restart: with (linalg):with(liesymm):with(difforms):

setup(x,y,z,t);defform(u=0,v=0,s=0,n=0,x=0,y=0,z=0,t=0,v=0,V=0,S=0,U=0,P=0,T=0,G=0,v1=0,v2=0,s1=0,s2=0,u1=0,u2=0,P=0,T=0,Vx=0,Vy=0,Vz=0,D1=0,D2=0,D3=0,Ax=0,Ay=0,Az=0,C=0,Phi=0,phi=0,a=const,b=0,c=const,q=const,Lx=0,Ly=0,Lz=0,E1=0,E2=0,E3=0,ACT=0,FF=0);

NyZJInhHNiJJInlHRiRJInpHRiRJInRHRiQ=

Omega:=d(x)&^d(y)&^d(z);V:=grad(alpha(x,y,z),[x,y,z]);W:=grad(beta(x,y,z),[x,y,z]);

LUkjJl5HNiI2JS1JImRHRiQ2I0kieEdGJC1GJzYjSSJ5R0YkLUYnNiNJInpHRiQ=

PTYiNiM7IiIiIiIkRVxbbCRGJi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSZhbHBoYUdGIzYlSSJ4R0YjSSJ5R0YjSSJ6R0YjRjAiIiMtRio2JEYtRjFGJy1GKjYkRi1GMg==

PTYiNiM7IiIiIiIkRVxbbCRGJi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSViZXRhR0YjNiVJInhHRiNJInlHRiNJInpHRiNGMCIiIy1GKjYkRi1GMUYnLUYqNiRGLUYy

JV:=V[1]*d(y)&^d(z)-V[2]*d(x)&^d(z)+V[3]*d(x)&^d(y);

LCgqJi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSZhbHBoYUc2IjYlSSJ4R0YqSSJ5R0YqSSJ6R0YqRiwiIiItSSMmXkdGKjYkLUkiZEdGKjYjRi0tRjQ2I0YuRi9GLyomLUYlNiRGKEYtRi8tRjE2JC1GNDYjRixGNkYvISIiKiYtRiU2JEYoRi5GLy1GMTYkRj1GM0YvRi8=

JVW:=wcollect(W[2]*V[1]*&^d(z)-W[3]*V[2]*d(y) -W[1]*V[2]*d(z)+W[3]*W[2]*d(x)+W[1]*V[3]*d(y)-W[2]*V[3]*d(x));

LCgqJiwmKiYtSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUklYmV0YUc2IjYlSSJ4R0YsSSJ5R0YsSSJ6R0YsRjAiIiItRic2JEYqRi9GMUYxKiZGMkYxLUYnNiQtSSZhbHBoYUdGLEYtRjBGMSEiIkYxLUkiZEdGLDYjRi5GMUYxKiYsJiomRiZGMS1GJzYkRjdGL0YxRjkqJi1GJzYkRipGLkYxRjVGMUYxRjEtRjs2I0YvRjFGMSomLCYqJkYyRjEtRic2JEY3Ri5GMUYxKiZGQ0YxRkBGMUY5RjEtRjs2I0YwRjFGMQ==

V:=[V1,V2,V3,V4];A:=[Ax(x,y,z,t),Ay(x,y,z,t),Az(x,y,z,t),-phi(x,y,z,t)];setup(x,y,z,t);

NyZJI1YxRzYiSSNWMkdGJEkjVjNHRiRJI1Y0R0Yk

NyYtSSNBeEc2IjYmSSJ4R0YlSSJ5R0YlSSJ6R0YlSSJ0R0YlLUkjQXlHRiVGJi1JI0F6R0YlRiYsJC1JJHBoaUdGJUYmISIi

NyZJInhHNiJJInlHRiRJInpHRiRJInRHRiQ=

R:=[x,y,z,t];d(R);d(A[1]);ACT:=A[1]*d(x)+A[2]*d(y)+A[3]*d(z)+A[4]*d(t);FF:=d(ACT);

