x =  f ξ η Cos[ϕ]

y  =  f ξ η Sin[ϕ]

z =  f Sqrt[(ξ^2 - 1) (1 - η^2)]

hξ  = 
      Simplify[ Sqrt[D[x, ξ]^2  +  D[y, ξ]^2  +  D[z, ξ]^2], 
        Assumptions ->  f > 0]

hη  = 
      Simplify[ 
        Sqrt[D[x, η]^2  +  D[y, η]^2  +  D[z, η]^2], 
        Assumptions ->  f > 0]

hϕ  = 
      Simplify[ 
        Sqrt[D[x, ϕ]^2  +  D[y, ϕ]^2  +  D[z, ϕ]^2], 
        Assumptions ->  f > 0]

V  =  f[ξ, η, ϕ]

D2V  =  
      Simplify[(D[( hη hϕ/hξ) D[V, ξ], ξ] +  
              D[( hϕ hξ/hη) D[V, η], η] + 
              D[( hξ hη/hϕ) D[
                    V, ϕ], ϕ])/(hξ hη hϕ)]

Vξ  = {D[x, ξ], D[y, ξ], D[z, ξ]}

Vη  = {D[x, η], D[y, η], D[z, η]}

Vϕ  = {D[x, ϕ], D[y, ϕ], D[z, ϕ]}

V  =  {Vξ, Vη, Vϕ}

Simplify[Vξ  .  Vξ]

Simplify[Vξ  .  Vη]

Simplify[Vξ  .  Vϕ]

Simplify[Vη  .  Vξ]

Simplify[Vη  .  Vη]

Simplify[Vη  .  Vϕ]

Simplify[Vϕ .  Vξ]

Simplify[Vϕ  .  Vη]

Simplify[Vϕ  .  Vϕ]

P  =  
      Table[Simplify[Sum[V[[i, k]] V[[j, k]], {k, 1, 3}]], {i, 1, 
            3}, {j, 1, 3}] // TableForm

H =  Simplify[Inverse[V]]

Simplify[H  .  V]