with(plots):with(LinearAlgebra):

EFG:=proc(X)
> local Xu,Xv,E,F,G;
> Xu:=<diff(X[1],u),diff(X[2],u),diff(X[3],u)>;
> Xv:=<diff(X[1],v),diff(X[2],v),diff(X[3],v)>;
> E:=DotProduct(Xu,Xu,conjugate=false);
> F:=DotProduct(Xu,Xv,conjugate=false);
> G:=DotProduct(Xv,Xv,conjugate=false);
> simplify([E,F,G]);
> end:

UN:=proc(X)
> local Xu,Xv,Z,s;
> Xu:=<diff(X[1],u),diff(X[2],u),diff(X[3],u)>;
> Xv:=<diff(X[1],v),diff(X[2],v),diff(X[3],v)>;
> Z:=CrossProduct(Xu,Xv);
> s:=VectorNorm(Z,Euclidean,conjugate=false);
> simplify(<Z[1]/s|Z[2]/s|Z[3]/s>,sqrt,trig,symbolic);
> end:

lmn:=proc(X)
> local Xu,Xv,Xuu,Xuv,Xvv,U,l,m,n;
> Xu:=<diff(X[1],u),diff(X[2],u),diff(X[3],u)>;
> Xv:=<diff(X[1],v),diff(X[2],v),diff(X[3],v)>;
> Xuu:=<diff(Xu[1],u),diff(Xu[2],u),diff(Xu[3],u)>;
> Xuv:=<diff(Xu[1],v),diff(Xu[2],v),diff(Xu[3],v)>;
> Xvv:=<diff(Xv[1],v),diff(Xv[2],v),diff(Xv[3],v)>;
> U:=UN(X);
> l:=DotProduct(U,Xuu,conjugate=false);
> m:=DotProduct(U,Xuv,conjugate=false);
> n:=DotProduct(U,Xvv,conjugate=false);
> simplify([l,m,n],sqrt,trig,symbolic);
> end:

MK:=proc(X)
> local E,F,G,l,m,n,S,T;
> S:=EFG(X);T:=lmn(X);
> E:=S[1];F:=S[2];G:=S[3];
> l:=T[1];m:=T[2];n:=T[3];
> simplify((G*l+E*n-2*F*m)/(2*E*G-2*F^2),sqrt,trig,symbolic);
> end:

scherk1:=<u|v|ln(cos(v)/cos(u))>;

MK(scherk1);

plot3d(scherk1,u=-1.57..1.57,v=-1.57..1.57,shading=xy,
> lightmodel=light4,orientation=[51,66],grid=[20,20]);