By successive applications of the Weingarten equations and the Gauss equations, it is possible to express all partial derivatives of X and as linear combinations of X₁, X₂, and . The fact that the mixed partial derivatives are independent of the order of differentiation leads to the integrability conditions that must be satisfies by the coefficients of the first and second fundamental forms. The resulting relationships are amon the most fundamental in the theory of surfaces.

By differentiating _i = -h_i^jX_j we obtain _ik = -h_i^j/u^k X_j -h_i^jX_jk = -[h_i^j/u^k + h_i^m _m^j_k ] X_j -h_i^jh_jk . The coefficient of is -h_i^jh_jk = -g^jmh_ijh_mk so it is automatically symmetric in i and k. Since _ik = _ki, the coefficient of X_j is also symmetric in i and k. This fact gives the Codazzi equations h_i^j/u^k + h_i^m _m^j_k ⁼ h_k^j/uⁱ + h_k^m _m^j_i. These are usually expressed as h_i^j/u^k - h_k^j/uⁱ + h_i^m _m^j_k - h_k^m _m^j_i = 0.

It is also possible to express the Codazzi equations in an equivalent form, using h_ij instead of h_i^j. For we have h_ij/u^k = [h_i^mg_mj]/u^k = h_i^m/u^k g_mj + h_i^m (_mjk + _jmk) = (h_i^m/u^k + h_iⁿ_n^m_k) g_mj + h_{in j}ⁿ_k. Applying the above result then leads to h_ij/u^k - h_kj/uⁱ - h_{in j}ⁿ_k + h_{kn j}ⁿ_i= 0.