? rinvariant;

Calculate

? rinvariant;

Calculate

### rinvariant

Run Example
```(%i1)/*  Schwarzschild simplesave as name.macLoad from the file menu - >
File|Batch file*/kill(all);
(%o0)                                done
(%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
(%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) ```

### Related Help

Help for Rinvariant