reduce-historical/r37/log/avector.rlg

Sun Aug 18 16:26:50 2002 run on Windows

*** ^ redefined 
% Vector test routine

% Author: David Harper (algebra@liverpool.ac.uk)
%         Computer Algebra Support Officer
%         University of Liverpool Computer Laboratory.

% Please compare carefully the output from running this test file with the
% log file provided to make sure your implementation is correct.

linelength 72;


80

off allfac;

 on div;


vec a,b,c;


matrix q;


a := avec(ax,ay,az);


vec(x) := ax

vec(y) := ay

vec(z) := az


b := avec(bx,by,bz);


vec(x) := bx

vec(y) := by

vec(z) := bz


q := mat((q11,q12,q13),(q21,q22,q23),(q31,q32,q33));


     [q11  q12  q13]
     [             ]
q := [q21  q22  q23]
     [             ]
     [q31  q32  q33]



c := a+b;


vec(x) := ax + bx

vec(y) := ay + by

vec(z) := az + bz


c := a-b;


vec(x) := ax - bx

vec(y) := ay - by

vec(z) := az - bz


c := a cross b;


vec(x) := ay*bz - az*by

vec(y) :=  - ax*bz + az*bx

vec(z) := ax*by - ay*bx


d := a dot b;


d := ax*bx + ay*by + az*bz

a dot c;


0
 b dot c;


0

q*a;


vec(x) := ax*q11 + ay*q12 + az*q13

vec(y) := ax*q21 + ay*q22 + az*q23

vec(z) := ax*q31 + ay*q32 + az*q33


c:=2*f*a - b/7;


                    1
vec(x) := 2*ax*f - ---*bx
                    7

                    1
vec(y) := 2*ay*f - ---*by
                    7

                    1
vec(z) := 2*az*f - ---*bz
                    7


c(0);


          1
2*ax*f - ---*bx
          7
 c(1);


          1
2*ay*f - ---*by
          7
 c(2);


          1
2*az*f - ---*bz
          7


1/vmod(a);


   2     2     2  - 1/2
(ax  + ay  + az )

b/vmod(a);


             2     2     2  - 1/2
vec(x) := (ax  + ay  + az )      *bx

             2     2     2  - 1/2
vec(y) := (ax  + ay  + az )      *by

             2     2     2  - 1/2
vec(z) := (ax  + ay  + az )      *bz


(a cross b)/(a dot b);


               ay*bz - az*by
vec(x) := -----------------------
           ax*bx + ay*by + az*bz

              - ax*bz + az*bx
vec(y) := -----------------------
           ax*bx + ay*by + az*bz

               ax*by - ay*bx
vec(z) := -----------------------
           ax*bx + ay*by + az*bz


2/3*vmod(a)*a*(a dot c)/(vmod(a cross c));


           28     2   2     2   2
vec(x) := ----*(ax *by  + ax *bz  - 2*ax*ay*bx*by - 2*ax*az*bx*bz
           3

                     2   2     2   2                     2   2     2   2
                 + ay *bx  + ay *bz  - 2*ay*az*by*bz + az *bx  + az *by

                     - 1         2     2     2    3      2     2   2
                )**------*sqrt(ax  + ay  + az )*ax *f - ---*(ax *by
                     2                                   3

                  2   2                                     2   2
              + ax *bz  - 2*ax*ay*bx*by - 2*ax*az*bx*bz + ay *bx

                  2   2                     2   2     2   2     - 1
              + ay *bz  - 2*ay*az*by*bz + az *bx  + az *by )**------
                                                                2

                  2     2     2    2       28     2   2     2   2
          *sqrt(ax  + ay  + az )*ax *bx + ----*(ax *by  + ax *bz
                                           3

                                                  2   2     2   2
              - 2*ax*ay*bx*by - 2*ax*az*bx*bz + ay *bx  + ay *bz

                                  2   2     2   2     - 1
              - 2*ay*az*by*bz + az *bx  + az *by )**------
                                                      2

                  2     2     2       2      2     2   2     2   2
          *sqrt(ax  + ay  + az )*ax*ay *f - ---*(ax *by  + ax *bz
                                             3

                                                  2   2     2   2
              - 2*ax*ay*bx*by - 2*ax*az*bx*bz + ay *bx  + ay *bz

                                  2   2     2   2     - 1
              - 2*ay*az*by*bz + az *bx  + az *by )**------
                                                      2

                  2     2     2              28     2   2     2   2
          *sqrt(ax  + ay  + az )*ax*ay*by + ----*(ax *by  + ax *bz
                                             3

