Sun Aug 18 17:59:10 2002 run on Windows
*** ^ redefined
% Problem: Calculate the PDE's for the isovector of the heat equation.
% --------
% (c.f. B.K. Harrison, f.B. Estabrook, "Geometric Approach...",
% J. Math. Phys. 12, 653, 1971)
% The heat equation @ psi = @ psi is equivalent to the set of exterior
% xx t
% equations (with u=@ psi, y=@ psi):
% T x
pform {psi,u,x,y,t}=0,a=1,{da,b}=2;
a := d psi - u*d t - y*d x;
a := d psi - d t*u - d x*y
da := - d u^d t - d y^d x;
da := d t^d u + d x^d y
b := u*d x^d t - d y^d t;
b := - d t^d x*u + d t^d y
% Now calculate the PDE's for the isovector.
tvector v;
pform {vpsi,vt,vu,vx,vy}=0;
fdomain vpsi=vpsi(psi,t,u,x,y),vt=vt(psi,t,u,x,y),vu=vu(psi,t,u,x,y),
vx=vx(psi,t,u,x,y),vy=vy(psi,t,u,x,y);
v := vpsi*@ psi + vt*@ t + vu*@ u + vx*@ x + vy*@ y;
v := @ *vpsi + @ *vt + @ *vu + @ *vx + @ *vy
psi t u x y
factor d;
on rat;
i1 := v |_ a - l*a;
i1 := d psi*(@ vpsi - @ vt*u - @ vx*y - l)
psi psi psi
+ d t*(@ vpsi - @ vt*u - @ vx*y + l*u - vu)
t t t
+ d u*(@ vpsi - @ vt*u - @ vx*y)
u u u
+ d x*(@ vpsi - @ vt*u - @ vx*y + l*y - vy)
x x x
+ d y*(@ vpsi - @ vt*u - @ vx*y)
y y y
pform o=1;
o := ot*d t + ox*d x + ou*d u + oy*d y;
o := d t*ot + d u*ou + d x*ox + d y*oy
fdomain f=f(psi,t,u,x,y);
i11 := v _| d a - l*a + d f;
i11 := - d psi*l + d t*(l*u - vu) + d u*vt + d x*(l*y - vy) + d y*vx
let vx=-@(f,y),vt=-@(f,u),vu=@(f,t)+u*@(f,psi),vy=@(f,x)+y*@(f,psi),
vpsi=f-u*@(f,u)-y*@(f,y);
factor ^;
i2 := v |_ b - xi*b - o^a + zeta*da;
i2 := d psi^d t*( - @ f*y - @ f - @ f*u + ot) + d psi^d u*ou
psi psi psi x psi y
+ d psi^d x*(@ f*u + ox) + d psi^d y*( - @ f + oy)
psi u psi u
+ d t^d u*(@ f*y + @ f + @ f*u - ou*u + zeta) + d t^d x*(
psi u u x u y
@ f*y - @ f*u + @ f*u - @ f + @ f + @ f*u + ot*y - ox*u
psi x psi t u t x x x y
+ u*xi)
+ d t^d y*(@ f*y + @ f - @ f + @ f + @ f*u - oy*u - xi)
psi y psi t u x y y y
+ d u^d x*(@ f*u + ou*y) - d u^d y*@ f
u u u u
+ d x^d y*( - @ f - @ f*u - oy*y + zeta)
u x u y
let ou=0,oy=@(f,u,psi),ox=-u*@(f,u,psi),
ot=@(f,x,psi)+u*@(f,y,psi)+y*@(f,psi,psi);
i2;
2
d t^d u*(@ f*y + @ f + @ f*u + zeta) + d t^d x*(@ f*y
psi u u x u y psi psi
2
+ @ f*u + 2*@ f*y + @ f*u*y - @ f*u + @ f*u - @ f + @ f
psi u psi x psi y psi t u t x x
+ @ f*u + u*xi)
x y
+ d t^d y*( - @ f*u + @ f*y + @ f - @ f + @ f + @ f*u - xi)
psi u psi y psi t u x y y y
+ d u^d x*@ f*u - d u^d y*@ f
u u u u
+ d x^d y*( - @ f*y - @ f - @ f*u + zeta)
psi u u x u y
let zeta=-@(f,u,x)-@(f,u,y)*u-@(f,u,psi)*y;
i2;
2 2
d t^d x*(@ f*y + @ f*u + 2*@ f*y + @ f*u*y - @ f*u
psi psi psi u psi x psi y psi
+ @ f*u - @ f + @ f + @ f*u + u*xi)
t u t x x x y
+ d t^d y*( - @ f*u + @ f*y + @ f - @ f + @ f + @ f*u - xi)
psi u psi y psi t u x y y y
+ d u^d x*@ f*u - d u^d y*@ f - 2*d x^d y*(@ f*y + @ f + @ f*u)
u u u u psi u u x u y
let xi=-@(f,t,u)-u*@(f,u,psi)+@(f,x,y)+u*@(f,y,y)+y*@(f,y,psi)+@(f,psi);
i2;
2
d t^d x*(@ f*y + 2*@ f*y + 2*@ f*u*y - @ f + @ f + 2*@ f*u
psi psi psi x psi y t x x x y
2
+ @ f*u ) + d u^d x*@ f*u - d u^d y*@ f
y y u u u u
- 2*d x^d y*(@ f*y + @ f + @ f*u)
psi u u x u y
let @(f,u,u)=0;
i2;
2
d t^d x*(@ f*y + 2*@ f*y + 2*@ f*u*y - @ f + @ f + 2*@ f*u
psi psi psi x psi y t x x x y
2
+ @ f*u ) - 2*d x^d y*(@ f*y + @ f + @ f*u)
y y psi u u x u y
% These PDE's have to be solved.
