So for a 3D-Modelling exercise we need to create a smooth model, using Bezier surfaces. Now I've learned the recursive definition of a 1D Bezier curve, using this formula, where $P:[0,1]\rightarrow \mathbb{R}^3; t\mapsto(x,y,z)$
$$ P_i^{(j)}(t) = (1-t)P_{i-1}^{(j-1)} + tP_{i}^{(j-1)}$$
So how do I go from this formula to a recursive formula in two dimensions? So $S:[0,1]^2\rightarrow\mathbb{R}^3;(u,v)\mapsto(x,y,z)$ I saw some definitions (Page 16), which are non recursive, but I can't seem to make any sense of it
You can find a de Casteljau algorithm for tensor product (rectangular) surfaces on this web page.
But there's nothing magic about recursive formulae, and they're harder to understand and implement. If you fix the surface degree (to $3 \times 3$, for example). Then you can write explicit formulas for the basis function: \begin{align} B_0(t) &= (1-t)^3 \\ B_1(t) &= 3t(1-t)^2 \\ B_2(t) &= 3t^2(1-t) \\ B_3(t) &= t^3 \\ \end{align}
Then plug these into the surface equations on page 16 of your notes.