                                                  2   2     2   2
              - 2*ax*ay*bx*by - 2*ax*az*bx*bz + ay *bx  + ay *bz

                                  2   2     2   2     - 1
              - 2*ay*az*by*bz + az *bx  + az *by )**------
                                                      2

                  2     2     2       2      2     2   2     2   2
          *sqrt(ax  + ay  + az )*ax*az *f - ---*(ax *by  + ax *bz
                                             3

                                                  2   2     2   2
              - 2*ax*ay*bx*by - 2*ax*az*bx*bz + ay *bx  + ay *bz

                                  2   2     2   2     - 1
              - 2*ay*az*by*bz + az *bx  + az *by )**------
                                                      2

                  2     2     2
          *sqrt(ax  + ay  + az )*ax*az*bz

           28     2   2     2   2
vec(y) := ----*(ax *by  + ax *bz  - 2*ax*ay*bx*by - 2*ax*az*bx*bz
           3

                     2   2     2   2                     2   2     2   2
                 + ay *bx  + ay *bz  - 2*ay*az*by*bz + az *bx  + az *by

                     - 1         2     2     2    2         2     2   2
                )**------*sqrt(ax  + ay  + az )*ax *ay*f - ---*(ax *by
                     2                                      3

                  2   2                                     2   2
              + ax *bz  - 2*ax*ay*bx*by - 2*ax*az*bx*bz + ay *bx

                  2   2                     2   2     2   2     - 1
              + ay *bz  - 2*ay*az*by*bz + az *bx  + az *by )**------
                                                                2

                  2     2     2              28     2   2     2   2
          *sqrt(ax  + ay  + az )*ax*ay*bx + ----*(ax *by  + ax *bz
                                             3

                                                  2   2     2   2
              - 2*ax*ay*bx*by - 2*ax*az*bx*bz + ay *bx  + ay *bz

                                  2   2     2   2     - 1
              - 2*ay*az*by*bz + az *bx  + az *by )**------
                                                      2

                  2     2     2    3      2     2   2     2   2
          *sqrt(ax  + ay  + az )*ay *f - ---*(ax *by  + ax *bz
                                          3

                                                  2   2     2   2
              - 2*ax*ay*bx*by - 2*ax*az*bx*bz + ay *bx  + ay *bz

                                  2   2     2   2     - 1
              - 2*ay*az*by*bz + az *bx  + az *by )**------
                                                      2

                  2     2     2    2       28     2   2     2   2
          *sqrt(ax  + ay  + az )*ay *by + ----*(ax *by  + ax *bz
                                           3

                                                  2   2     2   2
              - 2*ax*ay*bx*by - 2*ax*az*bx*bz + ay *bx  + ay *bz

                                  2   2     2   2     - 1
              - 2*ay*az*by*bz + az *bx  + az *by )**------
                                                      2

                  2     2     2       2      2     2   2     2   2
          *sqrt(ax  + ay  + az )*ay*az *f - ---*(ax *by  + ax *bz
                                             3

                                                  2   2     2   2
              - 2*ax*ay*bx*by - 2*ax*az*bx*bz + ay *bx  + ay *bz

                                  2   2     2   2     - 1
              - 2*ay*az*by*bz + az *bx  + az *by )**------
                                                      2

                  2     2     2
          *sqrt(ax  + ay  + az )*ay*az*bz

           28     2   2     2   2
vec(z) := ----*(ax *by  + ax *bz  - 2*ax*ay*bx*by - 2*ax*az*bx*bz
           3

                     2   2     2   2                     2   2     2   2
                 + ay *bx  + ay *bz  - 2*ay*az*by*bz + az *bx  + az *by

                     - 1         2     2     2    2         2     2   2
                )**------*sqrt(ax  + ay  + az )*ax *az*f - ---*(ax *by
                     2                                      3

                  2   2                                     2   2
              + ax *bz  - 2*ax*ay*bx*by - 2*ax*az*bx*bz + ay *bx

                  2   2                     2   2     2   2     - 1
              + ay *bz  - 2*ay*az*by*bz + az *bx  + az *by )**------
                                                                2

                  2     2     2              28     2   2     2   2
          *sqrt(ax  + ay  + az )*ax*az*bx + ----*(ax *by  + ax *bz
                                             3

                                                  2   2     2   2
              - 2*ax*ay*bx*by - 2*ax*az*bx*bz + ay *bx  + ay *bz

                                  2   2     2   2     - 1
              - 2*ay*az*by*bz + az *bx  + az *by )**------
                                                      2

