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rinvariant

? rinvariant;

Calculate

rinvariant

? rinvariant;

Calculate

rinvariant

Run Example
(%i1)/*  Schwarzschild simplesave as name.macLoad from the file menu - >
 File|Batch file*/kill(all);
(%o0)                                done
(%i1) load(ctensor);
(%o1)          /usr/share/maxima/5.21.1/share/tensor/ctensor.mac
(%i2) /* set some flags */cframe_flag: false;
(%o2)                                false
(%i3) ratchristof: true;
(%o3)                                true
(%i4) ratriemann : true;
(%o4)                                true
(%i5) ratfac : true;
(%o5)                                true
(%i6) ctrgsimp: true;
(%o6)                                true
(%i7) /* define the dimension */dim: 4;
(%o7)                                  4
(%i8) /* list the coordinates */ct_coords: [t,x,y,z];
(%o8)                            [t, x, y, z]
(%i9) /* set up the metric *//* assign to lg a matrix of zeros  ':' means 'assign' */lg:zeromatrix(4,4);
                                [ 0  0  0  0 ]
                                [            ]
                                [ 0  0  0  0 ]
(%o9)                           [            ]
                                [ 0  0  0  0 ]
                                [            ]
                                [ 0  0  0  0 ]
(%i10) /* now add the Schwarzschild coefficients */lg[1,1]:1;
(%o10)                                 1
(%i11) lg[2,2]:-1;
(%o11)                                - 1
(%i12) lg[3,3]:-(1+f(t-x))^2;
                                               2
(%o12)                         - (f(t - x) + 1)
(%i13) lg[4,4]:-(1+g(t-x))^2;
                                               2
(%o13)                         - (g(t - x) + 1)
(%i14) /* make the inverse matrix */ug:invert(lg);
               [ 1   0           0                  0         ]
               [                                              ]
               [ 0  - 1          0                  0         ]
               [                                              ]
               [                  1                           ]
               [ 0   0   - ---------------          0         ]
(%o14)         [                         2                    ]
               [           (f(t - x) + 1)                     ]
               [                                              ]
               [                                     1        ]
               [ 0   0           0          - --------------- ]
               [                                            2 ]
               [                              (g(t - x) + 1)  ]
(%i15) /* get Christoffels */christof(mcs);
                                       d
                                       -- (f(t - x))
                                       dt
(%t15)                    mcs        = -------------
                             1, 3, 3   f(t - x) + 1

                                       d
                                       -- (g(t - x))
                                       dt
(%t16)                    mcs        = -------------
                             1, 4, 4   g(t - x) + 1

                                       d
                                       -- (f(t - x))
                                       dx
(%t17)                    mcs        = -------------
                             2, 3, 3   f(t - x) + 1

                                       d
                                       -- (g(t - x))
                                       dx
(%t18)                    mcs        = -------------
                             2, 4, 4   g(t - x) + 1

                                d
(%t19)            mcs        = (-- (f(t - x))) (f(t - x) + 1)
                     3, 3, 1    dt

                                 d
(%t20)           mcs        = - (-- (f(t - x))) (f(t - x) + 1)
                    3, 3, 2      dx

                                d
(%t21)            mcs        = (-- (g(t - x))) (g(t - x) + 1)
                     4, 4, 1    dt

                                 d
(%t22)           mcs        = - (-- (g(t - x))) (g(t - x) + 1)
                    4, 4, 2      dx

(%o22)                               done
(%i23) /* calculate and display Ricci tensor, ( 'dis' = true ) */ricci(true);
                      2                           2
                     d                           d
(%t23) ric     = - ((--- (f(t - x))) g(t - x) + (--- (g(t - x))) f(t - x)
          1, 1         2                           2
                     dt                          dt
                2                2
               d                d
             + --- (g(t - x)) + --- (f(t - x)))/((f(t - x) + 1) (g(t - x) + 1))
                 2                2
               dt               dt

                       2                             2
                      d                             d
(%t24) ric     = - ((----- (f(t - x))) g(t - x) + (----- (g(t - x))) f(t - x)
          1, 2       dt dx                         dt dx
             2                  2
            d                  d
         + ----- (g(t - x)) + ----- (f(t - x)))/((f(t - x) + 1) (g(t - x) + 1))
           dt dx              dt dx

