reduce-historical/r34/xlog/avector.log

Sat Jun 29 14:16:00 PDT 1991
REDUCE 3.4, 15-Jul-91 ...

1: 1: 
2: 2: 
*** ^ redefined 

3: 3: % 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*F*AX - ---*BX
                    7

                    1
VEC(Y) := 2*F*AY - ---*BY
                    7

                    1
VEC(Z) := 2*F*AZ - ---*BZ
                    7


c(0);


          1
2*F*AX - ---*BX
          7
 c(1);


          1
2*F*AY - ---*BY
          7
 c(2);


          1
2*F*AZ - ---*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
           3

                                     2   2     2   2
                 - 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    28     2   2
          *SQRT(AX  + AY  + AZ )*F*AX  + ----*(AX *BY
                                          3

                  2   2
              + AX *BZ  - 2*AX*AY*BX*BY - 2*AX*AZ*BX*BZ

                  2   2     2   2                     2   2
              + AY *BX  + AY *BZ  - 2*AY*AZ*BY*BZ + AZ *BX

                  2   2     - 1         2     2     2
              + AZ *BY )**------*SQRT(AX  + AY  + AZ )*F*AX
                            2

             2    28     2   2     2   2
          *AY  + ----*(AX *BY  + AX *BZ  - 2*AX*AY*BX*BY
                  3

                                  2   2     2   2
              - 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
          *SQRT(AX  + AY  + AZ )*F*AX*AZ  - ---*(AX *BY
                                             3

                  2   2
              + AX *BZ  - 2*AX*AY*BX*BY - 2*AX*AZ*BX*BZ

                  2   2     2   2                     2   2
              + AY *BX  + AY *BZ  - 2*AY*AZ*BY*BZ + AZ *BX

                  2   2     - 1         2     2     2    2
              + AZ *BY )**------*SQRT(AX  + AY  + AZ )*AX *BX
                            2

              2     2   2     2   2
           - ---*(AX *BY  + AX *BZ  - 2*AX*AY*BX*BY
              3

                                  2   2     2   2
              - 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
          *SQRT(AX  + AY  + AZ )*AX*AY*BY - ---*(AX *BY
                                             3

                  2   2
              + AX *BZ  - 2*AX*AY*BX*BY - 2*AX*AZ*BX*BZ

                  2   2     2   2                     2   2
              + AY *BX  + AY *BZ  - 2*AY*AZ*BY*BZ + AZ *BX

                  2   2     - 1         2     2     2
              + AZ *BY )**------*SQRT(AX  + AY  + AZ )*AX*AZ
                            2

          *BZ

           28     2   2     2   2
VEC(Y) := ----*(AX *BY  + AX *BZ  - 2*AX*AY*BX*BY
           3

                                     2   2     2   2
                 - 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
          *SQRT(AX  + AY  + AZ )*F*AX *AY + ----*(AX *BY
                                             3

                  2   2
              + AX *BZ  - 2*AX*AY*BX*BY - 2*AX*AZ*BX*BZ

                  2   2     2   2                     2   2
              + AY *BX  + AY *BZ  - 2*AY*AZ*BY*BZ + AZ *BX

                  2   2     - 1         2     2     2      3
              + AZ *BY )**------*SQRT(AX  + AY  + AZ )*F*AY  
                            2

             28     2   2     2   2
          + ----*(AX *BY  + AX *BZ  - 2*AX*AY*BX*BY
             3

                                  2   2     2   2
              - 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
          *SQRT(AX  + AY  + AZ )*F*AY*AZ  - ---*(AX *BY
                                             3

                  2   2
              + AX *BZ  - 2*AX*AY*BX*BY - 2*AX*AZ*BX*BZ

                  2   2     2   2                     2   2
              + AY *BX  + AY *BZ  - 2*AY*AZ*BY*BZ + AZ *BX

                  2   2     - 1         2     2     2
              + AZ *BY )**------*SQRT(AX  + AY  + AZ )*AX*AY
                            2

                 2     2   2     2   2
          *BX - ---*(AX *BY  + AX *BZ  - 2*AX*AY*BX*BY
                 3

                                  2   2     2   2
              - 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
          *SQRT(AX  + AY  + AZ )*AY *BY - ---*(AX *BY
                                           3

                  2   2
              + AX *BZ  - 2*AX*AY*BX*BY - 2*AX*AZ*BX*BZ

                  2   2     2   2                     2   2
              + AY *BX  + AY *BZ  - 2*AY*AZ*BY*BZ + AZ *BX

                  2   2     - 1         2     2     2
              + AZ *BY )**------*SQRT(AX  + AY  + AZ )*AY*AZ
                            2

