1 + 1

x[u_, v_]  := 
       {u  -  u^3/3  +  u v^2,  v  -  v^3/3  +  v u^2,  
        u^2  -  v^2}

ParametricPlot3D[x[u, v], {u, -3, 3}, {v, -3, 3}]

xu[x_, u_, v_]  :=  D[x[u, v], u]

xv [x_, u_, v_]  :=  D[x[u, v], v]

xu[x, u, v]

xv[x, u, v]

EE[x_, u_, v_]  :=  Simplify[xu[x, u, v]  .  xu[x, u, v]]

EE[x, u, v]

Simplify[%]

FF[x_, u_, v_ ]  :=  Simplify[xu[x, u, v]  .  xv[x, u, v]]

FF[x, u, v]

GG[x_, u_, v_]  :=  Simplify[xv[x, u, v] . xv[x, u, v]]

GG[x, u, v]

UU[x_, u_, v_]  :=  
      Cross[xu[x, u, v],  xv[x, u, v]]/
        Norm[Cross[xu[x, u, v],  xv[x, u, v]]]

Norm[v_]  :=  Sqrt[v . v]

UU[x, u, v]

Simplify[%]

PowerExpand[%]

xuu[x_, u_, v_]  :=  D[xu[x, u, v], u]

%25  .  xuu[x, u, v]

Simplify[%]

<<  curvature.m

KK[x, u, v]

HH[x, u, v]