Definition:Critical Point/Smooth Manifold

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Let $M$ be a smooth manifold.

Let $f \in \map {\CC^\infty} M : M \to \R$ be a smooth real-valued function.

Let $p \in M$ be a base point in $M$.

Let $\rd f_p$ be the differential of $f$ at $p$.

Suppose $\rd f_p = 0$.

Then $p$ is called a critical point of $f$.