NyZJInhHNiJJInlHRiRJInpHRiRJInRHRiQ=

NyYtSSJkRzYiNiNJInhHRiUtRiQ2I0kieUdGJS1GJDYjSSJ6R0YlLUYkNiNJInRHRiU=

LCoqJi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSNBeEc2IjYmSSJ4R0YqSSJ5R0YqSSJ6R0YqSSJ0R0YqRi8iIiItSSJkR0YqNiNGL0YwRjAqJi1GJTYkRihGLEYwLUYyNiNGLEYwRjAqJi1GJTYkRihGLUYwLUYyNiNGLUYwRjAqJi1GJTYkRihGLkYwLUYyNiNGLkYwRjA=

LCoqJi1JI0F4RzYiNiZJInhHRiZJInlHRiZJInpHRiZJInRHRiYiIiItSSJkR0YmNiNGKEYsRiwqJi1JI0F5R0YmRidGLC1GLjYjRilGLEYsKiYtSSNBekdGJkYnRiwtRi42I0YqRixGLComLUkkcGhpR0YmRidGLC1GLjYjRitGLCEiIg==

LC4qJiwmLUklZGlmZkclKnByb3RlY3RlZEc2JC1JI0F4RzYiNiZJInhHRitJInlHRitJInpHRitJInRHRitGMCIiIi1GJjYkLUkkcGhpR0YrRixGLUYxRjEtSSMmXkdGKzYkLUkiZEdGKzYjRjAtRjo2I0YtRjFGMSomLCYtRiY2JC1JI0F5R0YrRixGMEYxLUYmNiRGNEYuRjFGMS1GNzYkRjktRjo2I0YuRjFGMSomLCYtRiY2JC1JI0F6R0YrRixGMEYxLUYmNiRGNEYvRjFGMS1GNzYkRjktRjo2I0YvRjFGMSomLCYtRiY2JEYpRi5GMS1GJjYkRkJGLSEiIkYxLUY3NiRGSEY8RjFGMSomLCYtRiY2JEYpRi9GMS1GJjYkRk5GLUZmbkYxLUY3NiRGVEY8RjFGMSomLCYtRiY2JEZCRi9GMS1GJjYkRk5GLkZmbkYxLUY3NiRGVEZIRjFGMQ==

Ex:=diff(Ax(x, y, z, t), t)+diff(phi(x, y, z, t), x);

LCYtSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUkjQXhHNiI2JkkieEdGKUkieUdGKUkiekdGKUkidEdGKUYuIiIiLUYkNiQtSSRwaGlHRilGKkYrRi8=

FFe:=-Exx*d(t)&^d(x)-Eyy*d(t)&^d(y)-Ezz&^d(t)&^d(z);FFb:=Bzz*d(x)&^d(y)+Bxx*d(y)&^d(z)+Byy*d(z)&^d(x);

LCgqJkkkRXh4RzYiIiIiLUkjJl5HRiU2JC1JImRHRiU2I0kidEdGJS1GKzYjSSJ4R0YlRiYhIiIqJkkkRXl5R0YlRiYtRig2JEYqLUYrNiNJInlHRiVGJkYxLUYoNiVJJEV6ekdGJUYqLUYrNiNJInpHRiVGMQ==

LCgqJkkkQnp6RzYiIiIiLUkjJl5HRiU2JC1JImRHRiU2I0kieEdGJS1GKzYjSSJ5R0YlRiZGJiomSSRCeHhHRiVGJi1GKDYkRi4tRis2I0kiekdGJUYmRiYqJkkkQnl5R0YlRiYtRig2JEY1RipGJkYm

`i(V)dAe`:=wcollect(-Exx*V[4]*d(x)+Exx*V[1]*d(t)-Eyy*V[4]*d(y)+Eyy*V[2]*d(t)-Ezz*V[4]*d(z)+Ezz*V[3]*d(t));

