Can anyone help me compute the mean curvature of a paraboloid?

Question

Given a paraboloid with the following equation

\begin{align} z = P(x,y) = c_0 + c_1x + c_2y + c_3xy + c_4x^2 + c_5y^2, \end{align}

how would I derive an expression for the $\textbf{mean}$ curvature $H=\tfrac{1}{2}(\kappa_1+\kappa_2)$, where $\kappa_1$ and $\kappa_2$ are the principal curvatures at the point $\big(x,y,P(x,y)\big)$? I specifically want to evaluate the mean curvature at $\big(0,0,P(0,0)\big)$.

Answer 1

Hint: Mean curvature $H$ for a surface $z=P(x,y)$ satisfies

$$2H = -\nabla\cdot\left(\frac{\nabla(z-P)}{|\nabla(z-P)|}\right)$$ Can you compute this?