          *BZ

           28     2   2     2   2
VEC(Z) := ----*(AX *BY  + AX *BZ  - 2*AX*AY*BX*BY
           3

                                     2   2     2   2
                 - 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
          *SQRT(AX  + AY  + AZ )*F*AX *AZ + ----*(AX *BY
                                             3

                  2   2
              + AX *BZ  - 2*AX*AY*BX*BY - 2*AX*AZ*BX*BZ

                  2   2     2   2                     2   2
              + AY *BX  + AY *BZ  - 2*AY*AZ*BY*BZ + AZ *BX

                  2   2     - 1         2     2     2      2
              + AZ *BY )**------*SQRT(AX  + AY  + AZ )*F*AY
                            2

                 28     2   2     2   2
          *AZ + ----*(AX *BY  + AX *BZ  - 2*AX*AY*BX*BY
                 3

                                  2   2     2   2
              - 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
          *SQRT(AX  + AY  + AZ )*F*AZ  - ---*(AX *BY
                                          3

                  2   2
              + AX *BZ  - 2*AX*AY*BX*BY - 2*AX*AZ*BX*BZ

                  2   2     2   2                     2   2
              + AY *BX  + AY *BZ  - 2*AY*AZ*BY*BZ + AZ *BX

                  2   2     - 1         2     2     2
              + AZ *BY )**------*SQRT(AX  + AY  + AZ )*AX*AZ
                            2

                 2     2   2     2   2
          *BX - ---*(AX *BY  + AX *BZ  - 2*AX*AY*BX*BY
                 3

                                  2   2     2   2
              - 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
          *SQRT(AX  + AY  + AZ )*AY*AZ*BY - ---*(AX *BY
                                             3

                  2   2
              + AX *BZ  - 2*AX*AY*BX*BY - 2*AX*AZ*BX*BZ

                  2   2     2   2                     2   2
              + AY *BX  + AY *BZ  - 2*AY*AZ*BY*BZ + AZ *BX

                  2   2     - 1         2     2     2    2
              + AZ *BY )**------*SQRT(AX  + AY  + AZ )*AZ *BZ
                            2



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
DF(AX,X)*H1   + DF(AY,Y)*H2   + DF(AZ,Z)*H3

                 -1   -1                 -1   -1
 + DF(H1,Y)*AY*H1  *H2   + DF(H1,Z)*AZ*H1  *H3

                 -1   -1                 -1   -1
 + 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
VEC(X) :=  - DF(AY,Z)*H3   + DF(AZ,Y)*H2

                           -1   -1                 -1   -1
           - DF(H2,Z)*AY*H2  *H3   + DF(H3,Y)*AZ*H2  *H3

                     -1              -1
VEC(Y) := DF(AX,Z)*H3   - DF(AZ,X)*H1

                           -1   -1                 -1   -1
           + DF(H1,Z)*AX*H1  *H3   - DF(H3,X)*AZ*H1  *H3

                        -1              -1
VEC(Z) :=  - DF(AX,Y)*H2   + DF(AY,X)*H1

                           -1   -1                 -1   -1
           - DF(H1,Y)*AX*H1  *H2   + DF(H2,X)*AY*H1  *H2


dp1 := delsq p;


                   -2                      -3
DP1 := DF(P,X,2)*H1   - DF(P,X)*DF(H1,X)*H1

                             -2   -1
        + DF(P,X)*DF(H2,X)*H1  *H2

                             -2   -1               -2
        + DF(P,X)*DF(H3,X)*H1  *H3   + DF(P,Y,2)*H2

                             -1   -2                      -3
        + DF(P,Y)*DF(H1,Y)*H1  *H2   - DF(P,Y)*DF(H2,Y)*H2

                             -2   -1               -2
        + DF(P,Y)*DF(H3,Y)*H2  *H3   + DF(P,Z,2)*H3

                             -1   -2
        + DF(P,Z)*DF(H1,Z)*H1  *H3

                             -1   -2                      -3
        + DF(P,Z)*DF(H2,Z)*H2  *H3   - DF(P,Z)*DF(H3,Z)*H3

dp2 := div grad p;