                  2     2     2    2         2     2   2     2   2
          *sqrt(ax  + ay  + az )*ay *az*f - ---*(ax *by  + ax *bz
                                             3

                                                  2   2     2   2
              - 2*ax*ay*bx*by - 2*ax*az*bx*bz + ay *bx  + ay *bz

                                  2   2     2   2     - 1
              - 2*ay*az*by*bz + az *bx  + az *by )**------
                                                      2

                  2     2     2              28     2   2     2   2
          *sqrt(ax  + ay  + az )*ay*az*by + ----*(ax *by  + ax *bz
                                             3

                                                  2   2     2   2
              - 2*ax*ay*bx*by - 2*ax*az*bx*bz + ay *bx  + ay *bz

                                  2   2     2   2     - 1
              - 2*ay*az*by*bz + az *bx  + az *by )**------
                                                      2

                  2     2     2    3      2     2   2     2   2
          *sqrt(ax  + ay  + az )*az *f - ---*(ax *by  + ax *bz
                                          3

                                                  2   2     2   2
              - 2*ax*ay*bx*by - 2*ax*az*bx*bz + ay *bx  + ay *bz

                                  2   2     2   2     - 1
              - 2*ay*az*by*bz + az *bx  + az *by )**------
                                                      2

                  2     2     2    2
          *sqrt(ax  + ay  + az )*az *bz



a := avec(x**2*y**3,log(z+x),13*z-y);


           2  3
vec(x) := x *y

vec(y) := log(x + z)

vec(z) :=  - y + 13*z


df(a,x);


               3
vec(x) := 2*x*y

             1
vec(y) := -------
           x + z

vec(z) := 0


df(a,x,y);


               2
vec(x) := 6*x*y

vec(y) := 0

vec(z) := 0


int(a,x);


           1   3  3
vec(x) := ---*x *y
           3

vec(y) := log(x + z)*x + log(x + z)*z - x

vec(z) :=  - x*y + 13*x*z


exp(a);


            2  3
           x *y
vec(x) := e

vec(y) := x + z

            - y + 13*z
vec(z) := e


log sin b;


vec(x) := log(sin(bx))

vec(y) := log(sin(by))

vec(z) := log(sin(bz))



a := avec(ax,ay,az);


vec(x) := ax

vec(y) := ay

vec(z) := az


depend ax,x,y,z;

 depend ay,x,y,z;

 depend az,x,y,z;


depend p,x,y,z;


c := grad p;


vec(x) := df(p,x)

vec(y) := df(p,y)

vec(z) := df(p,z)


div c;


df(p,x,2) + df(p,y,2) + df(p,z,2)

delsq p;


df(p,x,2) + df(p,y,2) + df(p,z,2)

div a;


df(ax,x) + df(ay,y) + df(az,z)

curl a;


vec(x) :=  - df(ay,z) + df(az,y)

vec(y) := df(ax,z) - df(az,x)

vec(z) :=  - df(ax,y) + df(ay,x)


delsq a;


vec(x) := df(ax,x,2) + df(ax,y,2) + df(ax,z,2)

vec(y) := df(ay,x,2) + df(ay,y,2) + df(ay,z,2)

vec(z) := df(az,x,2) + df(az,y,2) + df(az,z,2)



depend h1,x,y,z;

 depend h2,x,y,z;

 depend h3,x,y,z;


scalefactors(h1,h2,h3);


grad p;


                    -1
vec(x) := df(p,x)*h1

                    -1
vec(y) := df(p,y)*h2

                    -1
vec(z) := df(p,z)*h3


div a;


           -1              -1              -1                 -1   -1
df(ax,x)*h1   + df(ay,y)*h2   + df(az,z)*h3   + df(h1,y)*ay*h1  *h2

                 -1   -1                 -1   -1                 -1   -1
 + df(h1,z)*az*h1  *h3   + df(h2,x)*ax*h1  *h2   + df(h2,z)*az*h2  *h3

                 -1   -1                 -1   -1
 + df(h3,x)*ax*h1  *h3   + df(h3,y)*ay*h2  *h3

curl a;


                        -1              -1                 -1   -1
vec(x) :=  - df(ay,z)*h3   + df(az,y)*h2   - df(h2,z)*ay*h2  *h3

                           -1   -1
           + df(h3,y)*az*h2  *h3

                     -1              -1                 -1   -1
vec(y) := df(ax,z)*h3   - df(az,x)*h1   + df(h1,z)*ax*h1  *h3