                      2                           2
                     d                           d
(%t25) ric     = - ((--- (f(t - x))) g(t - x) + (--- (g(t - x))) f(t - x)
          2, 2         2                           2
                     dx                          dx
                2                2
               d                d
             + --- (g(t - x)) + --- (f(t - x)))/((f(t - x) + 1) (g(t - x) + 1))
                 2                2
               dx               dx

                                      2
                                     d
(%t26) ric     = - ((f(t - x) + 1) ((--- (f(t - x))) g(t - x)
          3, 3                         2
                                     dx
     2
    d                           d               d
 - (--- (f(t - x))) g(t - x) + (-- (f(t - x))) (-- (g(t - x)))
      2                         dx              dx
    dt
                                      2                2
    d               d                d                d
 - (-- (f(t - x))) (-- (g(t - x))) + --- (f(t - x)) - --- (f(t - x))))
    dt              dt                 2                2
                                     dx               dt
/(g(t - x) + 1)

                       2                           2
                      d                           d
(%t27) ric     = - (((--- (g(t - x))) f(t - x) - (--- (g(t - x))) f(t - x)
          4, 4          2                           2
                      dx                          dt
    2                                                  2
   d                 d               d                d
 + --- (g(t - x)) + (-- (f(t - x))) (-- (g(t - x))) - --- (g(t - x))
     2               dx              dx                 2
   dx                                                 dt
    d               d
 - (-- (f(t - x))) (-- (g(t - x)))) (g(t - x) + 1))/(f(t - x) + 1)
    dt              dt

(%o27)                               done
(%i28) /* Riemann with all indexes down */lriemann(true);
                                   2
                                  d
(%t28)         lriem           = (--- (f(t - x))) (f(t - x) + 1)
                    3, 3, 1, 1      2
                                  dt

                                   2
                                  d
(%t29)        lriem           = (----- (f(t - x))) (f(t - x) + 1)
                   3, 3, 1, 2    dt dx

                                   2
                                  d
(%t30)        lriem           = (----- (f(t - x))) (f(t - x) + 1)
                   3, 3, 2, 1    dt dx

                                   2
                                  d
(%t31)         lriem           = (--- (f(t - x))) (f(t - x) + 1)
                    3, 3, 2, 2      2
                                  dx

                                   2
                                  d
(%t32)         lriem           = (--- (g(t - x))) (g(t - x) + 1)
                    4, 4, 1, 1      2
                                  dt

                                   2
                                  d
(%t33)        lriem           = (----- (g(t - x))) (g(t - x) + 1)
                   4, 4, 1, 2    dt dx

                                   2
                                  d
(%t34)        lriem           = (----- (g(t - x))) (g(t - x) + 1)
                   4, 4, 2, 1    dt dx

                                   2
                                  d
(%t35)         lriem           = (--- (g(t - x))) (g(t - x) + 1)
                    4, 4, 2, 2      2
                                  dx

                           d               d
(%t36) lriem           = ((-- (f(t - x))) (-- (g(t - x)))
            4, 4, 3, 3     dx              dx
                  d               d
               - (-- (f(t - x))) (-- (g(t - x)))) (f(t - x) + 1) (g(t - x) + 1)
                  dt              dt

