I have a parametric surface $S:\mathbb{R}^3\rightarrow\mathbb{R}^3$ defined by $S(x,z,t) = (x,h(x,z,t),z)$. This surface defined the surface of the water, the second component is the vertical axis and it represents the height of the surface of water at position $x$ and $z$ at time $t$.
I need to find the velocity vector at a given position of this surface. I tried to use partial derivative on $t$, but then the velocity vector will only have a value on $y$ component, which don't seem intuitive to me.
I think my problem is that I could not defined correctly what is the velocity of a parametric surface of this type... I don't even know if it exists with this type of object !
If someone have a definition of the velocity that I can rely on, I would appreciate it !