                           -1   -1
           - df(h3,x)*az*h1  *h3

                        -1              -1                 -1   -1
vec(z) :=  - df(ax,y)*h2   + df(ay,x)*h1   - df(h1,y)*ax*h1  *h2

                           -1   -1
           + df(h2,x)*ay*h1  *h2


dp1 := delsq p;


                             -3                      -1   -2
dp1 :=  - df(h1,x)*df(p,x)*h1   + df(h1,y)*df(p,y)*h1  *h2

                             -1   -2                      -2   -1
        + df(h1,z)*df(p,z)*h1  *h3   + df(h2,x)*df(p,x)*h1  *h2

                             -3                      -1   -2
        - df(h2,y)*df(p,y)*h2   + df(h2,z)*df(p,z)*h2  *h3

                             -2   -1                      -2   -1
        + df(h3,x)*df(p,x)*h1  *h3   + df(h3,y)*df(p,y)*h2  *h3

                             -3               -2               -2
        - df(h3,z)*df(p,z)*h3   + df(p,x,2)*h1   + df(p,y,2)*h2

                      -2
        + df(p,z,2)*h3

dp2 := div grad p;


                             -3                      -1   -2
dp2 :=  - df(h1,x)*df(p,x)*h1   + df(h1,y)*df(p,y)*h1  *h2

                             -1   -2                      -2   -1
        + df(h1,z)*df(p,z)*h1  *h3   + df(h2,x)*df(p,x)*h1  *h2

                             -3                      -1   -2
        - df(h2,y)*df(p,y)*h2   + df(h2,z)*df(p,z)*h2  *h3

                             -2   -1                      -2   -1
        + df(h3,x)*df(p,x)*h1  *h3   + df(h3,y)*df(p,y)*h2  *h3

                             -3               -2               -2
        - df(h3,z)*df(p,z)*h3   + df(p,x,2)*h1   + df(p,y,2)*h2

                      -2
        + df(p,z,2)*h3

dp1-dp2;


0

delsq a;


                       -2                       -3
vec(x) := df(ax,x,2)*h1   - df(ax,x)*df(h1,x)*h1

                                 -2   -1                       -2   -1
           + df(ax,x)*df(h2,x)*h1  *h2   + df(ax,x)*df(h3,x)*h1  *h3

                          -2                       -1   -2
           + df(ax,y,2)*h2   + df(ax,y)*df(h1,y)*h1  *h2

                                 -3                       -2   -1
           - df(ax,y)*df(h2,y)*h2   + df(ax,y)*df(h3,y)*h2  *h3

                          -2                       -1   -2
           + df(ax,z,2)*h3   + df(ax,z)*df(h1,z)*h1  *h3

                                 -1   -2                       -3
           + df(ax,z)*df(h2,z)*h2  *h3   - df(ax,z)*df(h3,z)*h3

                                   -2   -1
           + 2*df(ay,x)*df(h1,y)*h1  *h2

                                   -1   -2
           - 2*df(ay,y)*df(h2,x)*h1  *h2

                                   -2   -1
           + 2*df(az,x)*df(h1,z)*h1  *h3

                                   -1   -2                   -2   -1
           - 2*df(az,z)*df(h3,x)*h1  *h3   + df(h1,x,y)*ay*h1  *h2

                             -2   -1                          -3   -1
           + df(h1,x,z)*az*h1  *h3   - df(h1,x)*df(h1,y)*ay*h1  *h2

                                    -3   -1
           - df(h1,x)*df(h1,z)*az*h1  *h3

                                    -3   -1
           - df(h1,x)*df(h2,x)*ax*h1  *h2

                                    -3   -1                   -1   -2
           - df(h1,x)*df(h3,x)*ax*h1  *h3   + df(h1,y,2)*ax*h1  *h2

                     2      -2   -2                          -1   -3
           - df(h1,y) *ax*h1  *h2   - df(h1,y)*df(h2,y)*ax*h1  *h2

                                    -1   -2   -1
           + df(h1,y)*df(h3,y)*ax*h1  *h2  *h3

                             -1   -2           2      -2   -2
           + df(h1,z,2)*ax*h1  *h3   - df(h1,z) *ax*h1  *h3

                                    -1   -1   -2
           + df(h1,z)*df(h2,z)*ax*h1  *h2  *h3

                                    -1   -3                   -1   -2
           - df(h1,z)*df(h3,z)*ax*h1  *h3   - df(h2,x,y)*ay*h1  *h2

