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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:

  1. Let $f_x=0$ and $f_y=0$.

  2. 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.

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If you have a critical point and your function is differentiable then the directional derivative must be zero in all coordinate directions. Thus the partials $f_x$ and $f_y$ must be zero, so you are guaranteed to find all critical points by solving those equations. You could also use any other linearly independent pair of directions for the derivatives. The chain rule says that the derivative in any other direction will be a linear combination of the 2 basis directional derivatives you compute. Hence the derivative will be zero in all directions if it's zero for the 2 basis directions.

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