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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)? –  yasmar Aug 12 '11 at 6:15

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To say that $\phi \circ F$ has a critical point at $x = F^{-1}p$ is equivalent to saying that $\nabla (\phi \circ F) = 0$ at $x$, which by the chain rule is equivalent to saying that $\nabla \phi (F(x)) \cdot \partial_i F(x)= 0$ for all $i$. Since $F(x) = p$, this is the same as saying $\nabla \phi (p) \cdot \partial_i F(x)= 0$ for each $i$. But the various $\partial_i F(x)$ span the tangent space of $M$ at $p$, so the result follows.

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