], [16, 78, 1], [17, 4, 1], [17, 4, 2], [17, 5, 1], [17, 9, 1], [17, 14, 1], [17, 15, 1], [17, 15, 2], [17, 16, 1], [17, 20, 1], [17, 22, 1], [17, 23, 1], [17, 24, 1], [17, 24, 2], [17, 26, 1], [17, 27, 1], [17, 27, 2], [17, 28, 1], [17, 28, 2], [17, 29, 1], [17, 29, 2], [17, 30, 1], [17, 31, 1], [18, 2, 1], [18, 2, 2], [18, 2, 3], [18, 2, 4], [18, 2, 5], [18, 2, 6], [18, 2, 7], [18, 2, 8], [18, 48, 1], [18, 48, 2], [18, 49, 1], [18, 50, 1], [18, 64, 1], [18, 64, 2], [18, 64, 3], [18, 65, 1], [18, 66, 1], [18, 67, 1], [18, 71, 1], [18, 75, 1], [19, 17, 1], [19, 17, 2], [19, 21, 1], [20, 3, 1], [20, 4, 1], [20, 5, 1], [20, 7, 1], [20, 8, 1], [20, 9, 1], [20, 10, 1], [20, 11, 1], [20, 12, 1], [20, 13, 1], [20, 15, 1], [20, 16, 1], [20, 17, 1], [20, 20, 1], [20, 21, 1], [20, 23, 1], [20, 27, 1], [20, 28, 1], [20, 29, 1], [20, 32, 1], [20, 34, 1], [20, 36, 1], [20, 38, 1], [20, 38, 2], [20, 38, 3], [20, 44, 1], [20, 46, 1], [20, 54, 1], [20, 57, 1], [20, 57, 2], [28, 16, 1], [28, 17, 1], [28, 21, 1], [28, 21, 2], [28, 21, 3], [28, 21, 4], [28, 21, 5], [28, 21, 6], [28, 21, 7], [28, 24, 1], [28, 25, 1], [28, 26, 1], [28, 26, 2], [28, 26, 3], [28, 41, 1], [28, 43, 1], [28, 44, 1], [28, 44, 2], [28, 44, 3], [28, 44, 4], [28, 44, 5], [28, 44, 6], [28, 45, 1], [28, 45, 2], [28, 46, 1], [28, 46, 2], [28, 53, 1], [28, 53, 2], [28, 55, 1], [28, 55, 2], [28, 56.1, 1], [28, 56.2, 2], [28, 56.2, 3], [28, 56.3, 1], [28, 56.4, 1], [28, 56.5, 1], [28, 56.6, 1], [28, 58.2, 1], [28, 58.3, 1], [28, 58.3, 2], [28, 58.4, 1], [28, 60, 1], [28, 61, 1], [28, 64, 1], [28, 66, 1], [28, 67, 1], [28, 68, 1], [28, 72, 1], [28, 73, 1], [28, 74, 1]])

492

(25)

NULL

:)



Download Exact_Solutions_to_Einstein_Equations.mw

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

with(DynamicSystems)

T:= a vector

Iwant to make a single plot of:

DiscretePlot(T,x1,stile=stair); DisdretPlot(T,x2,stile=stair)

The usal way: DiscreePlot(T,[x1,x2]... ain't work

Thansk for helping

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