clear a,da,b,v,i1,i11,o,i2,xi,t;
remfdomain f,vpsi,vt,vu,vx,vy;
clear @(f,u,u);
% Problem:
% --------
% Calculate the integrability conditions for the system of PDE's:
% (c.f. B.F. Schutz, "Geometrical Methods of Mathematical Physics"
% Cambridge University Press, 1984, p. 156)
% @ z /@ x + a1*z + b1*z = c1
% 1 1 2
% @ z /@ y + a2*z + b2*z = c2
% 1 1 2
% @ z /@ x + f1*z + g1*z = h1
% 2 1 2
% @ z /@ y + f2*z + g2*z = h2
% 2 1 2 ;
pform w(k)=1,integ(k)=4,{z(k),x,y}=0,{a,b,c,f,g,h}=1,
{a1,a2,b1,b2,c1,c2,f1,f2,g1,g2,h1,h2}=0;
fdomain a1=a1(x,y),a2=a2(x,y),b1=b1(x,y),b2=b2(x,y),
c1=c1(x,y),c2=c2(x,y),f1=f1(x,y),f2=f2(x,y),
g1=g1(x,y),g2=g2(x,y),h1=h1(x,y),h2=h2(x,y);
a:=a1*d x+a2*d y$
b:=b1*d x+b2*d y$
c:=c1*d x+c2*d y$
f:=f1*d x+f2*d y$
g:=g1*d x+g2*d y$
h:=h1*d x+h2*d y$
% The equivalent exterior system:
factor d;
w(1) := d z(-1) + z(-1)*a + z(-2)*b - c;
1
w := d z + d x*(z *a1 + z *b1 - c1) + d y*(z *a2 + z *b2 - c2)
1 1 2 1 2
w(2) := d z(-2) + z(-1)*f + z(-2)*g - h;
2
w := d z + d x*(z *f1 + z *g1 - h1) + d y*(z *f2 + z *g2 - h2)
2 1 2 1 2
indexrange 1,2;
factor z;
% The integrability conditions:
integ(k) := d w(k) ^ w(1) ^ w(2);
1
integ := d z ^d z ^d x^d y*z *( - @ a1 + @ a2 + b1*f2 - b2*f1) +
1 2 1 y x
d z ^d z ^d x^d y*z *( - @ b1 + @ b2 + a1*b2 - a2*b1 + b1*g2 - b2*g1)
1 2 2 y x
+ d z ^d z ^d x^d y*(@ c1 - @ c2 - a1*c2 + a2*c1 - b1*h2 + b2*h1)
1 2 y x
2
integ := d z ^d z ^d x^d y*z *( - @ f1 + @ f2 - a1*f2 + a2*f1 - f1*g2 + f2*g1)
1 2 1 y x
+ d z ^d z ^d x^d y*z *( - @ g1 + @ g2 - b1*f2 + b2*f1)
1 2 2 y x
+ d z ^d z ^d x^d y*(@ h1 - @ h2 + c1*f2 - c2*f1 - g1*h2 + g2*h1)
1 2 y x
clear a,b,c,f,g,h,x,y,w(k),integ(k),z(k);
remfdomain a1,a2,b1,c1,c2,f1,f2,g1,g2,h1,h2;
% Problem:
% --------
% Calculate the PDE's for the generators of the d-theta symmetries of
% the Lagrangian system of the planar Kepler problem.
% c.f. W.Sarlet, F.Cantrijn, Siam Review 23, 467, 1981
% Verify that time translation is a d-theta symmetry and calculate the
% corresponding integral.
pform {t,q(k),v(k),lam(k),tau,xi(k),eta(k)}=0,theta=1,f=0,
{l,glq(k),glv(k),glt}=0;
tvector gam,y;
indexrange 1,2;
fdomain tau=tau(t,q(k),v(k)),xi=xi(t,q(k),v(k)),f=f(t,q(k),v(k));
l := 1/2*(v(1)**2 + v(2)**2) + m/r$
% The Lagrangian.
pform r=0;
fdomain r=r(q(k));
let @(r,q 1)=q(1)/r,@(r,q 2)=q(2)/r,q(1)**2+q(2)**2=r**2;
lam(k) := -m*q(k)/r;
1
1 - q *m
lam := ---------
r
2
2 - q *m
lam := ---------
r
% The force.
gam := @ t + v(k)*@(q(k)) + lam(k)*@(v(k))$
eta(k) := gam _| d xi(k) - v(k)*gam _| d tau$
y := tau*@ t + xi(k)*@(q(k)) + eta(k)*@(v(k))$
% Symmetry generator.
theta := l*d t + @(l,v(k))*(d q(k) - v(k)*d t)$
factor @;
s := y |_ theta - d f$
glq(k) := @(q k) _| s;
1 1 1 2
- @ (xi )*q *m - @ (xi )*q *m
1 2
1 1 1 1 2 v v
glq := 2*@ (xi )*v + @ (xi )*v + ------------------ + ------------------
1 2 r r
q q
1 2 2
+ @ (xi ) + @ (xi )*v - @ f
t 1 1
q q
1 2 2 2
@ tau*( - 3*(v ) *r - (v ) *r + 2*m)
1
q 1 2
+ --------------------------------------- - @ tau*v *v
2*r 2
q
1 1 2 1
@ tau*q *v *m @ tau*q *v *m
1 2
v v 1
+ ---------------- + ---------------- - @ tau*v
r r t
2 1
- @ (xi )*q *m
1
2 1 1 2 1 2 2 v
glq := @ (xi )*v + @ (xi )*v + 2*@ (xi )*v + ------------------
2 1 2 r
q q q
2 2
- @ (xi )*q *m
2
v 2 1 2
+ ------------------ + @ (xi ) - @ f - @ tau*v *v
r t 2 1
q q
1 2 2 2 1 2
@ tau*( - (v ) *r - 3*(v ) *r + 2*m) @ tau*q *v *m
2 1
q v
+ --------------------------------------- + ----------------
2*r r
2 2
@ tau*q *v *m
2
v 2
+ ---------------- - @ tau*v
r t
glv(k) := @(v k) _| s;
1 2 2 2
@ tau*( - (v ) *r - (v ) *r + 2*m)
1
1 1 1 2 2 v
glv := @ (xi )*v + @ (xi )*v - @ f + -------------------------------------
1 1 1 2*r
v v v
1 2 2 2
@ tau*( - (v ) *r - (v ) *r + 2*m)
2
2 1 1 2 2 v
glv := @ (xi )*v + @ (xi )*v - @ f + -------------------------------------
2 2 2 2*r
v v v
glt := @(t) _| s;
1 1 1
@ (xi )*q *v *m
1
1 1 2 1 1 2 v
glt := - @ (xi )*(v ) - @ (xi )*v *v + ------------------
1 2 r
q q
1 2 1
@ (xi )*q *v *m
2
v 2 1 2 2 2 2
+ ------------------ - @ (xi )*v *v - @ (xi )*(v )
r 1 2
q q
2 1 2 2 2 2
@ (xi )*q *v *m @ (xi )*q *v *m
1 2
v v
+ ------------------ + ------------------ - @ f
r r t
1 1 2 2 2 2 1 2 2 2
+ @ tau*v *((v ) + (v ) ) + @ tau*v *((v ) + (v ) )
1 2
q q
1 1 2 2 2 2 1 2 2 2
@ tau*q *m*((v ) + (v ) ) @ tau*q *m*((v ) + (v ) )
1 2
v v
- ----------------------------- - -----------------------------
r r
1 2 2 2
@ tau*((v ) *r + (v ) *r + 2*m) 1 1 2 2
t m*(q *xi + q *xi )
+ --------------------------------- - ---------------------
2*r 3
r
% Translation in time must generate a symmetry.
xi(k) := 0;
k
xi := 0
tau := 1;
tau := 1
glq k := glq k;
1
glq := - @ f
1
q
2
glq := - @ f
2
q
glv k := glv k;
1
glv := - @ f
1
v
2
glv := - @ f
2
v
glt;
- @ f
t
% The corresponding integral is of course the energy.
