Curvatura de Gauss y curvatura media de una superficie

Parametrización de la superficie

X[u_, v_]:=

Primera forma fundamental

EE[X_][u_, v_] := 
      Simplify[D[X[x, y], x] . D[X[x, y], x] /. {x -> u, 
            y -> v}]; FF[X_][u_, v_] := 
      Simplify[D[X[x, y], x] . D[X[x, y], y] /. {x -> u, 
            y -> v}];
GG[X_][u_, v_] := 
        Simplify[
          D[X[x, y], y] . D[X[x, y], y] /. {x -> u, 
              y -> v}];
W[X_][u_, v_] := 
      Simplify[EE[X][u, v] GG[X][u, v] - FF[X][u, 
              v]^2]

Segunda forma fundamental

ee[X_][u_, v_] := 
        Simplify[
          Det[{D[X[x, y], x], D[X[x, y], y], 
                D[D[X[x, y], x], x]}] /. {x -> u, 
              y -> v}];
ff[X_][u_, v_] := 
        Simplify[
          Det[{D[X[x, y], x], D[X[x, y], y], 
                D[D[X[x, y], x], y]}] /. {x -> u, 
              y -> v}];
gg[X_][u_, v_] := 
        Simplify[
          Det[{D[X[x, y], x], D[X[x, y], y], 
                D[D[X[x, y], y], y]}] /. {x -> u, 
              y -> v}];
e[X_][u_, v_] := ee[X][u, v]/
          Sqrt[W[X][u, v]];
f[X_][u_, v_] := ff[X][u, v]/
          Sqrt[W[X][u, v]];
g[X_][u_, v_] := gg[X][u, v]/Sqrt[W[X][u, v]]

Curvatura de Gauss de la superficie en X[u,v]

K[X_][u_, v_] := 
      Simplify[(ee[X][u, v] gg[X][u, v] - ff[X][u, v]^2)/W[
                X][u, v]^2]

Curvatura media de la superficie en X[u,v]

H[X_][u_, v_] := 
      Simplify[(e[X][u, v] GG[X][u, v] - 
              2  f[X][u, v] FF[X][u, v] + g[X][u, v] EE[X][
                  u, v])/(2  W[X][u, v])]

Gráfica de la curvatura de Gauss y curvatura media

u0=; u1= ; v0=; v1= ;
ParametricPlot3D[{u, v, K[X][u, v]}, {u, u0, u1}, {v, v0, 
        v1}]

u0=; u1= ; v0=; v1= ;
ParametricPlot3D[{u, v, H[X][u, v]}, {u, u0, u1}, {v, v0, 
        v1}]