                   -2                      -3
DP2 := DF(P,X,2)*H1   - DF(P,X)*DF(H1,X)*H1

                             -2   -1
        + DF(P,X)*DF(H2,X)*H1  *H2

                             -2   -1               -2
        + DF(P,X)*DF(H3,X)*H1  *H3   + DF(P,Y,2)*H2

                             -1   -2                      -3
        + DF(P,Y)*DF(H1,Y)*H1  *H2   - DF(P,Y)*DF(H2,Y)*H2

                             -2   -1               -2
        + DF(P,Y)*DF(H3,Y)*H2  *H3   + DF(P,Z,2)*H3

                             -1   -2
        + DF(P,Z)*DF(H1,Z)*H1  *H3

                             -1   -2                      -3
        + DF(P,Z)*DF(H2,Z)*H2  *H3   - DF(P,Z)*DF(H3,Z)*H3

dp1-dp2;


0

delsq a;


                       -2                       -3
VEC(X) := DF(AX,X,2)*H1   - DF(AX,X)*DF(H1,X)*H1

                                 -2   -1
           + DF(AX,X)*DF(H2,X)*H1  *H2

                                 -2   -1                -2
           + DF(AX,X)*DF(H3,X)*H1  *H3   + DF(AX,Y,2)*H2

                                 -1   -2
           + DF(AX,Y)*DF(H1,Y)*H1  *H2

                                 -3
           - DF(AX,Y)*DF(H2,Y)*H2

                                 -2   -1                -2
           + DF(AX,Y)*DF(H3,Y)*H2  *H3   + DF(AX,Z,2)*H3

                                 -1   -2
           + DF(AX,Z)*DF(H1,Z)*H1  *H3

                                 -1   -2
           + DF(AX,Z)*DF(H2,Z)*H2  *H3

                                 -3
           - 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*DF(AZ,Z)*DF(H3,X)*H1  *H3

                             -2   -1
           + DF(H1,X,Y)*AY*H1  *H2

                             -2   -1
           + DF(H1,X,Z)*AZ*H1  *H3

                                    -3   -1
           - 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

                             -1   -2           2      -2   -2
           + DF(H1,Y,2)*AX*H1  *H2   - DF(H1,Y) *AX*H1  *H2

                                    -1   -3
           - 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
           - DF(H1,Z)*DF(H3,Z)*AX*H1  *H3

                             -1   -2
           - DF(H2,X,Y)*AY*H1  *H2

                             -1   -1   -1
           + DF(H2,X,Z)*AZ*H1  *H2  *H3

                             -2   -1           2      -2   -2
           + DF(H2,X,2)*AX*H1  *H2   - DF(H2,X) *AX*H1  *H2

                                    -1   -3
           + 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
           + DF(H3,X,Y)*AY*H1  *H2  *H3

                             -1   -2
           - 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
           - DF(AY,X)*DF(H1,X)*H1

                                 -2   -1
           + 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
           + DF(AY,Y)*DF(H1,Y)*H1  *H2

                                 -3
           - 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
           + DF(AY,Z)*DF(H1,Z)*H1  *H3

                                 -1   -2
           + DF(AY,Z)*DF(H2,Z)*H2  *H3

                                 -3
           - DF(AY,Z)*DF(H3,Z)*H3

                                   -2   -1
           + 2*DF(AZ,Y)*DF(H2,Z)*H2  *H3

                                   -1   -2
           - 2*DF(AZ,Z)*DF(H3,Y)*H2  *H3

                             -2   -1
           - 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
           + DF(H1,Y,Z)*AZ*H1  *H2  *H3

                             -1   -2           2      -2   -2
           + DF(H1,Y,2)*AY*H1  *H2   - 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
           + DF(H2,X,Y)*AX*H1  *H2

                             -2   -1           2      -2   -2
           + DF(H2,X,2)*AY*H1  *H2   - DF(H2,X) *AY*H1  *H2

                                    -1   -3
           - DF(H2,X)*DF(H2,Y)*AX*H1  *H2

                                    -2   -1   -1
           + DF(H2,X)*DF(H3,X)*AY*H1  *H2  *H3

                             -2   -1
           + DF(H2,Y,Z)*AZ*H2  *H3

                                    -3   -1
           - DF(H2,Y)*DF(H2,Z)*AZ*H2  *H3

                                    -3   -1
           - DF(H2,Y)*DF(H3,Y)*AY*H2  *H3

                             -1   -2           2      -2   -2
           + DF(H2,Z,2)*AY*H2  *H3   - DF(H2,Z) *AY*H2  *H3

                                    -1   -3
           - 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
           - DF(H3,Y,Z)*AZ*H2  *H3

                             -2   -1           2      -2   -2
           + DF(H3,Y,2)*AY*H2  *H3   - DF(H3,Y) *AY*H2  *H3

                                    -1   -3
           + 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
           - DF(AZ,X)*DF(H1,X)*H1