integ := - y _| theta;
1 2 2 2
(v ) *r + (v ) *r - 2*m
integ := -------------------------
2*r
clear l,lam k,gam,eta k,y,theta,s,glq k,glv k,glt,t,q k,v k,tau,xi k;
remfdomain r,f,tau,xi;
% Problem:
% --------
% Calculate the "gradient" and "Laplacian" of a function and the "curl"
% and "divergence" of a one-form in elliptic coordinates.
coframe e u = sqrt(cosh(v)**2 - sin(u)**2)*d u,
e v = sqrt(cosh(v)**2 - sin(u)**2)*d v,
e phi = cos u*sinh v*d phi;
pform f=0;
fdomain f=f(u,v,phi);
factor e,^;
on rat,gcd;
order cosh v, sin u;
% The gradient:
d f;
phi u v
e *@ f e *@ f e *@ f
phi u v
---------------- + -------------------------- + --------------------------
cos(u)*sinh(v) 2 2 2 2
sqrt(cosh(v) - sin(u) ) sqrt(cosh(v) - sin(u) )
factor @;
% The Laplacian:
# d # d f;
@ f @ f @ f*sin(u)
phi phi u u u
------------------ + -------------------- - -----------------------------
2 2 2 2 2 2
cos(u) *sinh(v) cosh(v) - sin(u) cos(u)*(cosh(v) - sin(u) )
@ f @ f*cosh(v)
v v v
+ -------------------- + ------------------------------
2 2 2 2
cosh(v) - sin(u) sinh(v)*(cosh(v) - sin(u) )
% Another way of calculating the Laplacian:
-#vardf(1/2*d f^#d f,f);
@ f @ f @ f*sin(u)
phi phi u u u
------------------ + -------------------- - -----------------------------
2 2 2 2 2 2
cos(u) *sinh(v) cosh(v) - sin(u) cos(u)*(cosh(v) - sin(u) )
@ f @ f*cosh(v)
v v v
+ -------------------- + ------------------------------
2 2 2 2
cosh(v) - sin(u) sinh(v)*(cosh(v) - sin(u) )
remfac @;
% Now calculate the "curl" and the "divergence" of a one-form.
pform w=1,a(k)=0;
fdomain a=a(u,v,phi);
w := a(-k)*e k;
phi u v
w := e *a + e *a + e *a
phi u v
% The curl:
x := # d w;
phi 2 2
x := (e *( - cosh(v) *@ (a ) + cosh(v) *@ (a ) - cosh(v)*a *sinh(v)
v u u v u
2 2
+ sin(u) *@ (a ) - sin(u) *@ (a ) - sin(u)*a *cos(u)))/(
v u u v v
2 2 2 2 u
sqrt(cosh(v) - sin(u) )*(cosh(v) - sin(u) )) + (e *(
cosh(v)*a *cos(u) + cos(u)*@ (a )*sinh(v)
phi v phi
2 2 2 2
- sqrt(cosh(v) - sin(u) )*@ (a )))/(sqrt(cosh(v) - sin(u) )
phi v
v
*cos(u)*sinh(v)) + (e *(sin(u)*a *sinh(v) - cos(u)*@ (a )*sinh(v)
phi u phi
2 2 2 2
+ sqrt(cosh(v) - sin(u) )*@ (a )))/(sqrt(cosh(v) - sin(u) )
phi u
*cos(u)*sinh(v))
factor @;
% The divergence:
y := # d # w;
@ (a ) @ (a ) @ (a )
phi phi u u v v
y := ---------------- + -------------------------- + --------------------------
cos(u)*sinh(v) 2 2 2 2
sqrt(cosh(v) - sin(u) ) sqrt(cosh(v) - sin(u) )
3 2
+ (cosh(v) *a *cos(u) - cosh(v) *sin(u)*a *sinh(v)
v u
2 2
- cosh(v)*sin(u) *a *cos(u) + cosh(v)*a *cos(u)*sinh(v)
v v
3 2
+ sin(u) *a *sinh(v) - sin(u)*a *cos(u) *sinh(v))/(
u u
2 2 2 2
sqrt(cosh(v) - sin(u) )*cos(u)*sinh(v)*(cosh(v) - sin(u) ))
remfac @;
clear x,y,w,u,v,phi,e k,a k;
remfdomain a,f;
% Problem:
% --------
% Calculate in a spherical coordinate system the Navier Stokes equations.
coframe e r=d r, e theta =r*d theta, e phi = r*sin theta *d phi;
frame x;
fdomain v=v(t,r,theta,phi),p=p(r,theta,phi);
pform v(k)=0,p=0,w=1;
% We first calculate the convective derivative.
w := v(-k)*e(k)$
factor e;
on rat;
cdv := @(w,t) + (v(k)*x(-k)) |_ w - 1/2*d(v(k)*v(-k));
phi
cdv := (e *(cos(theta)*v *v + @ (v )*v
phi theta phi phi phi
+ @ (v )*sin(theta)*v *r + @ (v )*sin(theta)*r
r phi r t phi
+ @ (v )*sin(theta)*v + sin(theta)*v *v ))/(
theta phi theta phi r
r
sin(theta)*r) + (e *(@ (v )*v + @ (v )*sin(theta)*v *r
phi r phi r r r
+ @ (v )*sin(theta)*r + @ (v )*sin(theta)*v
t r theta r theta
2 2
- sin(theta)*(v ) - sin(theta)*(v ) ))/(sin(theta)*r) + (
phi theta
theta 2
e *( - cos(theta)*(v ) + @ (v )*v
phi phi theta phi
+ @ (v )*sin(theta)*v *r + @ (v )*sin(theta)*r
r theta r t theta
+ @ (v )*sin(theta)*v + sin(theta)*v *v ))/(
theta theta theta r theta
sin(theta)*r)
%next we calculate the viscous terms;
visc := nu*(d#d# w - #d#d w) + mu*d#d# w;
phi 2
visc := (e *( - cos(theta) *v *nu + cos(theta)*@ (v )*sin(theta)*nu
phi theta phi
+ cos(theta)*@ (v )*mu + 2*cos(theta)*@ (v )*nu
phi theta phi theta
+ @ (v )*mu + @ (v )*nu
phi phi phi phi phi phi
2 2 2
+ @ (v )*sin(theta) *nu*r + 2*@ (v )*sin(theta) *nu*r
r r phi r phi
2
+ @ (v )*sin(theta) *nu + @ (v )*sin(theta)*mu*r
theta theta phi phi r r
+ 2*@ (v )*sin(theta)*mu + 2*@ (v )*sin(theta)*nu
phi r phi r
2
+ @ (v )*sin(theta)*mu - sin(theta) *v *nu))/(
phi theta theta phi
2 2 r
sin(theta) *r ) + (e *(cos(theta)*@ (v )*sin(theta)*nu
theta r
+ cos(theta)*@ (v )*sin(theta)*mu*r
r theta
- cos(theta)*sin(theta)*v *mu
theta
- 2*cos(theta)*sin(theta)*v *nu
theta
+ @ (v )*sin(theta)*mu*r - @ (v )*sin(theta)*mu
phi r phi phi phi
- 2*@ (v )*sin(theta)*nu + @ (v )*nu
phi phi phi phi r
2 2 2 2
+ @ (v )*sin(theta) *mu*r + @ (v )*sin(theta) *nu*r
r r r r r r
2 2
+ 2*@ (v )*sin(theta) *mu*r + 2*@ (v )*sin(theta) *nu*r
r r r r
2
+ @ (v )*sin(theta) *nu
theta theta r
2
+ @ (v )*sin(theta) *mu*r
r theta theta
2 2
- @ (v )*sin(theta) *mu - 2*@ (v )*sin(theta) *nu
theta theta theta theta
2 2 2 2
- 2*sin(theta) *v *mu - 2*sin(theta) *v *nu))/(sin(theta) *r ) +
r r
theta 2 2
(e *( - cos(theta) *v *mu - cos(theta) *v *nu
theta theta
- cos(theta)*@ (v )*mu - 2*cos(theta)*@ (v )*nu
phi phi phi phi
+ cos(theta)*@ (v )*sin(theta)*mu
theta theta
+ cos(theta)*@ (v )*sin(theta)*nu
theta theta
+ @ (v )*sin(theta)*mu
phi theta phi
2 2
+ @ (v )*sin(theta) *mu*r + 2*@ (v )*sin(theta) *mu
r theta r theta r
2
+ 2*@ (v )*sin(theta) *nu + @ (v )*nu
theta r phi phi theta
2 2
+ @ (v )*sin(theta) *nu*r
r r theta
2
+ 2*@ (v )*sin(theta) *nu*r
r theta
2
+ @ (v )*sin(theta) *mu
theta theta theta
2 2
+ @ (v )*sin(theta) *nu - sin(theta) *v *mu
theta theta theta theta
2 2 2
- sin(theta) *v *nu))/(sin(theta) *r )
theta
% Finally we add the pressure term and print the components of the
% whole equation.
