Let $S$ be a regular surface in $\mathbb{R}^3$ and $p\in S$ a point on the surface.
By the implicit function theorem $S$ can be locally written as a graph of a function, e.g. $V\cap S = \{ (x,y,z) \in \mathbb{R}^3 : (x,y)\in U, z=f(x,y)\}$ for some open neighbourhood $V$ of $p$, open set $U\subset \mathbb{R}^2$ and some smooth function $f: U \rightarrow \mathbb{R}.$
By choosing local coordinates we can identify $U$ as part of the tangent plane of $S$ at $p$, furthermore we can set $f^{-1}(p)=(0,0)$. In this case, the mean curvature at $p$ is given by $H=\frac{f_{xx}+f_{yy}}{2}$ (average of second derivatives at $p$) and principal curvatures are $f_{xx},f_{yy}$.
MSE