(%o36)                               done
(%i37) /* Riemann with all indexes up */uriemann(false);
(%o37)                               done
(%i38) /* calculate Einstein tensor */einstein(false);
(%o38)                               done
(%i39) /* show Einstein tensor as a matrix */cdisplay(ein);
                   2                           2
                  d                           d
ein = matrix([- ((--- (f(t - x))) g(t - x) + (--- (g(t - x))) f(t - x)
                    2                           2
                  dx                          dx
    2
   d                 d               d
 + --- (g(t - x)) + (-- (f(t - x))) (-- (g(t - x)))
     2               dx              dx
   dx
                                      2
    d               d                d
 - (-- (f(t - x))) (-- (g(t - x))) + --- (f(t - x)))
    dt              dt                 2
                                     dx
                                      2
                                     d
/((f(t - x) + 1) (g(t - x) + 1)), ((----- (f(t - x))) g(t - x)
                                    dt dx
      2                            2                  2
     d                            d                  d
 + (----- (g(t - x))) f(t - x) + ----- (g(t - x)) + ----- (f(t - x)))
    dt dx                        dt dx              dt dx
/((f(t - x) + 1) (g(t - x) + 1)), 0, 0], 
       2                             2
      d                             d
[- ((----- (f(t - x))) g(t - x) + (----- (g(t - x))) f(t - x)
     dt dx                         dt dx
     2                  2
    d                  d
 + ----- (g(t - x)) + ----- (f(t - x)))/((f(t - x) + 1) (g(t - x) + 1)), 
   dt dx              dt dx
   2                           2
  d                           d
((--- (f(t - x))) g(t - x) + (--- (g(t - x))) f(t - x)
    2                           2
  dt                          dt
                                      2
    d               d                d
 - (-- (f(t - x))) (-- (g(t - x))) + --- (g(t - x))
    dx              dx                 2
                                     dt
                                      2
    d               d                d
 + (-- (f(t - x))) (-- (g(t - x))) + --- (f(t - x)))
    dt              dt                 2
                                     dt
/((f(t - x) + 1) (g(t - x) + 1)), 0, 0], 
          2                2
         d                d
         --- (g(t - x)) - --- (g(t - x))
           2                2
         dx               dt
[0, 0, - -------------------------------, 0], 
                  g(t - x) + 1
             2                2
            d                d
            --- (f(t - x)) - --- (f(t - x))
              2                2
            dx               dt
[0, 0, 0, - -------------------------------])
                     f(t - x) + 1

(%o39)                               done
(%i40) /* ricci curvature */scurvature();
             2                           2
            d                           d
(%o40) (2 ((--- (f(t - x))) g(t - x) - (--- (f(t - x))) g(t - x)
              2                           2
            dx                          dt
     2                           2                          2
    d                           d                          d
 + (--- (g(t - x))) f(t - x) - (--- (g(t - x))) f(t - x) + --- (g(t - x))
      2                           2                          2
    dx                          dt                         dx
                                      2
    d               d                d
 + (-- (f(t - x))) (-- (g(t - x))) - --- (g(t - x))
    dx              dx                 2
                                     dt
                                      2                2
    d               d                d                d
 - (-- (f(t - x))) (-- (g(t - x))) + --- (f(t - x)) - --- (f(t - x))))
    dt              dt                 2                2
                                     dx               dt
/((f(t - x) + 1) (g(t - x) + 1))
(%i41) /* Kretschmann scalar */rinvariant();
           d               d                 d               d              2
       4 ((-- (f(t - x))) (-- (g(t - x))) - (-- (f(t - x))) (-- (g(t - x))))
           dx              dx                dt              dt
(%o41) ----------------------------------------------------------------------
                                        2               2
                          (f(t - x) + 1)  (g(t - x) + 1)
       2                     2
      d              2      d              2        2
   4 (--- (g(t - x)))    4 (--- (g(t - x)))        d               2
        2                     2                8 (----- (g(t - x)))
      dx                    dt                    dt dx
 + ------------------- + ------------------- - ---------------------
                   2                     2                      2
     (g(t - x) + 1)        (g(t - x) + 1)         (g(t - x) + 1)
       2                     2
      d              2      d              2        2
   4 (--- (f(t - x)))    4 (--- (f(t - x)))        d               2
        2                     2                8 (----- (f(t - x)))
      dx                    dt                    dt dx
 + ------------------- + ------------------- - ---------------------
                   2                     2                      2
     (f(t - x) + 1)        (f(t - x) + 1)         (f(t - x) + 1)
(%i42) /* Geodesic equations */cgeodesic(true);
                d                        dz 2    d               dz 2
(%t42) geod  = (-- (g(t - x))) g(t - x) (--)  + (-- (g(t - x))) (--)
           1    dt                       ds      dt              ds
                                                                             2
                    d                        dy 2    d               dy 2   d t
                 + (-- (f(t - x))) f(t - x) (--)  + (-- (f(t - x))) (--)  + ---
                    dt                       ds      dt              ds       2
                                                                            ds

                   d                        dz 2    d               dz 2
(%t43) geod  = - ((-- (g(t - x))) g(t - x) (--)  + (-- (g(t - x))) (--)
           2       dx                       ds      dx              ds
                                                                            2
                   d                        dy 2    d               dy 2   d x
                + (-- (f(t - x))) f(t - x) (--)  + (-- (f(t - x))) (--)  - ---)
                   dx                       ds      dx              ds       2
                                                                           ds

