Parametergleichung links

X:=vector([R*cos(U)*cos(V),R*sin(U)*cos(V),R*sin(V)]);

Parametergleichung rechts

x:=vector([v*cos(u),v*sin(u)]);

Gaußsche Tangentenvektoren

links

XU:=vector([diff(X[1],U),diff(X[2],U),diff(X[3],U)]);

XV:=vector([diff(X[1],V),diff(X[2],V),diff(X[3],V)]);

rechts

xu:=vector([diff(x[1],u),diff(x[2],u)]);

xv:=vector([diff(x[1],v),diff(x[2],v)]);

erste Metriktensoren

links

Gl:=matrix(2,2):

Gl[1,1]:=simplify(dotprod(XU,XU,'orthogonal')):

Gl[1,2]:=simplify(dotprod(XU,XV,'orthogonal')):

Gl[2,1]:=Gl[1,2]:

Gl[2,2]:=simplify(dotprod(XV,XV,'orthogonal')):

print(Gl);

rechts

Gr:=matrix(2,2):

Gr[1,1]:=simplify(dotprod(xu,xu,'orthogonal')):

Gr[1,2]:=simplify(dotprod(xu,xv,'orthogonal')):

Gr[2,1]:=Gl[1,2]:

Gr[2,2]:=simplify(dotprod(xv,xv,'orthogonal')):

print(Gr);

Abbildungsgleichungen

fu:=U;

fv:=2*R*tan(Pi/4-V/2);

Linke Jacobimatrix

Jl:=matrix(2,2):

Jl[1,1]:=simplify(diff(fu,U)):

Jl[1,2]:=simplify(diff(fu,V)):

Jl[2,1]:=simplify(diff(fv,U)):

Jl[2,2]:=simplify(diff(fv,V)):

print(Jl);

Rechte Jacobimatrix incomplete! substitution muss angepasst werden. Besser isses, Jacobi so aufzustellen.

#Jr:=inverse(Jl);

#for i from 1 to 2 do

#for j from 1 to 2 do

#Gr[i,j]:=subs(v=2*R*tan(Pi/4-V/2),Gr[i,j]);

#od;

#od;

Cauchy Green Tensor

#for i from 1 to 2 do

#for j from 1 to 2 do

Gr[i,j]:=subs(v=2*R*tan(Pi/4-V/2),Gr[i,j]);

#od;

#od;

print(Gr);

Cl:=evalm(transpose(Jl)&*Gr&*Jl);

Hauptverzerrungen (Eigenwerte) bezüglich links

lambda[1]:=sqrt(1/2*(trace(evalm(Cl&*inverse(Gl))) + sqrt(trace(evalm(Cl&*inverse(Gl)))^2 - 4*det(evalm(Cl&*inverse(Gl))))));

lambda[2]:=sqrt(1/2*(trace(evalm(Cl&*inverse(Gl))) - sqrt(trace(evalm(Cl&*inverse(Gl)))^2 - 4*det(evalm(Cl&*inverse(Gl))))));

Eigenvektoren

EV[1]:=simplify(vector([(Cl[2,2]-lambda[1]^2*Gl[2,2])/sqrt( (Cl[2,2]-lambda[1]^2*Gl[2,2])^2*Gl[1,1] - 2*(Cl[1,2]-lambda[1]^2*Gl[1,2])*(Cl[2,2]-lambda[1]^2*Gl[2,2])*Gl[1,2] +  (Cl[1,2]-lambda[1]^2*Gl[1,2])^2*Gl[2,2] ) , -(Cl[1,2]-lambda[1]^2*Gl[1,2])/sqrt( (Cl[2,2]-lambda[1]^2*Gl[2,2])^2*Gl[1,1] - 2*(Cl[1,2]-lambda[1]^2*Gl[1,2])*(Cl[2,2]-lambda[1]^2*Gl[2,2])*Gl[1,2] +  (Cl[1,2]-lambda[1]^2*Gl[1,2])^2*Gl[2,2] )]));

EV[2]:=simplify(vector([-(Cl[1,2]-lambda[2]^2*Gl[1,2])/sqrt( (Cl[1,1]-lambda[2]^2*Gl[1,1])^2*Gl[2,2] - 2*(Cl[1,2]-lambda[2]^2*Gl[1,2])*(Cl[1,1]-lambda[2]^2*Gl[1,1])*Gl[1,2] +  (Cl[1,2]-lambda[2]^2*Gl[1,2])^2*Gl[1,1] ), (Cl[1,1]-lambda[2]^2*Gl[1,1])/sqrt( (Cl[1,1]-lambda[2]^2*Gl[1,1])^2*Gl[2,2] - 2*(Cl[1,2]-lambda[2]^2*Gl[1,2])*(Cl[1,1]-lambda[2]^2*Gl[1,1])*Gl[1,2] +  (Cl[1,2]-lambda[2]^2*Gl[1,2])^2*Gl[1,1] )]));