                                 -2   -1
           + 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
           + DF(AZ,Y)*DF(H1,Y)*H1  *H2

                                 -3
           - 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
           + DF(AZ,Z)*DF(H1,Z)*H1  *H3

                                 -1   -2
           + 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
           - DF(H2,Y,Z)*AY*H2  *H3

                                    -3   -1
           + DF(H2,Y)*DF(H2,Z)*AY*H2  *H3

                                    -3   -1
           - DF(H2,Y)*DF(H3,Y)*AZ*H2  *H3

                             -1   -2           2      -2   -2
           + DF(H2,Z,2)*AZ*H2  *H3   - DF(H2,Z) *AZ*H2  *H3

                                    -1   -3
           - DF(H2,Z)*DF(H3,Z)*AZ*H2  *H3

                             -1   -2
           + DF(H3,X,Z)*AX*H1  *H3

                             -2   -1           2      -2   -2
           + DF(H3,X,2)*AZ*H1  *H3   - DF(H3,X) *AZ*H1  *H3

                                    -1   -3
           - DF(H3,X)*DF(H3,Z)*AX*H1  *H3

                             -1   -2
           + DF(H3,Y,Z)*AY*H2  *H3

                             -2   -1           2      -2   -2
           + DF(H3,Y,2)*AZ*H2  *H3   - DF(H3,Y) *AZ*H2  *H3

                                    -1   -3
           - 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
           + DF(AX,X)*DF(H2,X)*H1  *H2

                                 -2   -1
           + DF(AX,X)*DF(H3,X)*H1  *H3

                          -1   -1
           + DF(AY,X,Y)*H1  *H2

                                 -2   -1
           + DF(AY,X)*DF(H1,Y)*H1  *H2

                                 -1   -1   -1
           + DF(AY,X)*DF(H3,Y)*H1  *H2  *H3

                                 -1   -2
           - DF(AY,Y)*DF(H2,X)*H1  *H2

                          -1   -1
           + 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
           - DF(AZ,Z)*DF(H3,X)*H1  *H3

                             -2   -1
           + DF(H1,X,Y)*AY*H1  *H2

                             -2   -1
           + DF(H1,X,Z)*AZ*H1  *H3

                                    -3   -1
           - 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
           + DF(H2,X,Z)*AZ*H1  *H2  *H3

                             -2   -1           2      -2   -2
           + DF(H2,X,2)*AX*H1  *H2   - 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
           + DF(H3,X,Y)*AY*H1  *H2  *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   -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
           + DF(AX,Y)*DF(H3,X)*H1  *H2  *H3

                          -2                       -1   -2
           + DF(AY,Y,2)*H2   + DF(AY,Y)*DF(H1,Y)*H1  *H2

                                 -3
           - DF(AY,Y)*DF(H2,Y)*H2

                                 -2   -1
           + DF(AY,Y)*DF(H3,Y)*H2  *H3

                          -1   -1
           + DF(AZ,Y,Z)*H2  *H3

                                 -1   -1   -1
           + DF(AZ,Y)*DF(H1,Z)*H1  *H2  *H3

                                 -2   -1
           + DF(AZ,Y)*DF(H2,Z)*H2  *H3

                                 -1   -2
           - DF(AZ,Z)*DF(H3,Y)*H2  *H3

                             -1   -1   -1
           + DF(H1,Y,Z)*AZ*H1  *H2  *H3

                             -1   -2           2      -2   -2
           + DF(H1,Y,2)*AY*H1  *H2   - 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
           + DF(H2,X,Y)*AX*H1  *H2

                                    -1   -3
           - DF(H2,X)*DF(H2,Y)*AX*H1  *H2

                             -2   -1
           + DF(H2,Y,Z)*AZ*H2  *H3

                                    -3   -1
           - 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
           + DF(AX,Z)*DF(H3,X)*H1  *H3

                          -1   -1
           + 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
           + DF(AZ,Z)*DF(H1,Z)*H1  *H3

                                 -1   -2
           + DF(AZ,Z)*DF(H2,Z)*H2  *H3

                                 -3
           - DF(AZ,Z)*DF(H3,Z)*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
           - 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
           - DF(H2,Z)*DF(H3,Z)*AZ*H2  *H3

                             -1   -2
           + DF(H3,X,Z)*AX*H1  *H3

                                    -1   -3
           - DF(H3,X)*DF(H3,Z)*AX*H1  *H3

                             -1   -2
           + 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) := Z*PP

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 *PP *PI
         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;

4: 4: 
Quitting
Sat Jun 29 14:16:18 PDT 1991