                             -1   -1   -1                   -2   -1
           + df(h2,x,z)*az*h1  *h2  *h3   + df(h2,x,2)*ax*h1  *h2

                     2      -2   -2                          -1   -3
           - df(h2,x) *ax*h1  *h2   + df(h2,x)*df(h2,y)*ay*h1  *h2

                                    -1   -2   -1
           - df(h2,x)*df(h2,z)*az*h1  *h2  *h3

                                      -1   -2   -1
           - 2*df(h2,x)*df(h3,y)*ay*h1  *h2  *h3

                                      -1   -1   -2
           - 2*df(h2,z)*df(h3,x)*az*h1  *h2  *h3

                             -1   -1   -1                   -1   -2
           + df(h3,x,y)*ay*h1  *h2  *h3   - df(h3,x,z)*az*h1  *h3

                             -2   -1           2      -2   -2
           + df(h3,x,2)*ax*h1  *h3   - df(h3,x) *ax*h1  *h3

                                    -1   -1   -2
           - df(h3,x)*df(h3,y)*ay*h1  *h2  *h3

                                    -1   -3
           + df(h3,x)*df(h3,z)*az*h1  *h3

                                   -2   -1
vec(y) :=  - 2*df(ax,x)*df(h1,y)*h1  *h2

                                   -1   -2                -2
           + 2*df(ax,y)*df(h2,x)*h1  *h2   + df(ay,x,2)*h1

                                 -3                       -2   -1
           - df(ay,x)*df(h1,x)*h1   + df(ay,x)*df(h2,x)*h1  *h2

                                 -2   -1                -2
           + df(ay,x)*df(h3,x)*h1  *h3   + df(ay,y,2)*h2

                                 -1   -2                       -3
           + df(ay,y)*df(h1,y)*h1  *h2   - df(ay,y)*df(h2,y)*h2

                                 -2   -1                -2
           + df(ay,y)*df(h3,y)*h2  *h3   + df(ay,z,2)*h3

                                 -1   -2                       -1   -2
           + df(ay,z)*df(h1,z)*h1  *h3   + df(ay,z)*df(h2,z)*h2  *h3

                                 -3                         -2   -1
           - df(ay,z)*df(h3,z)*h3   + 2*df(az,y)*df(h2,z)*h2  *h3

                                   -1   -2                   -2   -1
           - 2*df(az,z)*df(h3,y)*h2  *h3   - df(h1,x,y)*ax*h1  *h2

                                    -3   -1
           + df(h1,x)*df(h1,y)*ax*h1  *h2

                                    -3   -1
           - df(h1,x)*df(h2,x)*ay*h1  *h2

                             -1   -1   -1                   -1   -2
           + df(h1,y,z)*az*h1  *h2  *h3   + df(h1,y,2)*ay*h1  *h2

                     2      -2   -2
           - df(h1,y) *ay*h1  *h2

                                    -2   -1   -1
           - df(h1,y)*df(h1,z)*az*h1  *h2  *h3

                                    -1   -3
           - df(h1,y)*df(h2,y)*ay*h1  *h2

                                      -2   -1   -1
           - 2*df(h1,y)*df(h3,x)*ax*h1  *h2  *h3

                                    -1   -1   -2
           + df(h1,z)*df(h2,z)*ay*h1  *h2  *h3

                                      -1   -1   -2
           - 2*df(h1,z)*df(h3,y)*az*h1  *h2  *h3

                             -1   -2                   -2   -1
           + df(h2,x,y)*ax*h1  *h2   + df(h2,x,2)*ay*h1  *h2

                     2      -2   -2                          -1   -3
           - df(h2,x) *ay*h1  *h2   - df(h2,x)*df(h2,y)*ax*h1  *h2

                                    -2   -1   -1
           + df(h2,x)*df(h3,x)*ay*h1  *h2  *h3

                             -2   -1                          -3   -1
           + df(h2,y,z)*az*h2  *h3   - df(h2,y)*df(h2,z)*az*h2  *h3

                                    -3   -1                   -1   -2
           - df(h2,y)*df(h3,y)*ay*h2  *h3   + df(h2,z,2)*ay*h2  *h3

                     2      -2   -2                          -1   -3
           - df(h2,z) *ay*h2  *h3   - df(h2,z)*df(h3,z)*ay*h2  *h3

                             -1   -1   -1
           + df(h3,x,y)*ax*h1  *h2  *h3

                                    -1   -1   -2
           - df(h3,x)*df(h3,y)*ax*h1  *h2  *h3

                             -1   -2                   -2   -1
           - df(h3,y,z)*az*h2  *h3   + df(h3,y,2)*ay*h2  *h3

