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REDUCE 3.6, 15-Jul-95, patched to 6 Mar 96 ... *** ^ 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; t 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: avector 1460 1551)