pform nasteq=1,nast(k)=0;
nasteq := cdv - visc + 1/rho*d p$
factor @;
nast(-k) := x(-k) _| nasteq;
- @ (v )*mu @ (v )*(mu + 2*nu) - @ (v )*nu
phi r phi phi phi phi phi r
nast := -------------------- + ------------------------ + --------------------
r sin(theta)*r 2 2 2
sin(theta)*r sin(theta) *r
@ (v )*v @ (v )*(v *r - 2*mu - 2*nu)
phi r phi r r r
+ --------------- - @ (v )*(mu + nu) + -----------------------------
sin(theta)*r r r r r
- @ (v )*nu
theta theta r
+ @ (v ) + ------------------------
t r 2
r
@ (v )*( - cos(theta)*nu + sin(theta)*v *r)
theta r theta
+ -----------------------------------------------------
2
sin(theta)*r
- @ (v )*mu - @ (v )*cos(theta)*mu
r theta theta r theta
+ ------------------------ + -----------------------------
r sin(theta)*r
@ (v )*(mu + 2*nu) @ p
theta theta r
+ ---------------------------- + ----- + (cos(theta)*v *mu
2 rho theta
r
2
+ 2*cos(theta)*v *nu - sin(theta)*(v ) *r
theta phi
2
+ 2*sin(theta)*v *mu + 2*sin(theta)*v *nu - sin(theta)*(v ) *r)
r r theta
2
/(sin(theta)*r )
- @ (v )*mu @ (v )*cos(theta)*(mu + 2*nu)
phi theta phi phi phi
nast := ------------------------ + -----------------------------------
theta 2 2 2
sin(theta)*r sin(theta) *r
- @ (v )*mu 2*@ (v )*(mu + nu)
r theta r theta r
+ -------------------- - ------------------------
r 2
r
- @ (v )*nu @ (v )*v
phi phi theta phi theta phi
+ ------------------------ + ------------------- - @ (v )*nu
2 2 sin(theta)*r r r theta
sin(theta) *r
@ (v )*(v *r - 2*nu)
r theta r
+ -------------------------- + @ (v )
r t theta
@ (v )*(mu + nu)
theta theta theta
- -------------------------------- + (@ (v )
2 theta theta
r
*( - cos(theta)*mu - cos(theta)*nu + sin(theta)*v *r))/(
theta
@ p
2 theta 2
sin(theta)*r ) + --------- + (cos(theta) *v *mu
r*rho theta
2 2
+ cos(theta) *v *nu - cos(theta)*sin(theta)*(v ) *r
theta phi
2 2
+ sin(theta) *v *v *r + sin(theta) *v *mu
r theta theta
2 2 2
+ sin(theta) *v *nu)/(sin(theta) *r )
theta
@ (v )*(mu + nu) @ (v )*v
phi phi phi phi phi phi
nast := - -------------------------- + ----------------- - @ (v )*nu
phi 2 2 sin(theta)*r r r phi
sin(theta) *r
@ (v )*(v *r - 2*nu) - @ (v )*nu
r phi r theta theta phi
+ ------------------------ + @ (v ) + --------------------------
r t phi 2
r
@ (v )*( - cos(theta)*nu + sin(theta)*v *r)
theta phi theta
+ -------------------------------------------------------
2
sin(theta)*r
- @ (v )*mu 2*@ (v )*(mu + nu)
phi r r phi r
+ ------------------ - ----------------------
sin(theta)*r 2
sin(theta)*r
- @ (v )*mu
phi theta theta
+ --------------------------
2
sin(theta)*r
@ (v )*cos(theta)*( - mu - 2*nu) @ p
phi theta phi
+ ---------------------------------------- + ------------------ + (
2 2 sin(theta)*r*rho
sin(theta) *r
2
v *(cos(theta) *nu + cos(theta)*sin(theta)*v *r
phi theta
2 2 2 2
+ sin(theta) *v *r + sin(theta) *nu))/(sin(theta) *r )
r
remfac @,e;
clear v k,x k,nast k,cdv,visc,p,w,nasteq,e k;
remfdomain p,v;
% Problem:
% --------
% Calculate from the Lagrangian of a vibrating rod the equation of
% motion and show that the invariance under time translation leads
% to a conserved current.
pform {y,x,t,q,j}=0,lagr=2;
fdomain y=y(x,t),q=q(x),j=j(x);
factor ^;
lagr := 1/2*(rho*q*@(y,t)**2 - e*j*@(y,x,x)**2)*d x^d t;
2 2
d t^d x*( - @ y *q*rho + @ y *e*j)
t x x
lagr := --------------------------------------
2
vardf(lagr,y);
d t^d x*(@ j*@ y*e + 2*@ j*@ y*e + @ y*q*rho + @ y*e*j)
x x x x x x x x t t x x x x
% The Lagrangian does not explicitly depend on time; therefore the
% vector field @ t generates a symmetry. The conserved current is
pform c=1;
factor d;
c := noether(lagr,y,@ t);
c := d t*e*(@ j*@ y*@ y - @ y*@ y*j + @ y*@ y*j)
x t x x t x x x t x x x
2 2
d x*(@ y *q*rho + @ y *e*j)
t x x
- -------------------------------
2
% The exterior derivative of this must be zero or a multiple of the
% equation of motion (weak conservation law) to be a conserved current.
remfac d;
d c;
d t^d x*@ y*( - @ j*@ y*e - 2*@ j*@ y*e - @ y*q*rho - @ y*e*j)
t x x x x x x x x t t x x x x
% i.e. it is a multiple of the equation of motion.
clear lagr,c,j,y,q;
remfdomain y,q,j;
% Problem:
% --------
% Show that the metric structure given by Eguchi and Hanson induces a
% self-dual curvature.
