Functins

This question is a spin-off of a much wider question I asked on MSE: -is-a-pde-a-submanifold-and-not-just-a-subset. There I basically wanted to understand why defining a PDE to be submanifold is a good choice and used this one here as an example. I.e. I asked, why it is not too restrictive, as for example $\frac\partial u\partial x \frac\partial u\partial y = 0$ is not a PDE for functins $u: \Bbb R^2 \to \Bbb R$ in this sense.

functins