Lagrange Multipliers

In the setting of Functions of Class Cᵏ: Notation, let $f:U\to\mathbb{R}$ be of class $C^{1}$ on $U$, let $M\subseteq U$, and let $a\in M$ be such that the restriction $f|_{M}:M\to\mathbb{R}$ has a local maximum or a local minimum at $a$ relative to $M$, in the metric space $(\mathbb{R}^{n},d_{E})$.

(i) Several constraints. Let $m<n$, and let $g:U\to\mathbb{R}^{m}$ be of class $C^{1}$ on $U$ with $M=\{x\in U:g(x)=0\}$. Suppose that some $m$ columns of the Jacobian matrix $Dg(a)$ form an invertible matrix: there is an injective function $\sigma:[m]\to[n]$ such that the matrix in $\mathcal{M}_{m}(\mathbb{R})$ with entries $\bigl(Dg(a)\bigr)_{k\,\sigma(l)}$ ($k,l\in[m]$) is invertible. Then there is $\lambda\in\mathbb{R}^{m}$ with

$$\nabla f(a)=Dg(a)^{\top}\lambda=\sum_{k\in[m]}\lambda_{k}\,\nabla g_{k}(a).$$

(ii) One constraint. Let $n\ge2$, and let $\varphi:U\to\mathbb{R}$ be of class $C^{1}$ on $U$ with $M=\{x\in U:\varphi(x)=0\}$ and $\nabla\varphi(a)\neq0$. Then there is $\lambda\in\mathbb{R}$ with

$$\nabla f(a)=\lambda\,\nabla\varphi(a).$$