I am trying to derive the [Thin Plate Energy Functional][1]. Given a Thin Plate $z = z(x,y)$, how does one derive the energy function easily:

$$\iint_{\mathbb{R}^2} \,\left[\left(\frac{\partial ^2z}{\partial x^2}\right)^2+2\left(\frac{\partial^2 z}{\partial x\partial y}\right)^2+\left(\frac{\partial^2 z}{\partial y^2}\right)^2\right]\mathrm{d}x\,\mathrm{d}y$$


  [1]: https://en.wikipedia.org/wiki/Thin_plate_energy_functional