This is my first question posted here, I hope to make it as easy-to-answer as possible.
I'm currently studying Vector Calculus it is taught that to find critical points (over the entire surface, not over some domain), we do the following:
Let $f_x=0$ and $f_y=0$.
Solve the two resulting equations simultaneously if need be.
We are taking the partials along the coordinate axes, but what is the guarantee that these are the only critical points?
ie: If I take the partial derivatives along any two perpendicular vectors, could they yield critical points that would not be found by taking the partials along the coordinate axes?
Below is my 'explanation' based on my current understanding (which may be completely incorrect!)
Since we generally study 'friendly' surfaces, the partials (in any direction) will always tend to zero, so we take partials along the coordinate axes for the sake of convenience.