One of the most famous theorems of differential geometry is Gauss's Theorema Egrgium. It asserts that the Gaussian curvature of a surface is intrinsic: it can be computed from lengths and angles measured in the surface. It does not depend on how the surafce is embedded in $\R^3$!
Theorem 5.4.1. Let $D_r(\mathbf{p})$ be the set of all points $\mathbf{q}$ in a surface $S\subset \R^3$ such that there exists a curve of length $\leq r$ in $S$ joining $\mathbf{p}$ to $\mathbf{q}$. Then
$$ \text{Area}(D_r(\mathbf{p})) = \pi r^2- \frac{\pi K(\mathbf{p})}{12}r^4 + o(r^4). $$
Mean curvature measures how far a surface is from being minimal. The mean curvature vector $\vec H$ measures how the area of a surface $S$ varies as the surface is moved along a vector field.
Theorem 5.4.4. Let $\vec w$ be a normal vector field on surface $S$. Let $S_t$ be the family of surfaces that are images of $\varphi_t: S \to \R^3$ given by $\varphi_t(x) = x + t\vec w(x)$. The area of $S_t$ is given by the formula
$$ \text{Area}S_t = \text{Area}S - 2t\int_S\vec H(x) \cdot \vec w(x) |d^2x| + o(t). $$
Let $S \subset \R^3$ be a surface, and $\vec n$ a unit normal vector field. If the head is at the point of the unit sphere $S^2$ and its tail is at the origin, then $n: S \to S^2$ is the Gauss map. Its derivative is $[Dn(x)] : T_xS \to T_{n(x)}S^2$. Both the domain and codomain of this derivative are perpendicular to the normal.
Theorem 5.4.6 (Gaussian curvature and the Gauss map). Let $S\subset \R^3$ be a surface, and let $n: S \to S^2$ be the Gauss map. Then
$$ \int_S|K(x)||d^2x| = \int_{n(S)}|d^2x| = \text{Area } n(S). $$
If $S$ is a small neighborhood of $x$, where $K$ can be considered as a constant, equation 5.4.35 becomes
$$ |K(x)| \int_S|d^2x| \approx \int_{n(S)}|d^2x| , \text{i.e.,} |K(x)| \approx \frac{\int_{n(S)}|d^2x|}{\int_S|d^2x|}. $$
Thus the Gaussian curvature of $x$ is the limit, as the neighborhood $S$ of $x$ becomes small, of the ratio of the area of $n(S)$ to the area of $S$.
Example 5.4.7. Let $S\subset \R^3$ be an ellipsoid. Then $n(S) = S^2$ and