Extatic Determinant

Flow Curvature Manifold Equation

L1 = {x,y,z,(x^2),(y^2),(z^2),x y , x z, y z}

Lagutinskii Determinant

L2=FullSimplify[Dt[L1]/.{Dt[x]->f[x,y,z],Dt[y]->g[x,y,z],Dt[z]->h[x,y,z],Dt[ϵ]->0,Dt[(Subscript[c, 1])]->0,Dt[(Subscript[c, 2])]->0,Dt[μ]->0,Dt[a]->0,Dt[b]->0}]

L3=FullSimplify[Dt[L2]/.{Dt[x]->f[x,y,z],Dt[y]->g[x,y,z],Dt[z]->h[x,y,z],Dt[ϵ]->0,Dt[(Subscript[c, 1])]->0,Dt[(Subscript[c, 2])]->0,Dt[μ]->0,Dt[a]->0,Dt[b]->0}];

L4=Factor[Dt[L3]/.{Dt[x]->f[x,y,z],Dt[y]->g[x,y,z],Dt[z]->h[x,y,z],Dt[ϵ]->0,Dt[(Subscript[c, 1])]->0,Dt[(Subscript[c, 2])]->0,Dt[μ]->0,Dt[a]->0,Dt[b]->0}];

L5=Factor[Dt[L4]/.{Dt[x]->f[x,y,z],Dt[y]->g[x,y,z],Dt[z]->h[x,y,z],Dt[ϵ]->0,Dt[(Subscript[c, 1])]->0,Dt[(Subscript[c, 2])]->0,Dt[μ]->0,Dt[a]->0,Dt[b]->0}];

L6=Factor[Dt[L5]/.{Dt[x]->f[x,y,z],Dt[y]->g[x,y,z],Dt[z]->h[x,y,z],Dt[ϵ]->0,Dt[(Subscript[c, 1])]->0,Dt[(Subscript[c, 2])]->0,Dt[μ]->0,Dt[a]->0,Dt[b]->0}];

L7=Factor[Dt[L6]/.{Dt[x]->f[x,y,z],Dt[y]->g[x,y,z],Dt[z]->h[x,y,z],Dt[ϵ]->0,Dt[(Subscript[c, 1])]->0,Dt[(Subscript[c, 2])]->0,Dt[μ]->0,Dt[a]->0,Dt[b]->0}];

L8=Factor[Dt[L7]/.{Dt[x]->f[x,y,z],Dt[y]->g[x,y,z],Dt[z]->h[x,y,z],Dt[ϵ]->0,Dt[(Subscript[c, 1])]->0,Dt[(Subscript[c, 2])]->0,Dt[μ]->0,Dt[a]->0,Dt[b]->0}];

L9=Factor[Dt[L8]/.{Dt[x]->f[x,y,z],Dt[y]->g[x,y,z],Dt[z]->h[x,y,z],Dt[ϵ]->0,Dt[(Subscript[c, 1])]->0,Dt[(Subscript[c, 2])]->0,Dt[μ]->0,Dt[a]->0,Dt[b]->0}];

K={L1,L2,L3,L4,L5,L6,L7,L8,L9};

MatrixForm[K];

Factor[Det[K]]