Tue Feb 10 12:26:42 2004 run on Linux
*** ^ 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: 60 ms, plus GC time: 10 ms