% c.f. T. Eguchi, P.B. Gilkey, A.J. Hanson, "Gravitation, Gauge Theories
% and Differential Geometry", Physics Reports 66, 213, 1980
for all x let cos(x)**2=1-sin(x)**2;
pform f=0,g=0;
fdomain f=f(r), g=g(r);
coframe o(r) = f*d r,
o(theta) = (r/2)*(sin(psi)*d theta - sin(theta)*cos(psi)*d phi),
o(phi) = (r/2)*(-cos(psi)*d theta - sin(theta)*sin(psi)*d phi),
o(psi) = (r/2)*g*(d psi + cos(theta)*d phi);
frame e;
pform gamma(a,b)=1,curv2(a,b)=2;
index_symmetries gamma(a,b),curv2(a,b): antisymmetric;
factor o;
gamma(-a,-b) := -(1/2)*( e(-a) _| (e(-c) _| (d o(-b)))
-e(-b) _| (e(-a) _| (d o(-c)))
+e(-c) _| (e(-b) _| (d o(-a))) )*o(c)$
curv2(-a,b) := d gamma(-a,b) + gamma(-c,b)^gamma(-a,c)$
let f=1/g,g=sqrt(1-(a/r)**4);
pform chck(k,l)=2;
index_symmetries chck(k,l): antisymmetric;
% The following has to be zero for a self-dual curvature.
chck(k,l) := 1/2*eps(k,l,m,n)*curv2(-m,-n) + curv2(k,l);
r theta
chck := 0
r phi
chck := 0
theta phi
chck := 0
r psi
chck := 0
theta psi
chck := 0
phi psi
chck := 0
clear gamma(a,b),curv2(a,b),f,g,chck(a,b),o(k),e(k),r,phi,psi;
remfdomain f,g;
% Example: 6-dimensional FRW model with quadratic curvature terms in
% -------
% the Lagrangian (Lanczos and Gauss-Bonnet terms).
% cf. Henriques, Nuclear Physics, B277, 621 (1986)
for all x let cos(x)**2+sin(x)**2=1;
pform {r,s}=0;
fdomain r=r(t),s=s(t);
coframe o(t) = d t,
o(1) = r*d u/(1 + k*(u**2)/4),
o(2) = r*u*d theta/(1 + k*(u**2)/4),
o(3) = r*u*sin(theta)*d phi/(1 + k*(u**2)/4),
o(4) = s*d v1,
o(5) = s*sin(v1)*d v2
with metric g =-o(t)*o(t)+o(1)*o(1)+o(2)*o(2)+o(3)*o(3)
+o(4)*o(4)+o(5)*o(5);
frame e;
on nero;
factor o,^;
riemannconx om;
pform curv(k,l)=2,{riemann(a,b,c,d),ricci(a,b),riccisc}=0;
index_symmetries curv(k,l): antisymmetric,
riemann(k,l,m,n): antisymmetric in {k,l},{m,n}
symmetric in {{k,l},{m,n}},
ricci(k,l): symmetric;
curv(k,l) := d om(k,l) + om(k,-m)^om(m,l);
t 1
o ^o *@ r
t 1 t t
curv := -------------
r
t 2
o ^o *@ r
t 2 t t
curv := -------------
r
1 2 2
o ^o *(@ r + k)
1 2 t
curv := ------------------
2
r
t 3
o ^o *@ r
t 3 t t
curv := -------------
r
1 3 2
o ^o *(@ r + k)
1 3 t
curv := ------------------
2
r
2 3 2
o ^o *(@ r + k)
2 3 t
curv := ------------------
2
r
t 4
o ^o *@ s
t 4 t t
curv := -------------
s
1 4
o ^o *@ r*@ s
1 4 t t
curv := ---------------
r*s
2 4
o ^o *@ r*@ s
2 4 t t
curv := ---------------
r*s
3 4
o ^o *@ r*@ s
3 4 t t
curv := ---------------
r*s
t 5
o ^o *@ s
t 5 t t
curv := -------------
s
1 5
o ^o *@ r*@ s
1 5 t t
curv := ---------------
r*s
2 5
o ^o *@ r*@ s
2 5 t t
curv := ---------------
r*s
3 5
o ^o *@ r*@ s
3 5 t t
curv := ---------------
r*s
4 5 2
o ^o *(@ s + 1)
4 5 t
curv := ------------------
2
s
riemann(a,b,c,d) := e(d) _| (e (c) _| curv(a,b));
- @ r
t 1 t 1 t t
riemann := ----------
r
- @ r
t 2 t 2 t t
riemann := ----------
r
2
@ r + k
1 2 1 2 t
riemann := ----------
2
r
- @ r
t 3 t 3 t t
riemann := ----------
r
2
@ r + k
1 3 1 3 t
riemann := ----------
2
r
2
@ r + k
2 3 2 3 t
riemann := ----------
2
r
- @ s
t 4 t 4 t t
riemann := ----------
s
@ r*@ s
1 4 1 4 t t
riemann := ---------
r*s
@ r*@ s
2 4 2 4 t t
riemann := ---------
r*s
@ r*@ s
3 4 3 4 t t
riemann := ---------
r*s
- @ s
t 5 t 5 t t
riemann := ----------
s
@ r*@ s
1 5 1 5 t t
riemann := ---------
r*s
@ r*@ s
2 5 2 5 t t
riemann := ---------
r*s
@ r*@ s
3 5 3 5 t t
riemann := ---------
r*s
2
@ s + 1
4 5 4 5 t
riemann := ----------
2
s
% The rest is done in the Ricci calculus language,
ricci(-a,-b) := riemann(c,-a,-d,-b)*g(-c,d);
- 3*@ r*s - 2*@ s*r
t t t t
ricci := --------------------------
t t r*s
2
@ r*r*s + 2*@ r *s + 2*@ r*@ s*r + 2*k*s
t t t t t
ricci := --------------------------------------------
1 1 2
r *s
2
@ r*r*s + 2*@ r *s + 2*@ r*@ s*r + 2*k*s
t t t t t
ricci := --------------------------------------------
2 2 2
r *s
2
@ r*r*s + 2*@ r *s + 2*@ r*@ s*r + 2*k*s
t t t t t
