If x is a regular patch on a regular surface in R^3 with normal N^^, then x_(u u) | = | Γ_11^1 x_u + Γ_11^2 x_v + eN^^ x_(u v) | = | Γ_12^1 x_u + Γ_12^2 x_v + fN^^ x_(v v) | = | Γ_22^1 x_u + Γ_22^2 x_v + gN^^, where e, f, and g are coefficients of the second fundamental form and Γ_(i j)^k are Christoffel symbols of the second kind.

Christoffel symbol of the second kind | fundamental forms | Peterson-Mainardi-Codazzi equations