Next we consider a 2D surface embedded in a flat 3D space.
Consider the curvatures at <0, 0, 0> of the surface z = x^{2} + 2y^{2}.
In the x-z plane the curvature is 2 while in the y-z plane it is 4.
The principle curvatures are 2 and 4.

In general in n-dimensions at a point on a “smooth” n−1-surface you can find a coordinate system in which to 2nd order x_{0} = ∑k_{i}x_{i}^{2}/2 where the sum is over 1 to n−1.
The k_{i} are the principle curvatures.
They may be of different signs.
Some may be zero.

If the surface is the surface of a smooth body then we take the x_{0} coördinate to be pointing inwards.
If the body is smooth and convex then all of the k’s will be positive.