ricci := --------------------------------------------
3 3 2
r *s
2
3*@ r*@ s*s + @ s*r*s + @ s *r + r
t t t t t
ricci := --------------------------------------
4 4 2
r*s
2
3*@ r*@ s*s + @ s*r*s + @ s *r + r
t t t t t
ricci := --------------------------------------
5 5 2
r*s
riccisc := ricci(-a,-b)*g(a,b);
2 2 2 2 2 2
riccisc := (2*(3*@ r*r*s + 3*@ r *s + 6*@ r*@ s*r*s + 2*@ s*r *s + @ s *r
t t t t t t t t
2 2 2 2
+ 3*k*s + r ))/(r *s )
pform {laglanc,inv1,inv2} = 0;
index_symmetries riemc3(k,l),riemri(k,l),
hlang(k,l),einst(k,l): symmetric;
pform {riemc3(i,j),riemri(i,j)}=0;
riemc3(-i,-j) := riemann(-i,-k,-l,-m)*riemann(-j,k,l,m)$
inv1 := riemc3(-i,-j)*g(i,j);
2 2 4 4 4 2 2 2 2 2 4
inv1 := (4*(3*@ r *r *s + 3*@ r *s + 6*@ r *@ s *r *s + 6*@ r *k*s
t t t t t t
2 4 2 4 4 2 4 2 4 4 4 4
+ 2*@ s *r *s + @ s *r + 2*@ s *r + 3*k *s + r ))/(r *s )
t t t t
riemri(-i,-j) := 2*riemann(-i,-k,-j,-l)*ricci(k,l)$
inv2 := ricci(-a,-b)*ricci(a,b);
2 2 4 2 4 2 3
inv2 := (2*(6*@ r *r *s + 6*@ r*@ r *r*s + 6*@ r*@ r*@ s*r *s
t t t t t t t t t
3 3 4 4 4
+ 6*@ r*@ s*r *s + 6*@ r*k*r*s + 6*@ r *s
t t t t t t t
3 3 2 2 2 2 2 4
+ 12*@ r *@ s*r*s + 15*@ r *@ s *r *s + 12*@ r *k*s
t t t t t
3 2 3 3 3
+ 6*@ r*@ s*@ s*r *s + 6*@ r*@ s *r *s + 12*@ r*@ s*k*r*s
t t t t t t t t
3 2 4 2 2 4
+ 6*@ r*@ s*r *s + 3*@ s *r *s + 2*@ s*@ s *r *s
t t t t t t t
4 4 4 2 4 2 4 4 4 4
+ 2*@ s*r *s + @ s *r + 2*@ s *r + 6*k *s + r ))/(r *s )
t t t t
laglanc := (1/2)*(inv1 - 4*inv2 + riccisc**2);
2 2 2 2 2
laglanc := (12*(@ r*@ r *s + 4*@ r*@ r*@ s*r*s + @ r*@ s *r + @ r*k*s
t t t t t t t t t t t t
2 3 2 2 2
+ @ r*r + 2*@ r *@ s*s + 2*@ r *@ s*r*s + 3*@ r *@ s *r
t t t t t t t t t
2 2
+ @ r *r + 2*@ r*@ s*@ s*r + 2*@ r*@ s*k*s + 2*@ s*k*r*s
t t t t t t t t t
2 3 2
+ @ s *k*r + k*r))/(r *s )
t
pform {einst(a,b),hlang(a,b)}=0;
hlang(-i,-j) := 2*(riemc3(-i,-j) - riemri(-i,-j) -
2*ricci(-i,-k)*ricci(-j,K) +
riccisc*ricci(-i,-j) - (1/2)*laglanc*g(-i,-j));
hlang :=
t t
3 2 2 2 2
12*(2*@ r *@ s*s + 3*@ r *@ s *r + @ r *r + 2*@ r*@ s*k*s + @ s *k*r + k*r)
t t t t t t t t
-----------------------------------------------------------------------------
3 2
r *s
2
hlang := (4*( - 4*@ r*@ r*@ s*s - 2*@ r*@ s *r - 2*@ r*r
1 1 t t t t t t t t t
2 2 2 2
- 2*@ r *@ s*s - 3*@ r *@ s - @ r - 4*@ r*@ s*@ s*r
t t t t t t t t t t
2 2 2
- 2*@ s*k*s - @ s *k - k))/(r *s )
t t t
2
hlang := (4*( - 4*@ r*@ r*@ s*s - 2*@ r*@ s *r - 2*@ r*r
2 2 t t t t t t t t t
2 2 2 2
- 2*@ r *@ s*s - 3*@ r *@ s - @ r - 4*@ r*@ s*@ s*r
t t t t t t t t t t
2 2 2
- 2*@ s*k*s - @ s *k - k))/(r *s )
t t t
2
hlang := (4*( - 4*@ r*@ r*@ s*s - 2*@ r*@ s *r - 2*@ r*r
3 3 t t t t t t t t t
2 2 2 2
- 2*@ r *@ s*s - 3*@ r *@ s - @ r - 4*@ r*@ s*@ s*r
t t t t t t t t t t
2 2 2
- 2*@ s*k*s - @ s *k - k))/(r *s )
t t t
2 3
hlang := (12*( - @ r*@ r *s - 2*@ r*@ r*@ s*r - @ r*k*s - @ r *@ s
4 4 t t t t t t t t t t t
2 3
- @ r *@ s*r - @ r*@ s*k - @ s*k*r))/(r *s)
t t t t t t t
2 3
hlang := (12*( - @ r*@ r *s - 2*@ r*@ r*@ s*r - @ r*k*s - @ r *@ s
5 5 t t t t t t t t t t t
2 3
- @ r *@ s*r - @ r*@ s*k - @ s*k*r))/(r *s)
t t t t t t t
% The complete Einstein tensor:
einst(-i,-j) := (ricci(-i,-j) - (1/2)*riccisc*g(-i,-j))*alp1 +
hlang(-i,-j)*alp2$
alp1 := 1$
factor alp2;
einst(-i,-j) := einst(-i,-j);
3 2 2 2
einst := (12*alp2*(2*@ r *@ s*s + 3*@ r *@ s *r + @ r *r + 2*@ r*@ s*k*s
t t t t t t t t t
2 3 2
+ @ s *k*r + k*r))/(r *s )
t
2 2 2 2 2 2
3*@ r *s + 6*@ r*@ s*r*s + @ s *r + 3*k*s + r
t t t t
+ ---------------------------------------------------
2 2
r *s
2
einst := (4*alp2*( - 4*@ r*@ r*@ s*s - 2*@ r*@ s *r - 2*@ r*r
1 1 t t t t t t t t t
2 2 2 2
- 2*@ r *@ s*s - 3*@ r *@ s - @ r - 4*@ r*@ s*@ s*r
t t t t t t t t t t
2 2 2 2
- 2*@ s*k*s - @ s *k - k))/(r *s ) + ( - 2*@ r*r*s
t t t t t
2 2 2 2 2 2 2
- @ r *s - 4*@ r*@ s*r*s - 2*@ s*r *s - @ s *r - k*s - r )/
t t t t t t
2 2