                     2      -2   -2                          -1   -3
           - df(h3,y) *ay*h2  *h3   + df(h3,y)*df(h3,z)*az*h2  *h3

                                   -2   -1
vec(z) :=  - 2*df(ax,x)*df(h1,z)*h1  *h3

                                   -1   -2
           + 2*df(ax,z)*df(h3,x)*h1  *h3

                                   -2   -1
           - 2*df(ay,y)*df(h2,z)*h2  *h3

                                   -1   -2                -2
           + 2*df(ay,z)*df(h3,y)*h2  *h3   + df(az,x,2)*h1

                                 -3                       -2   -1
           - df(az,x)*df(h1,x)*h1   + df(az,x)*df(h2,x)*h1  *h2

                                 -2   -1                -2
           + df(az,x)*df(h3,x)*h1  *h3   + df(az,y,2)*h2

                                 -1   -2                       -3
           + df(az,y)*df(h1,y)*h1  *h2   - df(az,y)*df(h2,y)*h2

                                 -2   -1                -2
           + df(az,y)*df(h3,y)*h2  *h3   + df(az,z,2)*h3

                                 -1   -2                       -1   -2
           + df(az,z)*df(h1,z)*h1  *h3   + df(az,z)*df(h2,z)*h2  *h3

                                 -3                   -2   -1
           - df(az,z)*df(h3,z)*h3   - df(h1,x,z)*ax*h1  *h3

                                    -3   -1
           + df(h1,x)*df(h1,z)*ax*h1  *h3

                                    -3   -1
           - df(h1,x)*df(h3,x)*az*h1  *h3

                             -1   -1   -1
           + df(h1,y,z)*ay*h1  *h2  *h3

                                    -2   -1   -1
           - df(h1,y)*df(h1,z)*ay*h1  *h2  *h3

                                      -1   -2   -1
           - 2*df(h1,y)*df(h2,z)*ay*h1  *h2  *h3

                                    -1   -2   -1
           + df(h1,y)*df(h3,y)*az*h1  *h2  *h3

                             -1   -2           2      -2   -2
           + df(h1,z,2)*az*h1  *h3   - df(h1,z) *az*h1  *h3

                                      -2   -1   -1
           - 2*df(h1,z)*df(h2,x)*ax*h1  *h2  *h3

                                    -1   -3
           - df(h1,z)*df(h3,z)*az*h1  *h3

                             -1   -1   -1
           + df(h2,x,z)*ax*h1  *h2  *h3

                                    -1   -2   -1
           - df(h2,x)*df(h2,z)*ax*h1  *h2  *h3

                                    -2   -1   -1
           + df(h2,x)*df(h3,x)*az*h1  *h2  *h3

                             -2   -1                          -3   -1
           - df(h2,y,z)*ay*h2  *h3   + df(h2,y)*df(h2,z)*ay*h2  *h3

                                    -3   -1                   -1   -2
           - df(h2,y)*df(h3,y)*az*h2  *h3   + df(h2,z,2)*az*h2  *h3

                     2      -2   -2                          -1   -3
           - df(h2,z) *az*h2  *h3   - df(h2,z)*df(h3,z)*az*h2  *h3

                             -1   -2                   -2   -1
           + df(h3,x,z)*ax*h1  *h3   + df(h3,x,2)*az*h1  *h3

                     2      -2   -2                          -1   -3
           - df(h3,x) *az*h1  *h3   - df(h3,x)*df(h3,z)*ax*h1  *h3

                             -1   -2                   -2   -1
           + df(h3,y,z)*ay*h2  *h3   + df(h3,y,2)*az*h2  *h3

                     2      -2   -2                          -1   -3
           - df(h3,y) *az*h2  *h3   - df(h3,y)*df(h3,z)*ay*h2  *h3


curl grad p;


vec(x) := 0

vec(y) := 0

vec(z) := 0


grad div a;


                       -2                       -3
vec(x) := df(ax,x,2)*h1   - df(ax,x)*df(h1,x)*h1

                                 -2   -1                       -2   -1
           + df(ax,x)*df(h2,x)*h1  *h2   + df(ax,x)*df(h3,x)*h1  *h3

                          -1   -1                       -2   -1
           + df(ay,x,y)*h1  *h2   + df(ay,x)*df(h1,y)*h1  *h2

