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REDUCE 3.6, 15-Jul-95, patched to 6 Mar 96 ... *** ^ 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 @ f*e @ f*e @ f*e 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: 28540 ms plus GC time: 1791 ms end; (TIME: excalc 28560 30351)