(r *s )
2
einst := (4*alp2*( - 4*@ r*@ r*@ s*s - 2*@ r*@ s *r - 2*@ r*r
2 2 t t t t t t t t t
2 2 2 2
- 2*@ r *@ s*s - 3*@ r *@ s - @ r - 4*@ r*@ s*@ s*r
t t t t t t t t t t
2 2 2 2
- 2*@ s*k*s - @ s *k - k))/(r *s ) + ( - 2*@ r*r*s
t t t t t
2 2 2 2 2 2 2
- @ r *s - 4*@ r*@ s*r*s - 2*@ s*r *s - @ s *r - k*s - r )/
t t t t t t
2 2
(r *s )
2
einst := (4*alp2*( - 4*@ r*@ r*@ s*s - 2*@ r*@ s *r - 2*@ r*r
3 3 t t t t t t t t t
2 2 2 2
- 2*@ r *@ s*s - 3*@ r *@ s - @ r - 4*@ r*@ s*@ s*r
t t t t t t t t t t
2 2 2 2
- 2*@ s*k*s - @ s *k - k))/(r *s ) + ( - 2*@ r*r*s
t t t t t
2 2 2 2 2 2 2
- @ r *s - 4*@ r*@ s*r*s - 2*@ s*r *s - @ s *r - k*s - r )/
t t t t t t
2 2
(r *s )
2 3
einst := (12*alp2*( - @ r*@ r *s - 2*@ r*@ r*@ s*r - @ r*k*s - @ r *@ s
4 4 t t t t t t t t t t t
2 3
- @ r *@ s*r - @ r*@ s*k - @ s*k*r))/(r *s)
t t t t t t t
2 2
- 3*@ r*r*s - 3*@ r *s - 3*@ r*@ s*r - @ s*r - 3*k*s
t t t t t t t
+ ------------------------------------------------------------
2
r *s
2 3
einst := (12*alp2*( - @ r*@ r *s - 2*@ r*@ r*@ s*r - @ r*k*s - @ r *@ s
5 5 t t t t t t t t t t t
2 3
- @ r *@ s*r - @ r*@ s*k - @ s*k*r))/(r *s)
t t t t t t t
2 2
- 3*@ r*r*s - 3*@ r *s - 3*@ r*@ s*r - @ s*r - 3*k*s
t t t t t t t
+ ------------------------------------------------------------
2
r *s
clear o(k),e(k),riemc3(i,j),riemri(i,j),curv(k,l),riemann(a,b,c,d),
ricci(a,b),riccisc,t,u,v1,v2,theta,phi,r,om(k,l),einst(a,b),
hlang(a,b);
remfdomain r,s;
% Problem:
% --------
% Calculate for a given coframe and given torsion the Riemannian part and
% the torsion induced part of the connection. Calculate the curvature.
% For a more elaborate example see E.Schruefer, F.W. Hehl, J.D. McCrea,
% "Application of the REDUCE package EXCALC to the Poincare gauge field
% theory of gravity", GRG Journal, vol. 19, (1988) 197--218
pform {ff, gg}=0;
fdomain ff=ff(r), gg=gg(r);
coframe o(4) = d u + 2*b0*cos(theta)*d phi,
o(1) = ff*(d u + 2*b0*cos(theta)*d phi) + d r,
o(2) = gg*d theta,
o(3) = gg*sin(theta)*d phi
with metric g = -o(4)*o(1)-o(4)*o(1)+o(2)*o(2)+o(3)*o(3);
frame e;
pform {tor(a),gwt(a)}=2,gamma(a,b)=1,
{u1,u3,u5}=0;
index_symmetries gamma(a,b): antisymmetric;
fdomain u1=u1(r),u3=u3(r),u5=u5(r);
tor(4) := 0$
tor(1) := -u5*o(4)^o(1) - 2*u3*o(2)^o(3)$
tor(2) := u1*o(4)^o(2) + u3*o(4)^o(3)$
tor(3) := u1*o(4)^o(3) - u3*o(4)^o(2)$
gwt(-a) := d o(-a) - tor(-a)$
% The following is the combined connection.
% The Riemannian part could have equally well been calculated by the
% RIEMANNCONX statement.
gamma(-a,-b) := (1/2)*( e(-b) _| (e(-c) _| gwt(-a))
+e(-c) _| (e(-a) _| gwt(-b))
-e(-a) _| (e(-b) _| gwt(-c)) )*o(c);
4
gamma := o *(@ ff - u5)
4 1 r
2
o *(@ gg*ff + gg*u1) 3 2
r o *( - b0*ff + gg *u3)
gamma := - ---------------------- + ------------------------
4 2 gg 2
gg
2
o *@ gg 3
r - o *b0
gamma := --------- + ----------
1 2 gg 2
gg
3
2 2 o *(@ gg*ff + gg*u1)
o *(b0*ff - gg *u3) r
gamma := --------------------- - ----------------------
4 3 2 gg
gg
3
2 o *@ gg
o *b0 r
gamma := ------- + ---------
1 3 2 gg
gg
1 3 4 2
o *b0 o *cos(theta) o *(b0*ff - 2*gg *u3)
gamma := ------- + --------------- + -----------------------
2 3 2 sin(theta)*gg 2
gg gg
pform curv(a,b)=2;
index_symmetries curv(a,b): antisymmetric;
factor ^;
curv(-a,b) := d gamma(-a,b) + gamma(-c,b)^gamma(-a,c);
4 2 3
curv := (2*o ^o
4
2
*(@ ff*b0*gg - 2*@ gg*b0*ff + @ gg*gg *u3 - b0*gg*u1 - b0*gg*u5))/
r r r
3 4 1
gg + o ^o *(@ ff - @ u5)
r r r
1 2 3 2
o ^o *(@ gg*gg - b0 )
4 r r 4 2 3 3
curv := -------------------------- + (o ^o *( - @ ff*@ gg*gg - @ gg*ff*gg
2 4 r r r r
gg
3 2 2 4
+ @ gg*gg *u5 - b0 *ff + 2*b0*gg *u3))/gg
r
4 3 2
o ^o *(@ ff*b0*gg - 2*@ gg*b0*ff + 2*@ gg*gg *u3 - b0*gg*u5)
r r r
+ --------------------------------------------------------------
3
gg
1 3 3 2
o ^o *(@ gg*gg - b0 )
4 r r
curv := --------------------------
3 4
gg
4 2 2