                                 -1   -1   -1
           + df(ay,x)*df(h3,y)*h1  *h2  *h3

                                 -1   -2                -1   -1
           - df(ay,y)*df(h2,x)*h1  *h2   + df(az,x,z)*h1  *h3

                                 -2   -1
           + df(az,x)*df(h1,z)*h1  *h3

                                 -1   -1   -1
           + df(az,x)*df(h2,z)*h1  *h2  *h3

                                 -1   -2                   -2   -1
           - df(az,z)*df(h3,x)*h1  *h3   + df(h1,x,y)*ay*h1  *h2

                             -2   -1                          -3   -1
           + df(h1,x,z)*az*h1  *h3   - df(h1,x)*df(h1,y)*ay*h1  *h2

                                    -3   -1
           - df(h1,x)*df(h1,z)*az*h1  *h3

                                    -3   -1
           - df(h1,x)*df(h2,x)*ax*h1  *h2

                                    -3   -1
           - df(h1,x)*df(h3,x)*ax*h1  *h3

                                    -2   -2
           - df(h1,y)*df(h2,x)*ay*h1  *h2

                                    -2   -2
           - df(h1,z)*df(h3,x)*az*h1  *h3

                             -1   -1   -1                   -2   -1
           + df(h2,x,z)*az*h1  *h2  *h3   + df(h2,x,2)*ax*h1  *h2

                     2      -2   -2
           - df(h2,x) *ax*h1  *h2

                                    -1   -2   -1
           - df(h2,x)*df(h2,z)*az*h1  *h2  *h3

                                    -1   -2   -1
           - df(h2,x)*df(h3,y)*ay*h1  *h2  *h3

                                    -1   -1   -2
           - df(h2,z)*df(h3,x)*az*h1  *h2  *h3

                             -1   -1   -1                   -2   -1
           + df(h3,x,y)*ay*h1  *h2  *h3   + df(h3,x,2)*ax*h1  *h3

                     2      -2   -2
           - df(h3,x) *ax*h1  *h3

                                    -1   -1   -2
           - df(h3,x)*df(h3,y)*ay*h1  *h2  *h3

                       -1   -1                       -2   -1
vec(y) := df(ax,x,y)*h1  *h2   - df(ax,x)*df(h1,y)*h1  *h2

                                 -1   -2
           + df(ax,y)*df(h2,x)*h1  *h2

                                 -1   -1   -1                -2
           + df(ax,y)*df(h3,x)*h1  *h2  *h3   + df(ay,y,2)*h2

                                 -1   -2                       -3
           + df(ay,y)*df(h1,y)*h1  *h2   - df(ay,y)*df(h2,y)*h2

                                 -2   -1                -1   -1
           + df(ay,y)*df(h3,y)*h2  *h3   + df(az,y,z)*h2  *h3

                                 -1   -1   -1
           + df(az,y)*df(h1,z)*h1  *h2  *h3

                                 -2   -1                       -1   -2
           + df(az,y)*df(h2,z)*h2  *h3   - df(az,z)*df(h3,y)*h2  *h3

                             -1   -1   -1                   -1   -2
           + df(h1,y,z)*az*h1  *h2  *h3   + df(h1,y,2)*ay*h1  *h2

                     2      -2   -2
           - df(h1,y) *ay*h1  *h2

                                    -2   -1   -1
           - df(h1,y)*df(h1,z)*az*h1  *h2  *h3

                                    -2   -2
           - df(h1,y)*df(h2,x)*ax*h1  *h2

                                    -1   -3
           - df(h1,y)*df(h2,y)*ay*h1  *h2

                                    -2   -1   -1
           - df(h1,y)*df(h3,x)*ax*h1  *h2  *h3

                                    -1   -1   -2
           - df(h1,z)*df(h3,y)*az*h1  *h2  *h3

                             -1   -2                          -1   -3
           + df(h2,x,y)*ax*h1  *h2   - df(h2,x)*df(h2,y)*ax*h1  *h2

                             -2   -1                          -3   -1
           + df(h2,y,z)*az*h2  *h3   - df(h2,y)*df(h2,z)*az*h2  *h3

                                    -3   -1
           - df(h2,y)*df(h3,y)*ay*h2  *h3

                                    -2   -2
           - df(h2,z)*df(h3,y)*az*h2  *h3

                             -1   -1   -1
           + df(h3,x,y)*ax*h1  *h2  *h3

                                    -1   -1   -2
           - df(h3,x)*df(h3,y)*ax*h1  *h2  *h3

                             -2   -1           2      -2   -2
           + df(h3,y,2)*ay*h2  *h3   - df(h3,y) *ay*h2  *h3