                   d                        dz 2    d               dz 2
(%t44) geod  = - ((-- (g(t - x))) g(t - x) (--)  + (-- (g(t - x))) (--)
           3       dy                       ds      dy              ds
                                                                           2
      d                       dy dz      d              dy dz    2        d y
 - 2 (-- (f(t - x))) f(t - x) -- -- - 2 (-- (f(t - x))) -- -- - f (t - x) ---
      dz                      ds ds      dz             ds ds               2
                                                                          ds
               2     2
              d y   d y    d                        dy 2
 - 2 f(t - x) --- - --- - (-- (f(t - x))) f(t - x) (--)
                2     2    dy                       ds
              ds    ds
    d               dy 2      d                       dx dy
 - (-- (f(t - x))) (--)  - 2 (-- (f(t - x))) f(t - x) -- --
    dy              ds        dx                      ds ds
      d              dx dy     dt  d                       dy
 - 2 (-- (f(t - x))) -- -- - 2 -- (-- (f(t - x))) f(t - x) --
      dx             ds ds     ds  dt                      ds
     dt  d              dy                2
 - 2 -- (-- (f(t - x))) --)/(f(t - x) + 1)
     ds  dt             ds

                           2                2     2
                 2        d z              d z   d z
(%t45) geod  = (g (t - x) --- + 2 g(t - x) --- + ---
           4                2                2     2
                          ds               ds    ds
    d                        dz 2    d               dz 2
 + (-- (g(t - x))) g(t - x) (--)  + (-- (g(t - x))) (--)
    dz                       ds      dz              ds
      d                       dy dz      d              dy dz
 + 2 (-- (g(t - x))) g(t - x) -- -- + 2 (-- (g(t - x))) -- --
      dy                      ds ds      dy             ds ds
      d                       dx dz      d              dx dz
 + 2 (-- (g(t - x))) g(t - x) -- -- + 2 (-- (g(t - x))) -- --
      dx                      ds ds      dx             ds ds
     dt  d                       dz     dt  d              dz
 + 2 -- (-- (g(t - x))) g(t - x) -- + 2 -- (-- (g(t - x))) --
     ds  dt                      ds     ds  dt             ds
    d                        dy 2    d               dy 2                2
 - (-- (f(t - x))) f(t - x) (--)  - (-- (f(t - x))) (--) )/(g(t - x) + 1)
    dz                       ds      dz              ds

(%o45)                               done
(%i46) 
Run Example
/*  Schwarzschild simplesave as name.macLoad from the file menu - >
 File|Batch file*/kill(all);
(%o0)                                done
(%i1) load(ctensor);
(%o1)          /usr/share/maxima/5.21.1/share/tensor/ctensor.mac
(%i2) /* set some flags */cframe_flag: false;
(%o2)                                false
(%i3) ratchristof: true;
(%o3)                                true
(%i4) ratriemann : true;
(%o4)                                true
(%i5) ratfac : true;
(%o5)                                true
(%i6) ctrgsimp: true;
(%o6)                                true
(%i7) /* define the dimension */dim: 2;
(%o7)                                  2
(%i8) /* list the coordinates */ct_coords: [theta,phi];
(%o8)                            [theta, phi]
(%i9) /* set up the metric *//* assign to lg a matrix of zeros  ':' means 'assign' */lg:zeromatrix(2,2);
                                   [ 0  0 ]
(%o9)                              [      ]
                                   [ 0  0 ]
(%i10) /* now add the Schwarzschild coefficients */lg[1,1]:1;
(%o10)                                 1
(%i11) lg[2,2]:sin(theta)^2;
                                     2
(%o11)                            sin (theta)
(%i12) /* make the inverse matrix */ug:invert(lg);
                              [ 1       0      ]
                              [                ]
(%o12)                        [         1      ]
                              [ 0  ----------- ]
                              [       2        ]
                              [    sin (theta) ]
(%i13) /* get Christoffels */christof(mcs);
                                         cos(theta)
(%t13)                      mcs        = ----------
                               1, 2, 2   sin(theta)

(%t14)               mcs        = - cos(theta) sin(theta)
                        2, 2, 1

