a problem on geometry of hypersurfaces

Question

Recently I am reading book on mean curvature flow by carlo mantegazza.There I found a problem on hypersurfaces stated below : Show that if the hypersurface $M \subset {R}^{n+1}$ is locally the graph of a function $f:R^n\rightarrow R$,then we have $$g_{ij}=\delta_{ij}+f_if_j, $$ $$\nu=-{(\nabla f,-1)\over \sqrt{1+|\nabla f |^2}},$$ $$h_{ij}={Hess_{ij}f\over\sqrt{1+|\nabla f |^2}},$$ $$H={\Delta f \over\sqrt{1+|\nabla f |^2}}-{Hess f(\nabla f,\nabla f)\over (\sqrt{1+|\nabla f |^2})^3}=div\left({\nabla f \over\sqrt{1+|\nabla f |^2}}\right) $$where $f_i=\partial_i f$ and $Hess f$ is the Hessian of the function $f$. I guess that it is a simple problem and I can feel the intuition behind it.But please anybody help me the explicit calculation associated with the problem.