LCoqJiwoKiZJJEV4eEc2IiIiIkkjVjFHRidGKEYoKiZJJEV5eUdGJ0YoSSNWMkdGJ0YoRigqJkkkRXp6R0YnRihJI1YzR0YnRihGKEYoLUkiZEdGJzYjSSJ0R0YnRihGKCooRiZGKEkjVjRHRidGKC1GMTYjSSJ4R0YnRighIiIqKEYrRihGNUYoLUYxNiNJInlHRidGKEY5KihGLkYoRjVGKC1GMTYjSSJ6R0YnRihGOQ==

`i(V)dAb`:=wcollect(Bzz*V[1]*d(y)-Bzz*V[2]*d(x)+Bxx*V[2]*d(z)-Bxx*V[3]*d(y)+Byy*V[3]*d(x)-Byy*V[1]*d(z));

LCgqJiwmKiZJJEJ5eUc2IiIiIkkjVjNHRidGKEYoKiZJJEJ6ekdGJ0YoSSNWMkdGJ0YoISIiRigtSSJkR0YnNiNJInhHRidGKEYoKiYsJiomRitGKEkjVjFHRidGKEYoKiZJJEJ4eEdGJ0YoRilGKEYtRigtRi82I0kieUdGJ0YoRigqJiwmKiZGJkYoRjVGKEYtKiZGN0YoRixGKEYoRigtRi82I0kiekdGJ0YoRig=

FFeb:=wcollect(`i(V)dAe`+`i(V)dAb`);

LCoqJiwoKiZJJEV4eEc2IiIiIkkjVjFHRidGKEYoKiZJJEV5eUdGJ0YoSSNWMkdGJ0YoRigqJkkkRXp6R0YnRihJI1YzR0YnRihGKEYoLUkiZEdGJzYjSSJ0R0YnRihGKComLCgqJkkkQnl5R0YnRihGL0YoRigqJkkkQnp6R0YnRihGLEYoISIiKiZGJkYoSSNWNEdGJ0YoRjpGKC1GMTYjSSJ4R0YnRihGKComLCgqJkY5RihGKUYoRigqJkkkQnh4R0YnRihGL0YoRjoqJkYrRihGPEYoRjpGKC1GMTYjSSJ5R0YnRihGKComLCgqJkY3RihGKUYoRjoqJkZERihGLEYoRigqJkYuRihGPEYoRjpGKC1GMTYjSSJ6R0YnRihGKA==

subs(Exx*V1+Eyy*V2+Ezz*V3=`EdotB`,Byy*V3-Bzz*V2-Exx*V4=`-{V4*E+(V X B)}x`,FFeb);

LCoqJkkmRWRvdEJHNiIiIiItSSJkR0YlNiNJInRHRiVGJkYmKiZJMS18ZnJWNCpFKyhWflh+Qil8aHJ4R0YlRiYtRig2I0kieEdGJUYmRiYqJiwoKiZJJEJ6ekdGJUYmSSNWMUdGJUYmRiYqJkkkQnh4R0YlRiZJI1YzR0YlRiYhIiIqJkkkRXl5R0YlRiZJI1Y0R0YlRiZGOEYmLUYoNiNJInlHRiVGJkYmKiYsKComSSRCeXlHRiVGJkY0RiZGOComRjZGJkkjVjJHRiVGJkYmKiZJJEV6ekdGJUYmRjtGJkY4RiYtRig2I0kiekdGJUYmRiY=

FF:=d(ACT);

LCgqJi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSNBeEc2IjYmSSJ4R0YqSSJ5R0YqSSJ6R0YqSSJ0R0YqRi8iIiItSSMmXkdGKjYkLUkiZEdGKjYjRi8tRjU2I0YsRjBGMComLUYlNiRGKEYtRjAtRjI2JC1GNTYjRi1GN0YwRjAqJi1GJTYkRihGLkYwLUYyNiQtRjU2I0YuRjdGMEYw

hook(FF,V);

Error, (in hook) case not handled d(Ax(x, y, z, t)), Ax(x, y, z, t)