(%o14)                               done
(%i15) /* calculate and display Ricci tensor, ( 'dis' = true ) */ricci(true);
(%t15)                            ric     = 1
                                     1, 1

                                          2
(%t16)                       ric     = sin (theta)
                                2, 2

(%o16)                               done
(%i17) /* Riemann with all indexes down */lriemann(true);
                                              2
(%t17)                   lriem           = sin (theta)
                              2, 2, 1, 1

(%o17)                               done
(%i18) /* Riemann with all indexes up */uriemann(false);
(%o18)                               done
(%i19) /* calculate Einstein tensor */einstein(false);
(%o19)                               done
(%i20) /* show Einstein tensor as a matrix */cdisplay(ein);
                                      [ 0  0 ]
                                ein = [      ]
                                      [ 0  0 ]

(%o20)                               done
(%i21) /* ricci curvature */scurvature();
(%o21)                                 2
(%i22) /* Kretschmann scalar */rinvariant();
(%o22)                                 4
(%i23) /* Geodesic equations */cgeodesic(true);
                         2
                        d theta    dphi 2
(%t23)          geod  = ------- - (----)  cos(theta) sin(theta)
                    1       2       ds
                          ds

                                                  2
                        dphi            dtheta   d phi
                      2 ---- cos(theta) ------ + ----- sin(theta)
                         ds               ds        2
                                                  ds
(%t24)        geod  = -------------------------------------------
                  2                   sin(theta)

(%o24)                               done
(%i25) 
Run Example
load(ctensor);
(%o1)          /usr/share/maxima/5.21.1/share/tensor/ctensor.mac
(%i2) ct_coords:[t,r,theta,phi];
(%o2)                         [t, r, theta, phi]
(%i3) lg:matrix([-f(r),-2*H*r/g,0,0],[-2*H*r/g,1/g^2,0,0],[0,0,r^2,0],[0,0,0,r^2*sin(theta)^2]);
                   [            2 r H                     ]
                   [ - f(r)   - -----  0         0        ]
                   [              g                       ]
                   [                                      ]
                   [   2 r H    1                         ]
                   [ - -----    --     0         0        ]
(%o3)              [     g       2                        ]
                   [            g                         ]
                   [                                      ]
                   [                    2                 ]
                   [    0        0     r         0        ]
                   [                                      ]
                   [                        2    2        ]
                   [    0        0     0   r  sin (theta) ]
(%i4) cmetric();
(%o4)                                done
(%i5) lriemann(dis);
(%t5) lriem           = 
           2, 2, 1, 1
          2                                           2
      2  d                  d            2           d              d         2
  (8 r  (--- (f(r))) - 8 r (-- (f(r)))) H  + 2 f(r) (--- (f(r))) - (-- (f(r)))
           2                dr                         2            dr
         dr                                          dr
  -----------------------------------------------------------------------------
                                    2  2
                                16 r  H  + 4 f(r)

                                       2         d
                                      g  r f(r) (-- (f(r)))
                                                 dr
(%t6)               lriem           = ---------------------
                         3, 3, 1, 1        2  2
                                        8 r  H  + 2 f(r)

                                          2  d
                                       g r  (-- (f(r))) H
                                             dr
(%t7)                lriem           = ------------------
                          3, 3, 1, 2        2  2
                                         4 r  H  + f(r)

                                          2  d
                                       g r  (-- (f(r))) H
                                             dr
(%t8)                lriem           = ------------------
                          3, 3, 2, 1        2  2
                                         4 r  H  + f(r)

                                               2  2
                                            4 r  H
(%t9)                  lriem           = --------------
                            3, 3, 2, 2      2  2
                                         4 r  H  + f(r)

                                 2         d             2
                                g  r f(r) (-- (f(r))) sin (theta)
                                           dr
(%t10)        lriem           = ---------------------------------
                   4, 4, 1, 1              2  2
                                        8 r  H  + 2 f(r)

                                    2  d             2
                                 g r  (-- (f(r))) sin (theta) H
                                       dr
(%t11)         lriem           = ------------------------------
                    4, 4, 1, 2              2  2
                                         4 r  H  + f(r)