                       -1   -1                       -2   -1
vec(z) := df(ax,x,z)*h1  *h3   - df(ax,x)*df(h1,z)*h1  *h3

                                 -1   -1   -1
           + df(ax,z)*df(h2,x)*h1  *h2  *h3

                                 -1   -2                -1   -1
           + df(ax,z)*df(h3,x)*h1  *h3   + df(ay,y,z)*h2  *h3

                                 -2   -1
           - df(ay,y)*df(h2,z)*h2  *h3

                                 -1   -1   -1
           + df(ay,z)*df(h1,y)*h1  *h2  *h3

                                 -1   -2                -2
           + df(ay,z)*df(h3,y)*h2  *h3   + df(az,z,2)*h3

                                 -1   -2                       -1   -2
           + df(az,z)*df(h1,z)*h1  *h3   + df(az,z)*df(h2,z)*h2  *h3

                                 -3                   -1   -1   -1
           - df(az,z)*df(h3,z)*h3   + df(h1,y,z)*ay*h1  *h2  *h3

                                    -2   -1   -1
           - df(h1,y)*df(h1,z)*ay*h1  *h2  *h3

                                    -1   -2   -1
           - df(h1,y)*df(h2,z)*ay*h1  *h2  *h3

                             -1   -2           2      -2   -2
           + df(h1,z,2)*az*h1  *h3   - df(h1,z) *az*h1  *h3

                                    -2   -1   -1
           - df(h1,z)*df(h2,x)*ax*h1  *h2  *h3

                                    -2   -2
           - df(h1,z)*df(h3,x)*ax*h1  *h3

                                    -1   -3
           - df(h1,z)*df(h3,z)*az*h1  *h3

                             -1   -1   -1
           + df(h2,x,z)*ax*h1  *h2  *h3

                                    -1   -2   -1
           - df(h2,x)*df(h2,z)*ax*h1  *h2  *h3

                             -1   -2           2      -2   -2
           + df(h2,z,2)*az*h2  *h3   - df(h2,z) *az*h2  *h3

                                    -2   -2
           - df(h2,z)*df(h3,y)*ay*h2  *h3

                                    -1   -3                   -1   -2
           - df(h2,z)*df(h3,z)*az*h2  *h3   + df(h3,x,z)*ax*h1  *h3

                                    -1   -3                   -1   -2
           - df(h3,x)*df(h3,z)*ax*h1  *h3   + df(h3,y,z)*ay*h2  *h3

                                    -1   -3
           - df(h3,y)*df(h3,z)*ay*h2  *h3


div curl a;


0


% Examples of integration : (1) Volume integrals

getcsystem 'spherical;


(r theta phi)


% Example 1 : integration of r**n over a sphere

origin := avec(0,0,0);


vec(r) := 0

vec(theta) := 0

vec(phi) := 0


upperbound := avec(rr,pi,2*pi);


vec(r) := rr

vec(theta) := pi

vec(phi) := 2*pi


volintegral(r**n,origin,upperbound);


     n      3
 4*rr *pi*rr
--------------
    n + 3


% Substitute n=0 to get the volume of a sphere
sub(n=0,ws);


 4       3
---*pi*rr
 3


% Example 2 : volume of a right-circular cone

getcsystem 'cylindrical;


(r z phi)

upperbound := avec(pp*z,h,2*pi);


vec(r) := pp*z

vec(z) := h

vec(phi) := 2*pi


volintorder := avec(2,0,1);


vec(r) := 2

vec(z) := 0

vec(phi) := 1

      % Integrate in the order : phi, r, z
cone := volintegral(1,origin,upperbound);


         1   3      2
cone := ---*h *pi*pp
         3


% Now we replace P*Z by RR to get the result in the familiar form

let pp*h=rr;


cone := cone;


         1         2
cone := ---*h*pi*rr
         3
                    % This is the familiar form
clear pp*h;



% Example 3 : line integral to obtain the length of a line of latitude
%             on a sphere

getcsystem 'spherical;


(r theta phi)


a := avec(0,0,1);


vec(r) := 0

vec(theta) := 0

vec(phi) := 1

            % Function vector is the tangent to the
			     % line of latitude
curve := avec(rr,latitude,phi);


vec(r) := rr

vec(theta) := latitude

vec(phi) := phi

   % Path is round a line of latitude

deflineint(a,curve,phi,0,2*pi);


2*sin(latitude)*pi*rr


end;


Time for test: 1992 ms, plus GC time: 160 ms