I have seen the next theorem in A Lotka-Volterra three species food chain,
Theorem 0.1: Let S be a smooth surface in $\mathbb{R}^3$ and \begin{equation} \begin{cases} &\frac{dx}{dt}=f(x,y,z),\\ &\frac{dy}{dt}=g(x,y,z),\\ &\frac{dz}{dt}=h(x,y,z), \end{cases} \end{equation} where $f$, $g$ and $h$ are continuously differentiable. Suppose that $\textbf{n}$ is a normal vector to the surface S at $(x,y,z)$, and for all $(x,y,z)\in\mathbb{R}^3$ we have that \begin{equation*} \textbf{n}\cdot\left(\frac{dx}{dt},\frac{dy}{dt},\frac{dz}{dt}\right)=0. \end{equation*} Then, S is invariant with respect to the above system.
In the above reference, I have seen that they refer to V. Arnold's book "Ordinary Differential Equations" to search for the proof of this theorem, but I couldn't find it. Does someone know where could I find its proof? I don't mind if it is in Arnold's book or in another place.
Thank you very much for your help.
Take any Calc III book and find a proof (or better prove it yourself) that if the surface is defined by the equation $u(x)=C$ then vector gradient $\nabla u(x)$ at the point $x$ is exactly the normal to this surface at this point.