Thin plate energy functional
The exact thin plate energy functional (TPEF) for a function is where and are the principal curvatures of the surface mapping at the point This is the surface integral of hence the in the integrand. Minimizing the exact thin plate energy functional would result in a system of non-linear equations. So in practice, an approximation that results in linear systems of equations is often used. The approximation is derived by assuming that the gradient of is 0. At any point where the first fundamental form of the surface mapping is the identity matrix and the second fundamental form is . We can use the formula for mean curvature to determine that and the formula for Gaussian curvature (where and are the determinants of the second and first fundamental forms, respectively) to determine that Since and the integrand of the exact TPEF equals The expressions we just computed for the mean curvature and Gaussian curvature as functions of partial derivatives of show that the integrand of the exact TPEF is So the approximate thin plate energy functional is (Wikipedia).
