Working toward understanding Euler's theorem on page 70 of Eisenhart, Luther P. (2004), A Treatise on the Differential Geometry of Curves and Surfaces the following is stated:
A linear element on a parametrized surface $S$ of the form
$$\begin{align} x=f_1(u,v)\\ y=f_2(u,v)\\ z=f_3(u,v) \end{align}$$
on any curve will have equations of the form $\phi(u,v)=0$ will have linear elements of the curve given by
$$ds^2 = dx^2 + dy^2 + dz^2$$
where
$$\begin{align} dx=\frac{\partial x}{\partial u}du + \frac{\partial x}{\partial v}dv\\ dy=\frac{\partial y}{\partial u}du+\frac{\partial y}{\partial v}dv\\ dz=\frac{\partial z}{\partial u}du+\frac{\partial z}{\partial v}dv\\ \end{align}$$
the differentials $du,dv$ satisfying the condition $$\frac{\partial \phi}{\partial u}du+\frac{\partial \phi}{\partial v}dv=0$$
How can I see that this last line (this condition) makes sense or is true?
You have $\phi(u,v)=0$, so the differential of $\phi$ must also be zero. The differential is $$d\phi=\frac{\partial \phi}{\partial u}du+\frac{\partial \phi}{\partial v}dv.$$