Sat May 30 16:26:41 PDT 1992
REDUCE 3.4.1, 15-Jul-92 ...
1: 1:
2: 2:
*** ^ redefined
(AVECTOR)
3: 3:
Time: 68 ms
4: 4: % 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;
5: 5:
Time: 4216 ms plus GC time: 136 ms
6: 6:
Quitting
Sat May 30 16:26:46 PDT 1992