Definition 3.9.1 (Curvature of a curve in $\R^2$). let a curve in $\R^2$ be locally the graph of a function $g$, with a Taylor polynomial
$$ g(X) = \frac{A_2}{2}X^2 + \frac{A_3}{6}X^3 + \dots $$
Then the curvature $\kappa$ of a curve at $\mathbf{0}$ is
$$ \kappa(\mathbf{0}) = |A_2|. $$
Proposition 3.9.2 (Computing the curvature of a plane curve known as a graph). The curvature $\kappa$ of the curve $y = f(x)$ at $\begin{pmatrix} a\\f(a)\end{pmatrix}$ is
$$ \kappa\begin{pmatrix} a\\f(a)\end{pmatrix} = \frac{|f''(a)|}{(1 + (f'(a))^2)^{3/2}}. $$
To choose the best coordinates to measure the curvature of a surface, we anchor them to a coordinate system. $X,Y,Z$ with repsect to an orthonormal basis $v_1, v_2, v_3$ anchored at $a$ are adapted to $S$ at $a$ if $v_1, v_2$ span $T_aS$ so that $v_3$ is a unit vector orthogonal to $S$ at $a$, often called the unit normal $\vec n$. Our Taylor approximation for this surface is
$$ Z = f\begin{pmatrix}X\\Y\end{pmatrix} = \frac{1}{2}(A_{2,0}X^2 + 2A_{1,1}XY + A_{0,2}Y^2) + o(x^2). $$
The quadratic form
$$ \begin{pmatrix}X\\Y\end{pmatrix} \mapsto A_{2,0}X^2 + 2A_{1,1}XY + A_{0,2}Y^2 $$
is known as the second fundamental form of the surface $S$ at point $a$.
Definition 3.9.7 (Mean curvature, mean curvature vector). The mean curvature $H$ of a surface at point $a$ is
$$ H(a) = \frac{1}{2}(A_{2,0} + A_{0,2}). $$
The mean curvature vector is $\vec H(a) = H(a)v_3$, where $v_3$ is the unit normal.
Definition 3.9.8 (Gaussian curvature of a surface). The Gaussian curvature $K$ **of a surface at point $a$ is
$$ K(a) = A_{2,0}A_{0,2} - A^2_{1,1}. $$