Lagrange multipliers and critical points (differential form description). On $M \times V^*$, where $M$ is a differentiable manifold (not necessarily equipped with a metric) and $V^*$ is dual to a vector space $V$, one can define a Lagrange function $F = f +v^*h(x)$ using functions $f: M \to \mathbb{R}$ and $h: M \to V$, and with $v^* \in V^*$. In this set up what is the standard argument for showing that the critical point set of $F$ is in bijection with the critical points of $f$ restricted to $h^{-1}(0)$?
In general case, let $N=h^{-1}(0)$ and $x\in N$ is a critical point of $f\big|_N$, that is, $df_x(\alpha)=0, \forall \alpha\in T_xN\subset T_xM$. Since $h\equiv 0$ on $N$, $dh_x(\alpha)=0$ too. Choose any $\beta\in T_xM-T_xN$ such that $df_x(\beta)=1$, denote $\gamma=dh_x(\beta)\in V$ and choose $v^*$ such that $v^*(\gamma)=-1$ then we have $df_x=-v^*\circ dh_x$, that is $df_x+v^*\circ dh_x=0$, implying that $(x, v^*)$ is a critical point of $F$.
In David Bressoud's text on calculus (from celestial mechanics to special relativity), there is an elegant and annotated proof (section 9.7).