From this definition we have the following relations for the gradients of a scalar field ϕ ϕ , a vector field v, and a second-order tensor field S . ∇ ∇ ϕ ϕ = ∂ ∂ ϕ ϕ ∂ ∂ ξ ξ i g i ∇ ∇ v = ∂ ∂ ( v j g j ) ∂ ∂ ξ ξ i ⊗ ⊗ g i = ( ∂ ∂ v j ∂ ∂ ξ ξ i + v k Γ Γ i k j ) g j ⊗ ⊗ g i = ( ∂ ∂ v j ∂ ∂ ξ ξ i − − v k Γ Γ i j k ) g j ⊗ ⊗ g i ∇ ∇ S = ∂ ∂ ( S j k g j ⊗ ⊗ g k ) ∂ ∂ ξ ξ i ⊗ ⊗ g i = ( ∂ ∂ S j k ∂ ∂ ξ ξ i − − S l k Γ Γ i j l − − S j l Γ Γ i k l ) g j ⊗ ⊗ g k ⊗ ⊗ g i