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If there is a function $f : M \to \mathbb R$ then the critical point is given as a point where

$$d f = 0$$

$df$ being 1-form (btw am I right here?). Is there a coordinate independent formulation of a criteria to determine if this point is a local maximum or minimum (or a saddle point)?

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    There are coordinate free definitions of second derivatives. Positive/negative definiteness of this object is a coordinate free criterion.2012-07-18

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