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Here is the Problem 11-11, page 301 in John Lee's book:

Let $M$ be a smooth manifold, and $C \subset M$ be an embedded submanifold. Let $f \in C^{\infty}(M)$, and suppose $p\in C$ is a point at which $f$ attains a local maximum or minimum value among points in C. Given a smooth local dening function $\Phi: U \to \mathbb R^k$ for C on a neighborhood $U$ of $p$ in $M$, there are real numbers $\lambda_1, \cdots, \lambda_k$ (called Lagrange multipliers) such that $$df_p = \lambda_1 d\Phi^1|_p + \cdots + \lambda_k d\Phi^k|_p.$$

I got confused when I was trying to solve it. Here are my questions:

(1) He didn't say anything about the dimension of $M$ and $C$, nor did he put corrections here. Is it necessary to assume that $dim M=n>k$ and that $dim C=n-k$ or these results implicitly follow from the conditions in this problem?

(2) Why do we need the condition that $C$ is an embedded submanifold? Assume $dim C=n-k$, then theorem 5.8 on page 102 tells us that $C$ satisfies local k-slice condition(This is the only theorem I can think of that is related to this condition). But what good can this condition do to us?

(3) I think I need to apply the Lagrange multipliers theorem(see page 113) in multi-variable calculus. But we need to make sure that the rank of the Jacobian matrix of $\Phi$ is of rank $k$ at the point $p$. However, there isn't any extra conditions on $\Phi$.

You can either answer my questions separately or show me a detailed proof of it. Thank you in advance!

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Here are a few comments that might be helpful.

(1) He didn't say anything about the dimension of $M$ and $C$, nor did he put corrections [here][1]. Is it necessary to assume that $dim > M=n>k$ and that $dim C=n-k$ or these results implicitly follow from the conditions in this problem?

The definition of a local defining function (page 107) specifies that $C\cap U$ is a regular level set of $\Phi$, which implies that $d\Phi$ has rank $k$ everywhere on $C\cap U$, and therefore $C$ has codimension $k$ in $M$.

(2) Why do we need the condition that $C$ is an embedded submanifold?

Embedded submanifolds are the only ones that admit local defining functions in a neighborhood of each point.

(3) I think I need to apply the Lagrange multipliers theorem in multi-variable calculus. But we need to make sure that the rank of the Jacobian matrix of $\Phi$ is of rank $k$ at the point $p$. However, there isn't any extra conditions on $\Phi$.

The point of this problem is to prove the Lagrange multiplier theorem, albeit in a more general setting than the one usually introduced in advanced calculus courses. The fact that the Jacobian of $\Phi$ in coordinates has maximal rank is an immediate consequence of the definition of a local defining function.

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Assume that $\dim M=n$. Since $C$ is a regular submanifold of $M$ defined as a level set of $\Phi$ and $p\in C$, there exists an adapted chart, also called a slice chart, $(V,\varphi)$ on some neighbhorhood $V$ of $p$, so that on $C$, $n-k$ coordinate functions vanish. Let $(r^1,\ldots,r^n)$ be the standard coordinates on $\mathbb{R}^n$. We can write $\varphi=(x^1,\ldots,x^n)$, where $x^i=r^i\circ\varphi$. Without loss of generality, we may assume that the last $k$ coordinates, $x^{k+1},\ldots,x^n$, vanish on $C\cap V$.

Let $(r^1,\ldots,r^k)$ be the standard coordinates on $\mathrm{d}^k$. Since $C\cap V=\Phi^{-1}(a)$ for some $a\in\mathrm{d}^k$ and $p\in C$, we have \begin{equation*} \left.\frac{\partial \Phi^i}{\partial x^j}\right|_{p}= \left.\frac{\partial (r^i\circ\Phi)}{\partial x^j}\right|_{p}= \left.\frac{\partial (r^i\circ\Phi\circ\varphi^{-1})}{\partial r^j}\right|_{\varphi(p)}\neq 0, \end{equation*} for all $i\in\{1,\ldots,k\}$ and all $j\in\{k+1,\ldots,n\}$. It follows from the Implicit Function Theorem that $(x^1,\ldots,x^{n-k},\Phi^1,\ldots,\Phi^k)$ are smooth coordinates in some neighborhood $W\subset V$ of $q$. Hence $\mathrm{d} f_p$ has coordinate representation, \begin{equation*} \mathrm{d} f_p=\lambda_1 \mathrm{d} \Phi^1_p+\cdots+\lambda_k\mathrm{d}\Phi^k_p+ \lambda_{k+1}\mathrm{d} x^1_p + \cdots + \lambda_{n}\mathrm{d} x^{n-k}_p. \end{equation*} Since $f|_C$ attains local extremum at $p$ and $(x^1,\ldots,x^{n-k})$ are smooth local coordinates for the regular submanifold $C$ in some neighborhood $V$ of $p$. \begin{equation*} \lambda_i=\mathrm{d} f_p\left(\left.\frac{\partial}{\partial x^{i-k}}\right|_p\right)= \left.\frac{\partial f}{\partial x^{i-k}}\right|_p=0, \quad i\in\{k+1,\ldots,n\}. \end{equation*} Hence \begin{equation*} \mathrm{d} f_p=\lambda_1\mathrm{d}\Phi^1_p+\cdots +\lambda_k\mathrm{d}\Phi^k_p. \end{equation*}

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