o ^o *( - @ ff*b0*gg + 2*@ gg*b0*ff - 2*@ gg*gg *u3 + b0*gg*u5)
r r r
+ -----------------------------------------------------------------
3
gg
4 3 3 3 3 2
+ (o ^o *( - @ ff*@ gg*gg - @ gg*ff*gg + @ gg*gg *u5 - b0 *ff
r r r r r
2 4
+ 2*b0*gg *u3))/gg
1 2 3
curv := (2*o ^o
1
2
*( - @ ff*b0*gg + 2*@ gg*b0*ff - @ gg*gg *u3 + b0*gg*u1 + b0*gg*u5))
r r r
3 4 1
/gg + o ^o *( - @ ff + @ u5)
r r r
1 1 2 3 3 3 4
curv := (o ^o *( - @ ff*@ gg*gg - @ gg*ff*gg - @ gg*gg *u1 - @ u1*gg
2 r r r r r r
2 2 4 1 3
- b0 *ff + b0*gg *u3))/gg + (o ^o *( - @ ff*b0*gg
r
2 3 3
+ 2*@ gg*b0*ff + @ gg*gg *u3 + @ u3*gg + b0*gg*u1))/gg + (
r r r
4 2 4 2 3 3
o ^o *( - @ ff*gg *u1 + @ gg*ff *gg + @ gg*ff*gg *u1
r r r r
3 4 2 2 2
+ @ gg*ff*gg *u5 + @ u1*ff*gg - b0 *ff + 3*b0*ff*gg *u3
r r
4 4 2 4 4 3 2
+ gg *u1*u5 - 2*gg *u3 ))/gg + (o ^o *(@ ff*gg *u3
r
2
- 3*@ gg*ff*gg*u3 - @ u3*ff*gg + b0*ff*u1 + b0*ff*u5
r r
2 2 2
- 2*gg *u1*u3 - gg *u3*u5))/gg
1 1 2
curv := (o ^o
3
2 3
*(@ ff*b0*gg - 2*@ gg*b0*ff - @ gg*gg *u3 - @ u3*gg - b0*gg*u1))/
r r r r
3 1 3 3 3 3
gg + (o ^o *( - @ ff*@ gg*gg - @ gg*ff*gg - @ gg*gg *u1
r r r r r
4 2 2 4 4 2 2
- @ u1*gg - b0 *ff + b0*gg *u3))/gg + (o ^o *( - @ ff*gg *u3
r r
2
+ 3*@ gg*ff*gg*u3 + @ u3*ff*gg - b0*ff*u1 - b0*ff*u5
r r
2 2 2 4 3 4
+ 2*gg *u1*u3 + gg *u3*u5))/gg + (o ^o *( - @ ff*gg *u1
r
2 3 3 3
+ @ gg*ff *gg + @ gg*ff*gg *u1 + @ gg*ff*gg *u5
r r r r
4 2 2 2 4
+ @ u1*ff*gg - b0 *ff + 3*b0*ff*gg *u3 + gg *u1*u5
r
4 2 4
- 2*gg *u3 ))/gg
2 1 2 3 3 3 4
curv := (o ^o *( - @ ff*@ gg*gg - @ gg*ff*gg - @ gg*gg *u1 - @ u1*gg
4 r r r r r r
2 2 4 1 3
- b0 *ff + b0*gg *u3))/gg + (o ^o *( - @ ff*b0*gg
r
2 3 3
+ 2*@ gg*b0*ff + @ gg*gg *u3 + @ u3*gg + b0*gg*u1))/gg + (
r r r
4 2 4 2 3 3
o ^o *( - @ ff*gg *u1 + @ gg*ff *gg + @ gg*ff*gg *u1
r r r r
3 4 2 2 2
+ @ gg*ff*gg *u5 + @ u1*ff*gg - b0 *ff + 3*b0*ff*gg *u3
r r
4 4 2 4 4 3 2
+ gg *u1*u5 - 2*gg *u3 ))/gg + (o ^o *(@ ff*gg *u3
r
2
- 3*@ gg*ff*gg*u3 - @ u3*ff*gg + b0*ff*u1 + b0*ff*u5
r r
2 2 2
- 2*gg *u1*u3 - gg *u3*u5))/gg
1 2 3 2
o ^o *(@ gg*gg - b0 )
2 r r 4 2 3 3
curv := -------------------------- + (o ^o *( - @ ff*@ gg*gg - @ gg*ff*gg
1 4 r r r r
gg
3 2 2 4
+ @ gg*gg *u5 - b0 *ff + 2*b0*gg *u3))/gg
r
4 3 2
o ^o *(@ ff*b0*gg - 2*@ gg*b0*ff + 2*@ gg*gg *u3 - b0*gg*u5)
r r r
+ --------------------------------------------------------------
3
gg
2 2 3
curv := (o ^o
3
2 2 3 2 2 2
*( - 2*@ gg *ff*gg - 2*@ gg*gg *u1 + 6*b0 *ff - 6*b0*gg *u3 + gg ))
r r
4 1 3
2*o ^o *(@ ff*b0*gg - 2*@ gg*b0*ff - @ u3*gg )
4 r r r
/gg + ------------------------------------------------
3
gg
3 1 2
curv := (o ^o
4
2 3
*(@ ff*b0*gg - 2*@ gg*b0*ff - @ gg*gg *u3 - @ u3*gg - b0*gg*u1))/
r r r r
3 1 3 3 3 3
gg + (o ^o *( - @ ff*@ gg*gg - @ gg*ff*gg - @ gg*gg *u1
r r r r r
4 2 2 4 4 2 2
- @ u1*gg - b0 *ff + b0*gg *u3))/gg + (o ^o *( - @ ff*gg *u3
r r
2
+ 3*@ gg*ff*gg*u3 + @ u3*ff*gg - b0*ff*u1 - b0*ff*u5
r r
2 2 2 4 3 4
+ 2*gg *u1*u3 + gg *u3*u5))/gg + (o ^o *( - @ ff*gg *u1
r
2 3 3 3
+ @ gg*ff *gg + @ gg*ff*gg *u1 + @ gg*ff*gg *u5
r r r r
4 2 2 2 4
+ @ u1*ff*gg - b0 *ff + 3*b0*ff*gg *u3 + gg *u1*u5
r
4 2 4
- 2*gg *u3 ))/gg
1 3 3 2
o ^o *(@ gg*gg - b0 )
3 r r
curv := --------------------------
1 4
gg
4 2 2
o ^o *( - @ ff*b0*gg + 2*@ gg*b0*ff - 2*@ gg*gg *u3 + b0*gg*u5)
r r r
+ -----------------------------------------------------------------
3
gg
4 3 3 3 3 2
+ (o ^o *( - @ ff*@ gg*gg - @ gg*ff*gg + @ gg*gg *u5 - b0 *ff
r r r r r
2 4
+ 2*b0*gg *u3))/gg
3 2 3
curv := (o ^o
2
2 2 3 2 2 2
*(2*@ gg *ff*gg + 2*@ gg*gg *u1 - 6*b0 *ff + 6*b0*gg *u3 - gg ))/
r r
4 1 3
2*o ^o *( - @ ff*b0*gg + 2*@ gg*b0*ff + @ u3*gg )
4 r r r
gg + ---------------------------------------------------
3
gg
clear o(k),e(k),curv(a,b),gamma(a,b),theta,phi,x,y,z,r,s,t,u,v,p,q,c,cs;
remfdomain u1,u3,u5,ff,gg;
showtime;
Time: 46882 ms plus GC time: 1061 ms
end;
Time for test: 46982 ms, plus GC time: 1061 ms