                                    2  d             2
                                 g r  (-- (f(r))) sin (theta) H
                                       dr
(%t12)         lriem           = ------------------------------
                    4, 4, 2, 1              2  2
                                         4 r  H  + f(r)

                                          2    2         2
                                       4 r  sin (theta) H
(%t13)               lriem           = -------------------
                          4, 4, 2, 2        2  2
                                         4 r  H  + f(r)

                            4    2         2         2   2         2
                         4 r  sin (theta) H  + (1 - g ) r  f(r) sin (theta)
(%t14) lriem           = --------------------------------------------------
            4, 4, 3, 3                        2  2
                                           4 r  H  + f(r)

(%o14)                               done
(%i15) uriemann(dis);
                                     2
                              4  2  d                4    d            2
(%t15) uriem           = ((8 g  r  (--- (f(r))) - 8 g  r (-- (f(r)))) H
            2, 2, 1, 1                2                   dr
                                    dr
               2
      4       d              4  d         2
 + 2 g  f(r) (--- (f(r))) - g  (-- (f(r))) )
                2               dr
              dr
       6  6        4       4       2  2     2      3
/(256 r  H  + 192 r  f(r) H  + 48 r  f (r) H  + 4 f (r))

(%t16) uriem           = 
            3, 3, 1, 1
                         2  3  4      2  2  d           2    2       d
                     32 g  r  H  + 8 g  r  (-- (f(r))) H  + g  f(r) (-- (f(r)))
                                            dr                       dr
                     ----------------------------------------------------------
                           9  6       7       4       5  2     2      3  3
                      128 r  H  + 96 r  f(r) H  + 24 r  f (r) H  + 2 r  f (r)

                                     3    d               3        3
                                 (4 g  r (-- (f(r))) - 8 g  f(r)) H
                                          dr
(%t17) uriem           = ---------------------------------------------------
            3, 3, 1, 2       7  6       5       4       3  2     2      3
                         64 r  H  + 48 r  f(r) H  + 12 r  f (r) H  + r f (r)

                                     3    d               3        3
                                 (4 g  r (-- (f(r))) - 8 g  f(r)) H
                                          dr
(%t18) uriem           = ---------------------------------------------------
            3, 3, 2, 1       7  6       5       4       3  2     2      3
                         64 r  H  + 48 r  f(r) H  + 12 r  f (r) H  + r f (r)

                                    4         d               4  2      2
                                (2 g  r f(r) (-- (f(r))) - 4 g  f (r)) H
                                              dr
(%t19) uriem           = - ----------------------------------------------------
            3, 3, 2, 2         8  6       6       4       4  2     2    2  3
                           64 r  H  + 48 r  f(r) H  + 12 r  f (r) H  + r  f (r)

                              2  3  4      2  2  d           2
(%t20) uriem           = (32 g  r  H  + 8 g  r  (-- (f(r))) H
            4, 4, 1, 1                           dr
    2       d                 9    2         6       7         2         4
 + g  f(r) (-- (f(r))))/(128 r  sin (theta) H  + 96 r  f(r) sin (theta) H
            dr
       5  2       2         2      3  3       2
 + 24 r  f (r) sin (theta) H  + 2 r  f (r) sin (theta))

                              3    d               3        3
(%t21) uriem           = ((4 g  r (-- (f(r))) - 8 g  f(r)) H )
            4, 4, 1, 2             dr
      7    2         6       5         2         4       3  2       2         2
/(64 r  sin (theta) H  + 48 r  f(r) sin (theta) H  + 12 r  f (r) sin (theta) H
      3       2
 + r f (r) sin (theta))

                              3    d               3        3
(%t22) uriem           = ((4 g  r (-- (f(r))) - 8 g  f(r)) H )
            4, 4, 2, 1             dr
      7    2         6       5         2         4       3  2       2         2
/(64 r  sin (theta) H  + 48 r  f(r) sin (theta) H  + 12 r  f (r) sin (theta) H
      3       2
 + r f (r) sin (theta))

                                4         d               4  2      2
(%t23) uriem           = - ((2 g  r f(r) (-- (f(r))) - 4 g  f (r)) H )
            4, 4, 2, 2                    dr
      8    2         6       6         2         4       4  2       2         2
/(64 r  sin (theta) H  + 48 r  f(r) sin (theta) H  + 12 r  f (r) sin (theta) H
    2  3       2
 + r  f (r) sin (theta))

