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Point of Inflection is the name given to the location at which a two dimensional curve changes from concave up to concave down.

Is there a similarly defined term or concept in three dimensions where the division need not be a single point?

for instance the concavity of
$z = cos(\sqrt{x^2 +y^2})$

seems to change when $ z= 0 $ but this results in circles.

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Do you mean a saddle point? –  Git Gud Mar 7 '13 at 21:56
    
@GitGud I don't think so. the jacobian at a saddle point is nonzero, whereas the first derivative vanishes at at inflexion point (and so does the second). I think OP is talking about a point where the jacobian and hessian vanish (or are degenerate). If so I don't know of any standard terminology –  Glougloubarbaki Mar 7 '13 at 21:59
    
or maybe a degenerate critical point ? –  Glougloubarbaki Mar 7 '13 at 22:00
    
@Glougloubarbaki that sounds right, but I've been out of working with any of this stuff in school for years and to be honest I'm having a little difficulty getting my head back around this. –  Mr.Mindor Mar 7 '13 at 22:40

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