# dg.differential geometry – The Derivation of Thin Plate Spline interpolation Energy function？

I am trying to derive the “Thin Plate Energy Functional”. Given a Thin Plate $$z = z(x,y)$$, how does one derive easily the energy functional

$$iint_{mathbb{R}^2} ,left(left(frac{partial ^2z}{partial x^2}right)^2+2left(frac{partial^2 z}{partial xpartial y}right)^2+left(frac{partial^2 z}{partial y^2}right)^2right)mathrm{d}x,mathrm{d}yquad ?$$