A0:=d(G);

LC4qJkkiU0c2IiIiIi1JImRHRiU2I0kiVEdGJUYmISIiKiZGKkYmLUYoNiNGJEYmRisqJkkiVkdGJUYmLUYoNiNJIlBHRiVGJkYmKiZGM0YmLUYoNiNGMEYmRiYqJkkiVUdGJUYmLUYoNiNJIm5HRiVGJkYrKiZGO0YmLUYoNiNGOEYmRis=

d(A0);

IiIh

Gibbs:=P*d(V)-T*d(S)-n*d(U);

LCgqJkkiUEc2IiIiIi1JImRHRiU2I0kiVkdGJUYmRiYqJkkiVEdGJUYmLUYoNiNJIlNHRiVGJiEiIiomSSJuR0YlRiYtRig2I0kiVUdGJUYmRjA=

Energy:=Vd(P)-Ud(n)-S(dT);

LCgtSSNWZEc2IjYjSSJQR0YlIiIiLUkjVWRHRiU2I0kibkdGJSEiIi1JIlNHRiU2I0kjZFRHRiVGLQ==

d(Gibbs);

LCgtSSMmXkc2IjYkLUkiZEdGJTYjSSJQR0YlLUYoNiNJIlZHRiUiIiItRiQ2JC1GKDYjSSJUR0YlLUYoNiNJIlNHRiUhIiItRiQ2JC1GKDYjSSJuR0YlLUYoNiNJIlVHRiVGNw==

Gibbs&^d(Gibbs);

LC4qJkkiUEc2IiIiIi1JIyZeR0YlNiUtSSJkR0YlNiNJIlZHRiUtRis2I0kiVEdGJS1GKzYjSSJTR0YlRiYhIiIqJkYkRiYtRig2JUYqLUYrNiNJIm5HRiUtRis2I0kiVUdGJUYmRjQqJkYwRiYtRig2JUYxLUYrNiNGJEYqRiZGNComRjBGJi1GKDYlRjFGOEY7RiZGJiomRjpGJi1GKDYlRjtGQUYqRiZGNComRjpGJi1GKDYlRjtGLkYxRiZGJg==

d(Gibbs)&^d(Gibbs);

LCgqJiIiIyIiIi1JIyZeRzYiNiYtSSJkR0YoNiNJIlBHRigtRis2I0kiVkdGKC1GKzYjSSJUR0YoLUYrNiNJIlNHRihGJSEiIiomRiRGJS1GJzYmRipGLi1GKzYjSSJuR0YoLUYrNiNJIlVHRihGJUY3KiZGJEYlLUYnNiZGMUY0RjtGPkYlRiU=

Action1:=simplify((P*d(V)-T*d(S)-n*d(U)));

LCgqJkkiUEc2IiIiIi1JImRHRiU2I0kiVkdGJUYmRiYqJkkiVEdGJUYmLUYoNiNJIlNHRiVGJiEiIiomLUYoNiNJIlVHRiVGJkkibkdGJUYmRjA=

FA1:=d(Action1);

LCgtSSMmXkc2IjYkLUkiZEdGJTYjSSJQR0YlLUYoNiNJIlZHRiUiIiItRiQ2JC1GKDYjSSJUR0YlLUYoNiNJIlNHRiUhIiItRiQ2JC1GKDYjSSJuR0YlLUYoNiNJIlVHRiVGNw==

Action1&^FA1;

FA1&^FA1;

LCgqJiIiIyIiIi1JIyZeRzYiNiYtSSJkR0YoNiNJIlBHRigtRis2I0kiVkdGKC1GKzYjSSJUR0YoLUYrNiNJIlNHRihGJSEiIiomRiRGJS1GJzYmRipGLi1GKzYjSSJuR0YoLUYrNiNJIlVHRihGJUY3KiZGJEYlLUYnNiZGMUY0RjtGPkYlRiU=

Action1&^FA1&^FA1;

