(* Surface Area Template *)

(* F[u_,v_]:=r*Cos[v]*Cos[u]; *)   (* Definition of Coordinate Mapping for Sphere *)
(* G[u_,v_]:=r*Cos[v]*Sin[u]; *)
(* H[u_,v_]:=r*Sin[v]; *)

F[u_,v_]:=a*Cos[v]*Cos[u];    (* Definition of Coordinate Mapping for Ellipsoid *)
G[u_,v_]:=b*Cos[v]*Sin[u]; 
H[u_,v_]:=c*Sin[v];       

(* F[u_,v_]:=(s+r*Cos[v])*Cos[u]; *)      (* Definition of Coordinate Mapping for Torus *)
(* G[u_,v_]:=(s+r*Cos[v])*Sin[u]; *)   
(* H[u_,v_]:=r*Sin[v]; *)  

(* F[u_,v_]:=f[u]*Cos[v]; *)	    (* Definition of Coordinate Mapping for Cylindrical *)
(* G[u_,v_]:=f[u]*Sin[v]; *)		(* Surface with Profile Functions f[u] and g[u] *) 
(* H[u_,v_]:=g[u]; *)

(* F[u_,v_]:=; *)   (* General Coordinate Mapping *)
(* G[u_,v_]:=; *)
(* H[u_,v_]:=; *)

Fu[u_,v_]:=D[F[u,v],u];		(* Jacobian of Coordinate Mapping *)
Fv[u_,v_]:=D[F[u,v],v];
Gu[u_,v_]:=D[G[u,v],u];
Gv[u_,v_]:=D[G[u,v],v];
Hu[u_,v_]:=D[H[u,v],u];
Hv[u_,v_]:=D[H[u,v],v];

FFFuu[u_,v_]:=Fu[u,v]*Fu[u,v]+Gu[u,v]*Gu[u,v]+Hu[u,v]*Hu[u,v];	(* First Fundamental Form *)
FFFuv[u_,v_]:=Fu[u,v]*Fv[u,v]+Gu[u,v]*Gv[u,v]+Hu[u,v]*Hv[u,v];
FFFvu[u_,v_]:=Fv[u,v]*Fu[u,v]+Gv[u,v]*Gu[u,v]+Hv[u,v]*Hu[u,v];
FFFvv[u_,v_]:=Fv[u,v]*Fv[u,v]+Gv[u,v]*Gv[u,v]+Hv[u,v]*Hv[u,v];

DD[u_,v_]:=FFFuu[u,v]*FFFvv[u,v]-FFFvu[u,v]*FFFuv[u,v]  (* Determinant *)

SD[u_,v_]:=Sqrt[DD[u,v]]  (* Surface Area Density *)

TWW = SD[u, v]

Simplify[TWW]

FullSimplify[%]