                                        2  2         2
                                     4 r  H  + (1 - g ) f(r)
(%t24)    uriem           = -----------------------------------------
               4, 4, 3, 3      8    2         2    6         2
                            4 r  sin (theta) H  + r  f(r) sin (theta)

(%o24)                               done
(%i25) rinvariant();
           2         d             2
(%o25) (4 g  r f(r) (-- (f(r))) sin (theta)
                     dr
      2  3  4      2  2  d           2    2       d
 (32 g  r  H  + 8 g  r  (-- (f(r))) H  + g  f(r) (-- (f(r)))))
                         dr                       dr
      2  2                 9    2         6       7         2         4       5  2       2         2      3  3       2
/((8 r  H  + 2 f(r)) (128 r  sin (theta) H  + 96 r  f(r) sin (theta) H  + 24 r  f (r) sin (theta) H  + 2 r  f (r) sin (theta)))
        2     4         d               4  2        2         4
 - (16 r  (2 g  r f(r) (-- (f(r))) - 4 g  f (r)) sin (theta) H )
                        dr
      2  2              8    2         6       6         2         4       4  2       2         2    2  3
/((4 r  H  + f(r)) (64 r  sin (theta) H  + 48 r  f(r) sin (theta) H  + 12 r  f (r) sin (theta) H  + r  f (r)
    2
 sin (theta)))
         2  d              3    d               3          2         4
 + (8 g r  (-- (f(r))) (4 g  r (-- (f(r))) - 8 g  f(r)) sin (theta) H )
            dr                  dr
      2  2              7    2         6       5         2         4       3  2       2         2      3
/((4 r  H  + f(r)) (64 r  sin (theta) H  + 48 r  f(r) sin (theta) H  + 12 r  f (r) sin (theta) H  + r f (r)
    2
 sin (theta)))
       2         d               2  3  4      2  2  d           2
 + (4 g  r f(r) (-- (f(r))) (32 g  r  H  + 8 g  r  (-- (f(r))) H
                 dr                                 dr
    2       d                 2  2                 9  6       7       4       5  2     2      3
 + g  f(r) (-- (f(r)))))/((8 r  H  + 2 f(r)) (128 r  H  + 96 r  f(r) H  + 24 r  f (r) H  + 2 r
            dr
  3
 f (r)))
                   2     4         d               4  2      4
               16 r  (2 g  r f(r) (-- (f(r))) - 4 g  f (r)) H
                                   dr
 - -----------------------------------------------------------------------
       2  2              8  6       6       4       4  2     2    2  3
   (4 r  H  + f(r)) (64 r  H  + 48 r  f(r) H  + 12 r  f (r) H  + r  f (r))
                2  d              3    d               3        4
           8 g r  (-- (f(r))) (4 g  r (-- (f(r))) - 8 g  f(r)) H
                   dr                  dr
 + ----------------------------------------------------------------------
       2  2              7  6       5       4       3  2     2      3
   (4 r  H  + f(r)) (64 r  H  + 48 r  f(r) H  + 12 r  f (r) H  + r f (r))
               2                                           2
           2  d                  d            2           d
 + (4 ((8 r  (--- (f(r))) - 8 r (-- (f(r)))) H  + 2 f(r) (--- (f(r)))
                2                dr                         2
              dr                                          dr
                             2
    d         2       4  2  d                4    d            2
 - (-- (f(r))) ) ((8 g  r  (--- (f(r))) - 8 g  r (-- (f(r)))) H
    dr                        2                   dr
                            dr
               2
      4       d              4  d         2
 + 2 g  f(r) (--- (f(r))) - g  (-- (f(r))) ))
                2               dr
              dr
       2  2                 6  6        4       4       2  2     2      3
/((16 r  H  + 4 f(r)) (256 r  H  + 192 r  f(r) H  + 48 r  f (r) H  + 4 f (r)))
          2  2         2            4    2         2
 + (4 (4 r  H  + (1 - g ) f(r)) (4 r  sin (theta) H
         2   2         2               2  2             8    2         2    6
 + (1 - g ) r  f(r) sin (theta)))/((4 r  H  + f(r)) (4 r  sin (theta) H  + r  f(r)
    2
 sin (theta)))
(%i26) 

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