LCgqKCIiIyIiIkkiVEc2IkYlLUkjJl5HRic2Jy1JImRHRic2I0kiU0dGJy1GLDYjSSJQR0YnLUYsNiNJIlZHRictRiw2I0kibkdGJy1GLDYjSSJVR0YnRiVGJSooRiRGJUY3RiUtRik2J0Y4Ri9GMi1GLDYjRiZGK0YlRiUqKEYkRiVGMUYlLUYpNidGMkY+RitGNUY4RiVGJQ==

FA1&^FA1&^FA1;

LCQqJiIiJyIiIi1JIyZeRzYiNigtSSJkR0YoNiNJIlBHRigtRis2I0kiVkdGKC1GKzYjSSJUR0YoLUYrNiNJIlNHRigtRis2I0kibkdGKC1GKzYjSSJVR0YoRiVGJQ==

3 independent functions. 1 triple, 3 pairs, 3 singles, 1 null 1-3+3-1 van der walls

1,5,4,

Action2:=simplify(P*d(v1+v2)-T*d(s1+s2)-d(u1+u2));

LC4qJkkiUEc2IiIiIi1JImRHRiU2I0kjdjFHRiVGJkYmKiZGJEYmLUYoNiNJI3YyR0YlRiZGJiomSSJUR0YlRiYtRig2I0kjczFHRiVGJiEiIiomRjBGJi1GKDYjSSNzMkdGJUYmRjQtRig2I0kjdTFHRiVGNC1GKDYjSSN1MkdGJUY0

FA2:=simplify(d(Action2));

LCotSSMmXkc2IjYkLUkiZEdGJTYjSSJQR0YlLUYoNiNJI3YxR0YlIiIiLUYkNiRGJy1GKDYjSSN2MkdGJUYuLUYkNiQtRig2I0kiVEdGJS1GKDYjSSNzMUdGJSEiIi1GJDYkRjYtRig2I0kjczJHRiVGPA==

simplify(Action2&^FA2);

LEIqJkkiVEc2IiIiIi1JIyZeR0YlNiUtSSJkR0YlNiNJI3MxR0YlLUYrNiNJIlBHRiUtRis2I0kjdjFHRiVGJiEiIiomRiRGJi1GKDYlRipGLi1GKzYjSSN2MkdGJUYmRjQqJkYkRiYtRig2JS1GKzYjSSNzMkdGJUYuRjFGJkY0KiZGJEYmLUYoNiVGPkYuRjhGJkY0LUYoNiUtRis2I0kjdTFHRiVGLkYxRjQtRig2JUZGRi5GOEY0LUYoNiVGRi1GKzYjRiRGKkYmLUYoNiVGRkZNRj5GJi1GKDYlLUYrNiNJI3UyR0YlRi5GMUY0LUYoNiVGU0YuRjhGNC1GKDYlRlNGTUYqRiYtRig2JUZTRk1GPkYmKiZGMEYmLUYoNiVGMUZNRipGJkY0KiZGMEYmLUYoNiVGMUZNRj5GJkY0KiZGMEYmLUYoNiVGOEZNRipGJkY0KiZGMEYmLUYoNiVGOEZNRj5GJkY0

simplify(FA2&^FA2);

LCoqJiIiIyIiIi1JIyZeRzYiNiYtSSJkR0YoNiNJIlBHRigtRis2I0kjdjFHRigtRis2I0kiVEdGKC1GKzYjSSNzMUdGKEYlISIiKiZGJEYlLUYnNiZGKkYuRjEtRis2I0kjczJHRihGJUY3KiZGJEYlLUYnNiZGKi1GKzYjSSN2MkdGKEYxRjRGJUY3KiZGJEYlLUYnNiZGKkZBRjFGO0YlRjc=

Action2&^FA2&^FA2;

