Solution
= Solution
Expanding the characteristic determinant gives
$$
(EG-F^2)\kappa^2-(LG+NE-2MF)\kappa+(LN-M^2)=0.
$$
The product and sum of its two roots therefore give
$$
\boxed{\kappa_1\kappa_2=\frac{LN-M^2}{EG-F^2}=K,\qquad H=\frac{\kappa_1+\kappa_2}{2}=\frac{LG+NE-2MF}{2(EG-F^2)}.}
$$
Reversing the normal changes both principal curvatures and the <mean curvature> in sign, while leaving their product unchanged.