Geometric interpretation of ${\partial f\over \partial x}= {\partial f \over \partial y}$

Question

I know that $${\partial f\over \partial x}= {\partial f \over \partial y}$$ iff there exists a differentiable function $g$ (of one variable) such that $g(x+y)=f(x,y)$ (where $f : D\subseteq \mathbb R^2 \to \mathbb R$ and $D$ is an open set), but I don´t know what is the geometric interpretation of this fact. So I would really appreciate if you can help me with this

Answer 1

Consider $f(x,y)$ as a sort of contour map, with the height $z$ in 3 dimensions equal to $f(x,y)$. The function defines a continuous surface in that space.

Now stare at that function from far off in the "South-East" direction, looking North-West.

Then if $\frac{\partial f}{\partial x} = \frac{\partial f}{\partial y}$, the projection of that surface that you would see is just a one-dimensional curve. And the shape of that curve is $g(x+y)$