Let $(x, y, f(x,y))$ be a surface in $\mathbb{R^3}$. It is written in a book without proof that all rotation invariant (rotating around $z$-axis) of $f$ are combinations of the following four quantities:

• $\Delta f = f_{xx}+f_{yy}$
• $||\nabla f||^2=f_x^2 + f_y ^2$
• $\det{H} = f_{xx} f_{yy} - f_{xy}^2$
• $(\nabla f, H \nabla f) = f_{xx}f_x^2 + 2f_{xy}f_xf_y+f_{yy}f_y^2$

How do I prove this?