Suppose we have a a graph of two functions in 3D, which I suppose can be called a 3D surface in 5D (?), via: $$ \Pi = \{ t(x,y,z)=(x,y,z,f(x,y,z),g(x,y,z))\,|\, x,y,z\in\mathbb{R}^3 \} $$ I want to compute the second fundamental form (in $x,y,z$ coordinates). Note that my actual goal is to compute the principal and mean curvatures.
The tangent (column) vectors can be $t_x=\partial_x t, t_y=\partial_y t, t_z=\partial_z t$. Let $J$ be the Jacobian (i.e. $J=[t_x,t_y,t_z]$).
Here is what is confusing me.
Normally, I would compute (for these coordinates) the second fundamental form via $S_{ij}=\partial_i n \cdot t_j$. But here $\text{null}(J^T)$ is 2D! So there are two linearly independent fields of normal vectors (which makes sense, intuitively). Every reference I look at computes $S$ using a field of normals $n$. What do I do when I have two fields, $n_1$ and $n_2$? Can I just choose one?
Thanks!