LDIqJiIiIyIiIi1JIyZeRzYiNictSSJkR0YoNiNJI3UxR0YoLUYrNiNJIlBHRigtRis2I0kjdjFHRigtRis2I0kiVEdGKC1GKzYjSSNzMUdGKEYlRiUqJkYkRiUtRic2J0YqRi5GMUY0LUYrNiNJI3MyR0YoRiVGJSomRiRGJS1GJzYnRipGLi1GKzYjSSN2MkdGKEY0RjdGJUYlKiZGJEYlLUYnNidGKkYuRkNGNEY9RiVGJSomRiRGJS1GJzYnLUYrNiNJI3UyR0YoRi5GMUY0RjdGJUYlKiZGJEYlLUYnNidGTEYuRjFGNEY9RiVGJSomRiRGJS1GJzYnRkxGLkZDRjRGN0YlRiUqJkYkRiUtRic2J0ZMRi5GQ0Y0Rj1GJUYl

simplify(FA2&^FA2&^FA2);

IiIh

So for 2 components, the max Pfaff dimension is 5, not six.!!! 2N(components)+1

1 5intersections, 10 doubles,

u=U/n,v=V/n,s=S/n, are per mole extensive quantities. P and T are intensive,

Action:=simplify((1*P(v,s,u)*d(v)-T(v,s,u)*d(s)-d(u)+(1*T(v,s,u)*d(u)-1*u*d(T(v,s,u)))/(u^2-T(v,s,u)^2)));

FF:=d(Action);

H:=Action&^FF;

JSFH

FF&^FF;

***

Equilibrium then implies PTD(Action)=1

there for FF =0 (no limit points)

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0 , 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

AND the constraint 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

Hence T and P are not dependent upon u,

and P could be dependent upon s, and T could be dependent on v if the constraint applies,

P is independent from sn = constant or d(n)=0

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=0

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkmbWZyYWNHRiQ2KC1GIzYkLUkjbW9HRiQ2LVErJlBhcnRpYWxEO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOS8lKXN0cmV0Y2h5R0Y5LyUqc3ltbWV0cmljR0Y5LyUobGFyZ2VvcEdGOS8lLm1vdmFibGVsaW1pdHNHRjkvJSdhY2NlbnRHRjkvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZIRjQtRiM2Ji1JI21pR0YkNiNRIUYnLUYjNiVGMC1GTjYlUSJ2RicvJSdpdGFsaWNHUSV0cnVlRicvRjVRJ2l0YWxpY0YnRjRGTUY0LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Zqbi8lKWJldmVsbGVkR0Y5LUYxNi1RMSZJbnZpc2libGVUaW1lcztGJ0Y0RjdGOkY8Rj5GQEZCRkRGRkZJLUYjNiYtRk42JVEiVEYnRlZGWS1GMTYtUTAmQXBwbHlGdW5jdGlvbjtGJ0Y0RjdGOkY8Rj5GQEZCRkRGRkZJLUkobWZlbmNlZEdGJDYkLUYjNihGUy1GMTYtUSIsRidGNEY3L0Y7RlhGPEY+RkBGQkZERkYvRkpRLDAuMzMzMzMzM2VtRictRk42JVEic0YnRlZGWUZfcC1GTjYlUSJ1RidGVkZZRjRGNEY0Rk1GNA===0

THIS IS THE FIXED POINT !??!!

****

Isolated implies PTD(Action) = 2,

therefor H=0, there is no constraint on n !!!!!

Two Possibilities:

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=0.

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

****

Closed means that

*********************

wcollect(simplify(Action));

AV:=simplify([0*P(v,s,u),-T(v,s,u)-u(T(v,s,u)/(u^2-T(v,s,u)^2)),-1+T(v,s,u)/(u^2-T(v,s,u)^2)]);AA:=innerprod(AV,[d(v),d(s),d(u)]);Action;

JAV:=array(jacobian(AV,[v,s,u]));

Xm:=simplify(trace(JAV));

Za:=det(JAV);

Yg:=trace(adjoint(JAV));

collect(charpoly(JAV,rho),rho);