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Let $\phi:M\rightarrow \mathbb{R}$ be smooth, M be a k-dimensional submanifold, and $F:U \rightarrow M$ be the inverse map of a local coordinate near $p \in M$ where $ U \subseteq \mathbb{R}^n$. How can I show that $\phi \circ F$ has a critical point at $F^{-1}p$ iff $\nabla \phi(p)$ is orthogonal to the tangent space to $M$ at $p$?

Thanks in advance.

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    I think you mean to define $\phi$ on the ambient manifold (which you haven't given a separate name)?2011-08-12

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